Decoherence in optimized quantum random-walk search algorithm
Zhang, Yu-Chao; Bao, Wan-Su; Wang, Xiang; Fu, Xiang-Qun
2015-08-01
This paper investigates the effects of decoherence generated by broken-link-type noise in the hypercube on an optimized quantum random-walk search algorithm. When the hypercube occurs with random broken links, the optimized quantum random-walk search algorithm with decoherence is depicted through defining the shift operator which includes the possibility of broken links. For a given database size, we obtain the maximum success rate of the algorithm and the required number of iterations through numerical simulations and analysis when the algorithm is in the presence of decoherence. Then the computational complexity of the algorithm with decoherence is obtained. The results show that the ultimate effect of broken-link-type decoherence on the optimized quantum random-walk search algorithm is negative. Project supported by the National Basic Research Program of China (Grant No. 2013CB338002).
White Noise in Quantum Random Walk Search Algorithm
MA Lei; DU Jiang-Feng; LI Yun; LI Hui; KWEK L. C.; OH C. H.
2006-01-01
@@ The quantum random walk is a possible approach to construct new quantum search algorithms. It has been shown by Shenvi et al. [Phys. Rev. A 67 (2003)52307] that a kind of algorithm can perform an oracle search on a database of N items with O(√N) calling to the oracle, yielding a speedup similar to other quantum search algorithms.
Limited Random Walk Algorithm for Big Graph Data Clustering
Zhang, Honglei; Kiranyaz, Serkan; Gabbouj, Moncef
2016-01-01
Graph clustering is an important technique to understand the relationships between the vertices in a big graph. In this paper, we propose a novel random-walk-based graph clustering method. The proposed method restricts the reach of the walking agent using an inflation function and a normalization function. We analyze the behavior of the limited random walk procedure and propose a novel algorithm for both global and local graph clustering problems. Previous random-walk-based algorithms depend on the chosen fitness function to find the clusters around a seed vertex. The proposed algorithm tackles the problem in an entirely different manner. We use the limited random walk procedure to find attracting vertices in a graph and use them as features to cluster the vertices. According to the experimental results on the simulated graph data and the real-world big graph data, the proposed method is superior to the state-of-the-art methods in solving graph clustering problems. Since the proposed method uses the embarrass...
Combinatorial Approximation Algorithms for MaxCut using Random Walks
Kale, Satyen
2010-01-01
We give the first combinatorial approximation algorithm for Maxcut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an O(n^{b}) algorithm that outputs a (0.5+delta)-approximation for Maxcut, where delta = delta(b) is some positive constant. One of the components of our algorithm is a weak local graph partitioning procedure that may be of independent interest. Given a starting vertex $i$ and a conductance parameter phi, unless a random walk of length ell = O(log n) starting from i mixes rapidly (in terms of phi and ell), we can find a cut of conductance at most phi close to the vertex. The work done per vertex found in the cut is sublinear in n.
A Novel Algorithm of Quantum Random Walk in Server Traffic Control and Task Scheduling
Dong Yumin
2014-01-01
Full Text Available A quantum random walk optimization model and algorithm in network cluster server traffic control and task scheduling is proposed. In order to solve the problem of server load balancing, we research and discuss the distribution theory of energy field in quantum mechanics and apply it to data clustering. We introduce the method of random walk and illuminate what the quantum random walk is. Here, we mainly research the standard model of one-dimensional quantum random walk. For the data clustering problem of high dimensional space, we can decompose one m-dimensional quantum random walk into m one-dimensional quantum random walk. In the end of the paper, we compare the quantum random walk optimization method with GA (genetic algorithm, ACO (ant colony optimization, and SAA (simulated annealing algorithm. In the same time, we prove its validity and rationality by the experiment of analog and simulation.
Quantization of Random Walks: Search Algorithms and Hitting Time
Santha, Miklos
Many classical search problems can be cast in the following abstract framework: Given a finite set X and a subset M ⊆ X of marked elements, detect if M is empty or not, or find an element in M if there is any. When M is not empty, a naive approach to the finding problem is to repeatedly pick a uniformly random element of X until a marked element is sampled. A more sophisticated approach might use a Markov chain, that is a random walk on the state space X in order to generate the samples. In that case the resources spent for previous steps are often reused to generate the next sample. Random walks also model spatial search in physical regions where the possible moves are expressed by the edges of some specific graph. The hitting time of a Markov chain is the number of steps necessary to reach a marked element, starting from the stationary distribution of the chain.
Optimized quantum random-walk search algorithm for multi-solution search
张宇超; 鲍皖苏; 汪翔; 付向群
2015-01-01
This study investigates the multi-solution search of the optimized quantum random-walk search algorithm on the hypercube. Through generalizing the abstract search algorithm which is a general tool for analyzing the search on the graph to the multi-solution case, it can be applied to analyze the multi-solution case of quantum random-walk search on the graph directly. Thus, the computational complexity of the optimized quantum random-walk search algorithm for the multi-solution search is obtained. Through numerical simulations and analysis, we obtain a critical value of the proportion of solutions q. For a given q, we derive the relationship between the success rate of the algorithm and the number of iterations when q is no longer than the critical value.
Ma, Tianren; Xia, Zhengyou
2017-05-01
Currently, with the rapid development of information technology, the electronic media for social communication is becoming more and more popular. Discovery of communities is a very effective way to understand the properties of complex networks. However, traditional community detection algorithms consider the structural characteristics of a social organization only, with more information about nodes and edges wasted. In the meanwhile, these algorithms do not consider each node on its merits. Label propagation algorithm (LPA) is a near linear time algorithm which aims to find the community in the network. It attracts many scholars owing to its high efficiency. In recent years, there are more improved algorithms that were put forward based on LPA. In this paper, an improved LPA based on random walk and node importance (NILPA) is proposed. Firstly, a list of node importance is obtained through calculation. The nodes in the network are sorted in descending order of importance. On the basis of random walk, a matrix is constructed to measure the similarity of nodes and it avoids the random choice in the LPA. Secondly, a new metric IAS (importance and similarity) is calculated by node importance and similarity matrix, which we can use to avoid the random selection in the original LPA and improve the algorithm stability. Finally, a test in real-world and synthetic networks is given. The result shows that this algorithm has better performance than existing methods in finding community structure.
A partially reflecting random walk on spheres algorithm for electrical impedance tomography
Maire, Sylvain; Simon, Martin
2015-12-01
In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute the voltage-to-current map in an embarrassingly parallel manner, which is an important issue with regard to the corresponding inverse problem. Our method uses the well-known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions. We analyze the global bias and the variance of the new estimator both theoretically and experimentally. Subsequently, the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms.
Diffusion Limits of the Random Walk Metropolis Algorithm in High Dimensions
Mattingly, Jonathan C; Stuart, Andrew M
2010-01-01
Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying computational complexity. In particular they lead directly to precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability. The purpose of this paper is to study diffusion limits for a class of naturally occuring high dimensional measures, found from the approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit of a random walk Metropolis algorithm to an infinite dimensional Hilbert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm.
Quantum walks and search algorithms
Portugal, Renato
2013-01-01
This book addresses an interesting area of quantum computation called quantum walks, which play an important role in building quantum algorithms, in particular search algorithms. Quantum walks are the quantum analogue of classical random walks. It is known that quantum computers have great power for searching unsorted databases. This power extends to many kinds of searches, particularly to the problem of finding a specific location in a spatial layout, which can be modeled by a graph. The goal is to find a specific node knowing that the particle uses the edges to jump from one node to the next. This book is self-contained with main topics that include: Grover's algorithm, describing its geometrical interpretation and evolution by means of the spectral decomposition of the evolution operater Analytical solutions of quantum walks on important graphs like line, cycles, two-dimensional lattices, and hypercubes using Fourier transforms Quantum walks on generic graphs, describing methods to calculate the limiting d...
An improved random walk algorithm for the implicit Monte Carlo method
Keady, Kendra P.; Cleveland, Mathew A.
2017-01-01
In this work, we introduce a modified Implicit Monte Carlo (IMC) Random Walk (RW) algorithm, which increases simulation efficiency for multigroup radiative transfer problems with strongly frequency-dependent opacities. To date, the RW method has only been implemented in "fully-gray" form; that is, the multigroup IMC opacities are group-collapsed over the full frequency domain of the problem to obtain a gray diffusion problem for RW. This formulation works well for problems with large spatial cells and/or opacities that are weakly dependent on frequency; however, the efficiency of the RW method degrades when the spatial cells are thin or the opacities are a strong function of frequency. To address this inefficiency, we introduce a RW frequency group cutoff in each spatial cell, which divides the frequency domain into optically thick and optically thin components. In the modified algorithm, opacities for the RW diffusion problem are obtained by group-collapsing IMC opacities below the frequency group cutoff. Particles with frequencies above the cutoff are transported via standard IMC, while particles below the cutoff are eligible for RW. This greatly increases the total number of RW steps taken per IMC time-step, which in turn improves the efficiency of the simulation. We refer to this new method as Partially-Gray Random Walk (PGRW). We present numerical results for several multigroup radiative transfer problems, which show that the PGRW method is significantly more efficient than standard RW for several problems of interest. In general, PGRW decreases runtimes by a factor of ∼2-4 compared to standard RW, and a factor of ∼3-6 compared to standard IMC. While PGRW is slower than frequency-dependent Discrete Diffusion Monte Carlo (DDMC), it is also easier to adapt to unstructured meshes and can be used in spatial cells where DDMC is not applicable. This suggests that it may be optimal to employ both DDMC and PGRW in a single simulation.
Identifying and Analyzing Novel Epilepsy-Related Genes Using Random Walk with Restart Algorithm
Wei Guo
2017-01-01
Full Text Available As a pathological condition, epilepsy is caused by abnormal neuronal discharge in brain which will temporarily disrupt the cerebral functions. Epilepsy is a chronic disease which occurs in all ages and would seriously affect patients’ personal lives. Thus, it is highly required to develop effective medicines or instruments to treat the disease. Identifying epilepsy-related genes is essential in order to understand and treat the disease because the corresponding proteins encoded by the epilepsy-related genes are candidates of the potential drug targets. In this study, a pioneering computational workflow was proposed to predict novel epilepsy-related genes using the random walk with restart (RWR algorithm. As reported in the literature RWR algorithm often produces a number of false positive genes, and in this study a permutation test and functional association tests were implemented to filter the genes identified by RWR algorithm, which greatly reduce the number of suspected genes and result in only thirty-three novel epilepsy genes. Finally, these novel genes were analyzed based upon some recently published literatures. Our findings implicate that all novel genes were closely related to epilepsy. It is believed that the proposed workflow can also be applied to identify genes related to other diseases and deepen our understanding of the mechanisms of these diseases.
Identifying and Analyzing Novel Epilepsy-Related Genes Using Random Walk with Restart Algorithm
Guo, Wei; Shang, Dong-Mei; Cao, Jing-Hui; Feng, Kaiyan; Wang, ShaoPeng
2017-01-01
As a pathological condition, epilepsy is caused by abnormal neuronal discharge in brain which will temporarily disrupt the cerebral functions. Epilepsy is a chronic disease which occurs in all ages and would seriously affect patients' personal lives. Thus, it is highly required to develop effective medicines or instruments to treat the disease. Identifying epilepsy-related genes is essential in order to understand and treat the disease because the corresponding proteins encoded by the epilepsy-related genes are candidates of the potential drug targets. In this study, a pioneering computational workflow was proposed to predict novel epilepsy-related genes using the random walk with restart (RWR) algorithm. As reported in the literature RWR algorithm often produces a number of false positive genes, and in this study a permutation test and functional association tests were implemented to filter the genes identified by RWR algorithm, which greatly reduce the number of suspected genes and result in only thirty-three novel epilepsy genes. Finally, these novel genes were analyzed based upon some recently published literatures. Our findings implicate that all novel genes were closely related to epilepsy. It is believed that the proposed workflow can also be applied to identify genes related to other diseases and deepen our understanding of the mechanisms of these diseases.
The adaptive dynamic community detection algorithm based on the non-homogeneous random walking
Xin, Yu; Xie, Zhi-Qiang; Yang, Jing
2016-05-01
With the changing of the habit and custom, people's social activity tends to be changeable. It is required to have a community evolution analyzing method to mine the dynamic information in social network. For that, we design the random walking possibility function and the topology gain function to calculate the global influence matrix of the nodes. By the analysis of the global influence matrix, the clustering directions of the nodes can be obtained, thus the NRW (Non-Homogeneous Random Walk) method for detecting the static overlapping communities can be established. We design the ANRW (Adaptive Non-Homogeneous Random Walk) method via adapting the nodes impacted by the dynamic events based on the NRW. The ANRW combines the local community detection with dynamic adaptive adjustment to decrease the computational cost for ANRW. Furthermore, the ANRW treats the node as the calculating unity, thus the running manner of the ANRW is suitable to the parallel computing, which could meet the requirement of large dataset mining. Finally, by the experiment analysis, the efficiency of ANRW on dynamic community detection is verified.
Aperiodic Quantum Random Walks
Ribeiro, P; Mosseri, R; Ribeiro, Pedro; Milman, Perola; Mosseri, Remy
2004-01-01
We generalize the quantum random walk protocol for a particle in a one-dimensional chain, by using several types of biased quantum coins, arranged in aperiodic sequences, in a manner that leads to a rich variety of possible wave function evolutions. Quasiperiodic sequences, following the Fibonacci prescription, are of particular interest, leading to a sub-ballistic wavefunction spreading. In contrast, random sequences leads to diffusive spreading, similar to the classical random walk behaviour. We also describe how to experimentally implement these aperiodic sequences.
Onoma, D P; Ruan, S; Thureau, S; Nkhali, L; Modzelewski, R; Monnehan, G A; Vera, P; Gardin, I
2014-12-01
A segmentation algorithm based on the random walk (RW) method, called 3D-LARW, has been developed to delineate small tumors or tumors with a heterogeneous distribution of FDG on PET images. Based on the original algorithm of RW [1], we propose an improved approach using new parameters depending on the Euclidean distance between two adjacent voxels instead of a fixed one and integrating probability densities of labels into the system of linear equations used in the RW. These improvements were evaluated and compared with the original RW method, a thresholding with a fixed value (40% of the maximum in the lesion), an adaptive thresholding algorithm on uniform spheres filled with FDG and FLAB method, on simulated heterogeneous spheres and on clinical data (14 patients). On these three different data, 3D-LARW has shown better segmentation results than the original RW algorithm and the three other methods. As expected, these improvements are more pronounced for the segmentation of small or tumors having heterogeneous FDG uptake.
Fractional random walk lattice dynamics
Michelitsch, Thomas; Riascos, Alejandro Perez; Nowakowski, Andrzeij; Nicolleau, Franck
2016-01-01
We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions.The fractional random walk dynamics is governed by a master equation involving {\\it fractional powers of Laplacian matrices $L^{\\frac{\\alpha}{2}}$}where $\\alpha=2$ recovers the normal walk.First we demonstrate thatthe interval $0\\textless{}\\alpha\\leq 2$ is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain thefundamental matrix $Z^{(\\alpha)}$, and the mean relaxation time (Kemeny constant) for the fractional random walk.The representation for the fundamental matrix $Z^{(\\alpha)}$ relates fractional random walks with normal random walks.We show that the fractional transition matrix elements exihibit for large cubic $n$-dimensional lattices a power law decay of an $n$-dimensional infinite spaceRiesz fractional deriva...
Snakes and perturbed random walks
Basak, Gopal
2011-01-01
In this paper we study some properties of random walks perturbed at extrema, which are generalizations of the walks considered e.g., in Davis (1999). This process can also be viewed as a version of {\\em excited random walk}, studied recently by many authors. We obtain a few properties related to the range of the process with infinite memory. We also prove the Strong law, Central Limit Theorem, and the criterion for the recurrence of the perturbed walk with finite memory.
Bartsch, Christian; Kochler, Thomas; Müller, Sebastian; Popov, Serguei
2011-01-01
We consider a branching random walk on $\\Z$, where the particles behave differently in visited and unvisited sites. Informally, each site on the positive half-line contains initially a cookie. On the first visit of a site its cookie is removed and particles at positions with a cookie reproduce and move differently from particles on sites without cookies. Therefore, the movement and the reproduction of the particles depend on the previous behaviour of the population of particles. We study the question if the process is recurrent or transient, i.e., whether infinitely many particles visit the origin or not.
Barlow, Martin T; Sousi, Perla
2010-01-01
A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $\\Z^d$ with $d \\ge 19$ and the uniform spanning tree in $\\Z^2$ all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.
Mak, Chi H; Pham, Phuong; Afif, Samir A; Goodman, Myron F
2015-09-01
Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C→U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics.
Mak, Chi H.; Pham, Phuong; Afif, Samir A.; Goodman, Myron F.
2015-09-01
Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C →U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics.
Mak, Chi H.; Pham, Phuong; Afif, Samir A.; Goodman, Myron F.
2015-01-01
Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C → U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics. PMID:26465508
On-line Viterbi Algorithm and Its Relationship to Random Walks
?rámek, Rastislav; Vina?, Tomá?
2007-01-01
In this paper, we introduce the on-line Viterbi algorithm for decoding hidden Markov models (HMMs) in much smaller than linear space. Our analysis on two-state HMMs suggests that the expected maximum memory used to decode sequence of length $n$ with $m$-state HMM can be as low as $\\Theta(m\\log n)$, without a significant slow-down compared to the classical Viterbi algorithm. Classical Viterbi algorithm requires $O(mn)$ space, which is impractical for analysis of long DNA sequences (such as complete human genome chromosomes) and for continuous data streams. We also experimentally demonstrate the performance of the on-line Viterbi algorithm on a simple HMM for gene finding on both simulated and real DNA sequences.
When Human Walking is a Random Walk
Hausdorff, J. M.
1998-03-01
The complex, hierarchical locomotor system normally does a remarkable job of controlling an inherently unstable, multi-joint system. Nevertheless, the stride interval --- the duration of a gait cycle --- fluctuates from one stride to the next, even under stationary conditions. We used random walk analysis to study the dynamical properties of these fluctuations under normal conditions and how they change with disease and aging. Random walk analysis of the stride-to-stride fluctuations of healthy, young adult men surprisingly reveals a self-similar pattern: fluctuations at one time scale are statistically similar to those at multiple other time scales (Hausdorff et al, J Appl Phsyiol, 1995). To study the stability of this fractal property, we analyzed data obtained from healthy subjects who walked for 1 hour at their usual pace, as well as at slower and faster speeds. The stride interval fluctuations exhibited long-range correlations with power-law decay for up to a thousand strides at all three walking rates. In contrast, during metronomically-paced walking, these long-range correlations disappeared; variations in the stride interval were uncorrelated and non-fractal (Hausdorff et al, J Appl Phsyiol, 1996). To gain insight into the mechanism(s) responsible for this fractal property, we examined the effects of aging and neurological impairment. Using detrended fluctuation analysis (DFA), we computed α, a measure of the degree to which one stride interval is correlated with previous and subsequent intervals over different time scales. α was significantly lower in healthy elderly subjects compared to young adults (p < .003) and in subjects with Huntington's disease, a neuro-degenerative disorder of the central nervous system, compared to disease-free controls (p < 0.005) (Hausdorff et al, J Appl Phsyiol, 1997). α was also significantly related to degree of functional impairment in subjects with Huntington's disease (r=0.78). Recently, we have observed that just as
Persistence of random walk records
Ben-Naim, E.; Krapivsky, P. L.
2014-06-01
We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk ‘superior’ if the record is always above average, and conversely, the walk is said to be ‘inferior’ if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ˜ t-β, in the limit t → ∞, and that the persistence exponent is nontrivial, β = 0.382 258…. The fraction of inferior walks, I, also decays as a power law, I ˜ t-α, but the persistence exponent is smaller, α = 0.241 608…. Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with a given record and position, while for inferior walks it suffices to study the density as a function of position.
Random walk term weighting for information retrieval
Blanco, R.; Lioma, Christina
2007-01-01
We present a way of estimating term weights for Information Retrieval (IR), using term co-occurrence as a measure of dependency between terms.We use the random walk graph-based ranking algorithm on a graph that encodes terms and co-occurrence dependencies in text, from which we derive term weights...... that represent a quantification of how a term contributes to its context. Evaluation on two TREC collections and 350 topics shows that the random walk-based term weights perform at least comparably to the traditional tf-idf term weighting, while they outperform it when the distance between co-occurring terms...
Crossover from random walk to self-avoiding walk
Rieger, Jens
1988-11-01
A one-dimensional n-step random walk on openZ1 which must not visit a vertex more than k times is studied via Monte Carlo methods. The dependences of the mean-square end-to-end distance of the walk and of the fraction of trapped walks on λ=(k-1)/n will be given for the range from λ=0 (self-avoiding walk) to λ=1 (unrestricted random walk). From the results it is conjectured that in the limit n-->∞ the walk obeys simple random walk statistics with respect to its static properties for all λ>0.
Discriminative Parameter Estimation for Random Walks Segmentation
Baudin, Pierre-Yves; Goodman, Danny; Kumar, Puneet; Azzabou, Noura; Carlier, Pierre G.; Paragios, Nikos; Pawan Kumar, M.
2013-01-01
International audience; The Random Walks (RW) algorithm is one of the most e - cient and easy-to-use probabilistic segmentation methods. By combining contrast terms with prior terms, it provides accurate segmentations of medical images in a fully automated manner. However, one of the main drawbacks of using the RW algorithm is that its parameters have to be hand-tuned. we propose a novel discriminative learning framework that estimates the parameters using a training dataset. The main challen...
Localization of reinforced random walks
Tarrès, Pierre
2011-01-01
We describe and analyze how reinforced random walks can eventually localize, i.e. only visit finitely many sites. After introducing vertex and edge self-interacting walks on a discrete graph in a general setting, and stating the main results and conjectures so far on the topic, we present martingale techniques that provide an alternative proof of the a.s. localization of vertex-reinforced random walks (VRRWs) on the integers on finitely many sites and, with positive probability, on five consecutive sites, initially proved by Pemantle and Volkov (1999). Next we introduce the continuous time-lines representation (sometimes called Rubin construction) and its martingale counterpart, and explain how it has been used to prove localization of some reinforced walks on one attracting edge. Then we show how a modified version of this construction enables one to propose a new short proof of the a.s. localization of VRRWs on five sites on Z.
Random walk models for top-N recommendation task
Yin ZHANG; Jiang-qin WU; Yue-ting ZHUANG
2009-01-01
Recently there has been an increasing interest in applying random walk based methods to recommender systems.We employ a Gaussian random field to model the top-N recommendation task as a semi-supervised learning problem.taking into account the degree of each node on the user-item bipartite graph,and induce an effective absorbing random walk (ARW) algorithm for the top-N recommendation task.Our random walk approach directly generates the top-N recommendations for individuals,rather than predicting the ratings of the recommendations.Experimental results on the two real data sets show that our random walk algorithm significantly outperforms the state-of-the-art random walk based personalized ranking algorithm as well as the popular item-based collaborative filtering method.
Implement Quantum Random Walks with Linear Optics Elements
Zhao, Z; Li, H; Yang, T; Chen, Z B; Pan, J W; Zhao, Zhi; Du, Jiangfeng; Li, Hui; Yang, Tao; Chen, Zeng-Bing; Pan, Jian-Wei
2002-01-01
The quantum random walk has drawn special interests because its remarkable features to the classical counterpart could lead to new quantum algorithms. In this paper, we propose a feasible scheme to implement quantum random walks on a line using only linear optics elements. With current single-photon interference technology, the steps that could be experimentally implemented can be extended to very large numbers. We also show that, by decohering the quantum states, our scheme for quantum random walk tends to be classical.
Near-Optimal Random Walk Sampling in Distributed Networks
Sarma, Atish Das; Pandurangan, Gopal
2012-01-01
Performing random walks in networks is a fundamental primitive that has found numerous applications in communication networks such as token management, load balancing, network topology discovery and construction, search, and peer-to-peer membership management. While several such algorithms are ubiquitous, and use numerous random walk samples, the walks themselves have always been performed naively. In this paper, we focus on the problem of performing random walk sampling efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds and messages required to obtain several random walk samples in a continuous online fashion. We present the first round and message optimal distributed algorithms that present a significant improvement on all previous approaches. The theoretical analysis and comprehensive experimental evaluation of our algorithms show that they perform very well in different types of networks of differing topologies. In particular, our results show h...
Random walk centrality for temporal networks
Rocha, Luis Enrique Correa
2014-01-01
Nodes can be ranked according to their relative importance within the network. Ranking algorithms based on random walks are particularly useful because they connect topological and diffusive properties of the network. Previous methods based on random walks, as for example the PageRank, have focused on static structures. However, several realistic networks are indeed dynamic, meaning that their structure changes in time. In this paper, we propose a centrality measure for temporal networks based on random walks which we call TempoRank. While in a static network, the stationary density of the random walk is proportional to the degree or the strength of a node, we find that in temporal networks, the stationary density is proportional to the in-strength of the so-called effective network. The stationary density also depends on the sojourn probability q which regulates the tendency of the walker to stay in the node. We apply our method to human interaction networks and show that although it is important for a node ...
基于Mean Shift和随机游走的图像分割算法%Image Segmentation Algorithm Based on Mean Shift and Random Walk
穆克; 程伟; 褚俊霞
2012-01-01
An improved random walk algorithm was proposed herein.First,Mean Shift algorithm was adopted to preprocess the image,which was partitioned into a series of homogeneous areas,so that the homogeneous areas were taken as nodes to walk at random,with noise inhibited while reducing the number of nodes.Second,PMD was used to define the weight between regions.Thirdly,seeds were improved to have added the auxiliary seeds,and the auxiliary and signed seeds were used to walk random,with region merging realized.The final image segmentation was reached.Experimental results expatiates that the proposed method highlights the segmentation accuracy.%提出了一种改进的随机游走算法。首先,采用Mean Shift算法对图像进行预处理,将图像划分成一些同质区域,用同质区域作为节点进行随机游走,在降低节点数的同时也抑制了噪声对分割的影响;其次,利用马氏距离定义区域之间的权值;对种子点进行了改进,增加了辅助种子点,利用辅助种子点和用户标记的种子点进行随机游走,实现同质区域的合并,实现图像的最终分割。实验结果表明,该算法提高了图像分割的精度。
Random walks, random fields, and disordered systems
Černý, Jiří; Kotecký, Roman
2015-01-01
Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disordered Systems. The specific topics covered include a study of Branching Brownian Motion from the perspective of disordered (spin-glass) systems, a detailed analysis of weakly self-avoiding random walks in four spatial dimensions via methods of field theory and the renormalization group, a study of phase transitions in disordered discrete structures using a rigorous version of the cavity method, a survey of recent work on interacting polymers in the ballisticity regime and, finally, a treatise on two-dimensional loop-soup models and their connection to conformally invariant systems and the Gaussian Free Field. The notes are aimed at early graduate students with a mod...
Korneta, W.; Pytel, Z.
1988-07-01
The random walk of a particle on a three-dimensional semi-infinite lattice is considered. In order to study the effect of the surface on the random walk, it is assumed that the velocity of the particle depends on the distance to the surface. Moreover it is assumed that at any point the particle may be absorbed with a certain probability. The probability of the return of the particle to the starting point and the average time of eventual return are calculated. The dependence of these quantities on the distance to the surface, the probability of absorption and the properties of the surface is discussed. The method of generating functions is used.
Groups, graphs and random walks
Salvatori, Maura; Sava-Huss, Ecaterina
2017-01-01
An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrödinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubted...
Convergence of a random walk method for the Burgers equation
Roberts, S.
1985-10-01
In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose of this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries.
Random Walks Estimate Land Value
Blanchard, Ph
2010-01-01
Expected urban population doubling calls for a compelling theory of the city. Random walks and diffusions defined on spatial city graphs spot hidden areas of geographical isolation in the urban landscape going downhill. First--passage time to a place correlates with assessed value of land in that. The method accounting the average number of random turns at junctions on the way to reach any particular place in the city from various starting points could be used to identify isolated neighborhoods in big cities with a complex web of roads, walkways and public transport systems.
Random walks on reductive groups
Benoist, Yves
2016-01-01
The classical theory of Random Walks describes the asymptotic behavior of sums of independent identically distributed random real variables. This book explains the generalization of this theory to products of independent identically distributed random matrices with real coefficients. Under the assumption that the action of the matrices is semisimple – or, equivalently, that the Zariski closure of the group generated by these matrices is reductive - and under suitable moment assumptions, it is shown that the norm of the products of such random matrices satisfies a number of classical probabilistic laws. This book includes necessary background on the theory of reductive algebraic groups, probability theory and operator theory, thereby providing a modern introduction to the topic.
Einstein's random walk and thermal diffusion
2013-01-01
Thermal diffusion has been studied for over 150 years. Despite of the long history and the increasing importance of the phenomenon, the physics of thermal diffusion remains poorly understood. In this paper Ludwig's thermal diffusion is explained using Einstein's random walk. The only new structure added is the spatial heterogeneity of the random walk to reflect the temperature gradient of thermal diffusion. Hence, the walk length and the walk speed are location dependent functions in this pap...
Random Walk Smooth Transition Autoregressive Models
2004-01-01
This paper extends the family of smooth transition autoregressive (STAR) models by proposing a specification in which the autoregressive parameters follow random walks. The random walks in the parameters can capture structural change within a regime switching framework, but in contrast to the time varying STAR (TV-STAR) speciifcation recently introduced by Lundbergh et al (2003), structural change in our random walk STAR (RW-STAR) setting follows a stochastic process rather than a determinist...
Exponential algorithmic speedup by quantum walk
Childs, A M; Deotto, E; Farhi, E; Gutmann, S; Spielman, D A; Childs, Andrew M.; Cleve, Richard; Deotto, Enrico; Farhi, Edward; Gutmann, Sam; Spielman, Daniel A.
2002-01-01
We construct an oracular problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time.
Excited random walks: results, methods, open problems
Kosygina, Elena
2012-01-01
We consider a class of self-interacting random walks in deterministic or random environments, known as excited random walks or cookie walks, on the d-dimensional integer lattice. The main purpose of this paper is two-fold: to give a survey of known results and some of the methods and to present several new results. The latter include functional limit theorems for transient one-dimensional excited random walks in bounded i.i.d. cookie environments as well as some zero-one laws. Several open problems are stated.
Near-Optimal Sublinear Time Bounds for Distributed Random Walks
Sarma, Atish Das; Pandurangan, Gopal; Tetali, Prasad
2009-01-01
We focus on the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample on an undirected network. Despite the widespread use of random walks in distributed computing, most algorithms that compute a random walk sample of length $\\ell$ naively, i.e., in $O(\\ell)$ rounds. Recently, the first sublinear time distributed algorithm was presented that ran in $\\tilde{O}(\\ell^{2/3}D^{1/3})$ rounds {$\\tilde{O}$ hides polylog factors in the number of nodes in the network} where $D$ is the diameter of the network [Das Sarma et al. PODC 2009]. This work further conjectured that a running time of $\\tilde{O}(\\sqrt{\\ell D})$ is possible and that this is essentially optimal. In this paper, we resolve these conjectures by showing almost tight bounds on distributed random walks. We present a distributed algorithm that performs a random walk of length $\\ell$ in $\\tilde{O}(\\sqrt{\\ell D})$ rounds, where...
A random walk with a branching system in random environments
Ying-qiu LI; Xu LI; Quan-sheng LIU
2007-01-01
We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.
Biased random walks on multiplex networks
Battiston, Federico; Latora, Vito
2015-01-01
Biased random walks on complex networks are a particular type of walks whose motion is biased on properties of the destination node, such as its degree. In recent years they have been exploited to design efficient strategies to explore a network, for instance by constructing maximally mixing trajectories or by sampling homogeneously the nodes. In multiplex networks, the nodes are related through different types of links (layers or communication channels), and the presence of connections at different layers multiplies the number of possible paths in the graph. In this work we introduce biased random walks on multiplex networks and provide analytical solutions for their long-term properties such as the stationary distribution and the entropy rate. We focus on degree-biased walks and distinguish between two subclasses of random walks: extensive biased walks consider the properties of each node separately at each layer, intensive biased walks deal instead with intrinsically multiplex variables. We study the effec...
Sub-Markov Random Walk for Image Segmentation.
Dong, Xingping; Shen, Jianbing; Shao, Ling; Van Gool, Luc
2016-02-01
A novel sub-Markov random walk (subRW) algorithm with label prior is proposed for seeded image segmentation, which can be interpreted as a traditional random walker on a graph with added auxiliary nodes. Under this explanation, we unify the proposed subRW and other popular random walk (RW) algorithms. This unifying view will make it possible for transferring intrinsic findings between different RW algorithms, and offer new ideas for designing novel RW algorithms by adding or changing auxiliary nodes. To verify the second benefit, we design a new subRW algorithm with label prior to solve the segmentation problem of objects with thin and elongated parts. The experimental results on both synthetic and natural images with twigs demonstrate that the proposed subRW method outperforms previous RW algorithms for seeded image segmentation.
Background Extraction Using Random Walk Image Fusion.
Hua, Kai-Lung; Wang, Hong-Cyuan; Yeh, Chih-Hsiang; Cheng, Wen-Huang; Lai, Yu-Chi
2016-12-23
It is important to extract a clear background for computer vision and augmented reality. Generally, background extraction assumes the existence of a clean background shot through the input sequence, but realistically, situations may violate this assumption such as highway traffic videos. Therefore, our probabilistic model-based method formulates fusion of candidate background patches of the input sequence as a random walk problem and seeks a globally optimal solution based on their temporal and spatial relationship. Furthermore, we also design two quality measures to consider spatial and temporal coherence and contrast distinctness among pixels as background selection basis. A static background should have high temporal coherence among frames, and thus, we improve our fusion precision with a temporal contrast filter and an optical-flow-based motionless patch extractor. Experiments demonstrate that our algorithm can successfully extract artifact-free background images with low computational cost while comparing to state-of-the-art algorithms.
Discriminative parameter estimation for random walks segmentation.
Baudin, Pierre-Yves; Goodman, Danny; Kumrnar, Puneet; Azzabou, Noura; Carlier, Pierre G; Paragios, Nikos; Kumar, M Pawan
2013-01-01
The Random Walks (RW) algorithm is one of the most efficient and easy-to-use probabilistic segmentation methods. By combining contrast terms with prior terms, it provides accurate segmentations of medical images in a fully automated manner. However, one of the main drawbacks of using the RW algorithm is that its parameters have to be hand-tuned. we propose a novel discriminative learning framework that estimates the parameters using a training dataset. The main challenge we face is that the training samples are not fully supervised. Specifically, they provide a hard segmentation of the images, instead of a probabilistic segmentation. We overcome this challenge by treating the optimal probabilistic segmentation that is compatible with the given hard segmentation as a latent variable. This allows us to employ the latent support vector machine formulation for parameter estimation. We show that our approach significantly outperforms the baseline methods on a challenging dataset consisting of real clinical 3D MRI volumes of skeletal muscles.
Quantum random walks - an introductory overview
Kempe, J
2003-01-01
This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions.
Some case studies of random walks in dynamic random environments
Soares dos Santos, Renato
2012-01-01
This thesis is dedicated to the study of random walks in dynamic random environments. These are models for the motion of a tracer particle in a disordered medium, which is called a static random environment if it stays constant in time, or dynamic otherwise. The evolution of the random walk is defi
Quantum random walks and decision making.
Shankar, Karthik H
2014-01-01
How realistic is it to adopt a quantum random walk model to account for decisions involving two choices? Here, we discuss the neural plausibility and the effect of initial state and boundary thresholds on such a model and contrast it with various features of the classical random walk model of decision making.
Random recursive trees and the elephant random walk
Kürsten, Rüdiger
2016-03-01
One class of random walks with infinite memory, so-called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce another model, the skew elephant random walk, and calculate the first and second moment of this process.
ENHANCED RANDOM WALK WITH CHOICE: AN EMPIRICAL STUDY
John Alexandris
2014-03-01
Full Text Available The Random Walk with d Choice RWC d( is a recently proposed variation of the simple Random Walk that first selects a subset of d neighbor nodes and then decides to move to the node which minimizes the value of a certain parameter; this parameter captures the number of past visits of the walk to that node. In this paper, we propose the Enhanced Random Walk with d Choice algorithm ERWC d h ( , which first selects a subset of d neighbor nodes and then decides to move to the node which minimizes a value H defined at every node; this H value depends on a parameter h and captures information about past visits of the walk to that node and - with a certain weight - to its neighbors. Simulations of the Enhanced Random Walk with d Choice algorithm on various types of graphs indicate beneficial results with respect to Cover Time and Load Balancing. The graph types used are the Random Geometric Graph, Torus, Grid, Hypercube, Lollipop and Bernoulli.
ON THE RANGE OF RANDOM WALKS IN RANDOM ENVIRONMENT
ZHOUXIANYIN
1995-01-01
The range of roaldom walk on Zd in symmetric random environment is investigated. As results, it is proved that the strong law of large numbers for the range of random walk oil Zd in some random environments holds if d > 3, and a weak law of large numbers holds for d = 1.
Cooper, M A
2000-01-01
We present various approximations for the angular distribution of particles emerging from an optically thick, purely isotropically scattering region into a vacuum. Our motivation is to use such a distribution for the Fleck-Canfield random walk method [1] for implicit Monte Carlo (IMC) [2] radiation transport problems. We demonstrate that the cosine distribution recommended in the original random walk paper [1] is a poor approximation to the angular distribution predicted by transport theory. Then we examine other approximations that more closely match the transport angular distribution.
Elements of random walk and diffusion processes
Ibe, Oliver C
2013-01-01
Presents an important and unique introduction to random walk theory Random walk is a stochastic process that has proven to be a useful model in understanding discrete-state discrete-time processes across a wide spectrum of scientific disciplines. Elements of Random Walk and Diffusion Processes provides an interdisciplinary approach by including numerous practical examples and exercises with real-world applications in operations research, economics, engineering, and physics. Featuring an introduction to powerful and general techniques that are used in the application of physical and dynamic
Scaling of random walk betweenness in networks
Narayan, O
2016-01-01
The betweenness centrality of graphs using random walk paths instead of geodesics is studied. A scaling collapse with no adjustable parameters is obtained as the graph size $N$ is varied; the scaling curve depends on the graph model. A normalized random betweenness, that counts each walk passing through a node only once, is also defined. It is argued to be more useful and seen to have simpler scaling behavior. In particular, the probability for a random walk on a preferential attachment graph to pass through the root node is found to tend to unity as $N\\rightarrow\\infty.$
Levy random walks on multiplex networks
Guo, Quantong; Zheng, Zhiming; Moreno, Yamir
2016-01-01
Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Levy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Levy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a ...
On a directionally reinforced random walk
Ghosh, Arka; Roitershtein, Alexander
2011-01-01
We consider a generalized version of a directionally reinforced random walk, which was originally introduced by Mauldin, Monticino, and von Weizs\\"{a}cker in \\cite{drw}. Our main result is a stable limit theorem for the position of the random walk in higher dimensions. This extends a result of Horv\\'{a}th and Shao \\cite{limits} that was previously obtained in dimension one only (however, in a more stringent functional form).
Quenched moderate deviations principle for random walk in random environment
无
2010-01-01
We derive a quenched moderate deviations principle for the one-dimensional nearest random walk in random environment,where the environment is assumed to be stationary and ergodic.The approach is based on hitting time decomposition.
Movie Recommendation using Random Walks over the Contextual Graph
Bogers, Toine
algorithm that makes it easy to include different types of contextual information. It models the browsing process of a user on a movie database website by taking random walks over the contextual graph. We present our approach in this paper and highlight a number of future extensions with additional...
Movie Recommendation using Random Walks over the Contextual Graph
Bogers, Toine
Recommender systems have become an essential tool in fighting information overload. However, the majority of recommendation algorithms focus only on using ratings information, while disregarding information about the context of the recommendation process. We present ContextWalk, a recommendation...... algorithm that makes it easy to include different types of contextual information. It models the browsing process of a user on a movie database website by taking random walks over the contextual graph. We present our approach in this paper and highlight a number of future extensions with additional...... contextual information....
Peres, Yuval; Sousi, Perla
2012-01-01
Let $\\mu_1,... \\mu_k$ be $d$-dimensional probability measures in $\\R^d$ with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.
Gaussian Networks Generated by Random Walks
Javarone, Marco Alberto
2014-01-01
We propose a random walks based model to generate complex networks. Many authors studied and developed different methods and tools to analyze complex networks by random walk processes. Just to cite a few, random walks have been adopted to perform community detection, exploration tasks and to study temporal networks. Moreover, they have been used also to generate scale-free networks. In this work, we define a random walker that plays the role of "edges-generator". In particular, the random walker generates new connections and uses these ones to visit each node of a network. As result, the proposed model allows to achieve networks provided with a Gaussian degree distribution, and moreover, some features as the clustering coefficient and the assortativity show a critical behavior. Finally, we performed numerical simulations to study the behavior and the properties of the cited model.
Feature Learning Based Random Walk for Liver Segmentation
Zheng, Yongchang; Ai, Danni; Zhang, Pan; Gao, Yefei; Xia, Likun; Du, Shunda; Sang, Xinting; Yang, Jian
2016-01-01
Liver segmentation is a significant processing technique for computer-assisted diagnosis. This method has attracted considerable attention and achieved effective result. However, liver segmentation using computed tomography (CT) images remains a challenging task because of the low contrast between the liver and adjacent organs. This paper proposes a feature-learning-based random walk method for liver segmentation using CT images. Four texture features were extracted and then classified to determine the classification probability corresponding to the test images. Seed points on the original test image were automatically selected and further used in the random walk (RW) algorithm to achieve comparable results to previous segmentation methods. PMID:27846217
The associated random walk and martingales in random walks with stationary increments
Grey, D R
2010-01-01
We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where the increments are Markovian or Gaussian, and an application in queueing is considered.
Scaling Argument of Anisotropic Random Walk
XU Bing-Zhen; JIN Guo-Jun; WANG Fei-Feng
2005-01-01
In this paper, we analytically discuss the scaling properties of the average square end-to-end distance for anisotropic random walk in D-dimensional space ( D ≥ 2), and the returning probability Pn(ro) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for and Pn (ro), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain ～ n, where ⊥ refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have ～ n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than the dimensions of the space, we must have ～ n2 for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.
Wang, Jeen-Shing; Lin, Che-Wei; Yang, Ya-Ting C; Ho, Yu-Jen
2012-10-01
This paper presents a walking pattern classification and a walking distance estimation algorithm using gait phase information. A gait phase information retrieval algorithm was developed to analyze the duration of the phases in a gait cycle (i.e., stance, push-off, swing, and heel-strike phases). Based on the gait phase information, a decision tree based on the relations between gait phases was constructed for classifying three different walking patterns (level walking, walking upstairs, and walking downstairs). Gait phase information was also used for developing a walking distance estimation algorithm. The walking distance estimation algorithm consists of the processes of step count and step length estimation. The proposed walking pattern classification and walking distance estimation algorithm have been validated by a series of experiments. The accuracy of the proposed walking pattern classification was 98.87%, 95.45%, and 95.00% for level walking, walking upstairs, and walking downstairs, respectively. The accuracy of the proposed walking distance estimation algorithm was 96.42% over a walking distance.
Memoryless Routing in Convex Subdivisions: Random Walks are Optimal
Chen, Dan; Dujmovic, Vida; Morin, Pat
2009-01-01
A memoryless routing algorithm is one in which the decision about the next edge on the route to a vertex t for a packet currently located at vertex v is made based only on the coordinates of v, t, and the neighbourhood, N(v), of v. The current paper explores the limitations of such algorithms by showing that, for any (randomized) memoryless routing algorithm A, there exists a convex subdivision on which A takes Omega(n^2) expected time to route a message between some pair of vertices. Since this lower bound is matched by a random walk, this result implies that the geometric information available in convex subdivisions is not helpful for this class of routing algorithms. The current paper also shows the existence of triangulations for which the Random-Compass algorithm proposed by Bose etal (2002,2004) requires 2^{\\Omega(n)} time to route between some pair of vertices.
From random walks to spin glasses
Derrida, B.
1997-02-01
The talk was a short review on systems which exhibit non-self-averaging effects: sums of random variables when the distribution has a long tail, mean field spin glasses, random map models and returns of a random walk to the origin. Non-self-averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach. Also we will see that for the random map models or for the problem of the returns of a random walk to the origin, the non-self-averaging effects coincide with the results of the replica approach when the number n of replica n = - {1}/{2} or n = -1.
Coverage maximization under resource constraints using proliferating random walks
Sudipta Saha; Niloy Ganguly; Abhijit Guria
2015-02-01
Dissemination of information has been one of the prime needs in almost every kind of communication network. The existing algorithms for this service, try to maximize the coverage, i.e., the number of distinct nodes to which a given piece of information could be conveyed under the constraints of time and energy. However, the problem becomes challenging for unstructured and decentralized environments. Due to its simplicity and adaptability, random walk (RW) has been a very useful tool for such environments. Different variants of this technique have been studied. In this paper, we study a history-based non-uniform proliferating random strategy where new walkers are dynamically introduced in the sparse regions of the network. Apart from this, we also study the breadth-first characteristics of the random walk-based algorithms through an appropriately designed metrics.
何昌保; 马秀丽; 余长明
2016-01-01
For dual source CT image with contrast media,due to heart soft tissue density and contrast media uneven distribution result in the CT value of heart tissues uneven and boundary fuzzy,taking a single image segmentation algorithm is too difficult to obtain satisfactory results,so morphological reconstruction and random walks hybrid method is proposed in this paper.Firstly,we used morphological reconstruction operation on image smoothing filtering, which makes the heart cavity gray information convergence and gray level differences with the surrounding tissue and get the left atrium area with the fuzzy boundary;Then the random walks algorithm sets the seed points for each region of the image,and gives the weight of each side,and takes the weight of the edge as the transfer probability.For each unlabeled point is calculated from the point of first arrival probability of seed points.Finally,according to the first hit probability to choose the maximum that a class as belonging to the class,attribute of the unlabeled points and finally get the accurate left atrial.%针对在传统的CT介入式治疗过程中，胸腔中软组织较多软组织的厚度和注射的造影剂在心脏中呈现的不均匀分布，导致在采用CT成像的图像中胸腔内部各组织之间存在边界模糊或者确实等状况，本文提出一种采用形态重构和随机行走相结合的分割方法。首先利用形态学开闭运算对图像进行化简，并使得心脏 CT腔体边界分离，进而使得各个组织组织分离，再结合Random walks算法。从而使得不需要标记太多种子点的情况下提高了分割的速度和准确性，实验证明该方法能够达到预期的目标。
Szybisz, L.; Zabolitzky, John G.
We describe a Monte-Carlo algorithm to solve exactly the ground-state problem for a system of up to four nucleons interacting via a scalar neutral meson field. The mesonic degrees of freedom are treated exactly without recourse to the potential approximation.
RENEWAL THEOREM FOR (L, 1)-RANDOM WALK IN RANDOM ENVIRONMENT
洪文明; 孙鸿雁
2013-01-01
We consider a random walk on Z in random environment with possible jumps{-L, · · · ,-1, 1}, in the case that the environment{ωi: i∈Z}are i.i.d.. We establish the renewal theorem for the Markov chain of “the environment viewed from the particle” in both annealed probability and quenched probability, which generalize partially the results of Kesten (1977) and Lalley (1986) for the nearest random walk in random environment on Z, respectively. Our method is based on the intrinsic branching structure within the (L, 1)-RWRE formulated in Hong and Wang (2013).
Exact Random Walk Distributions using Noncommutative Geometry
Bellissard, J; Barelli, A; Claro, F; Bellissard, Jean; Camacho, Carlos J; Barelli, Armelle; Claro, Francisco
1997-01-01
Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $ N $ on a two-dimensional square lattice for large $ N $, taking into account finite size contributions.
Iterated random walks with shape prior
Pujadas, Esmeralda Ruiz; Kjer, Hans Martin; Piella, Gemma;
2016-01-01
We propose a new framework for image segmentation using random walks where a distance shape prior is combined with a region term. The shape prior is weighted by a confidence map to reduce the influence of the prior in high gradient areas and the region term is computed with k-means to estimate th...
Random Walk Method for Potential Problems
Krishnamurthy, T.; Raju, I. S.
2002-01-01
A local Random Walk Method (RWM) for potential problems governed by Lapalace's and Paragon's equations is developed for two- and three-dimensional problems. The RWM is implemented and demonstrated in a multiprocessor parallel environment on a Beowulf cluster of computers. A speed gain of 16 is achieved as the number of processors is increased from 1 to 23.
RANDOM WALK HYPOTHESIS IN FINANCIAL MARKETS
Nicolae-Marius JULA
2017-05-01
Full Text Available Random walk hypothesis states that the stock market prices do not follow a predictable trajectory, but are simply random. If you are trying to predict a random set of data, one should test for randomness, because, despite the power and complexity of the used models, the results cannot be trustworthy. There are several methods for testing these hypotheses and the use of computational power provided by the R environment makes the work of the researcher easier and with a cost-effective approach. The increasing power of computing and the continuous development of econometric tests should give the potential investors new tools in selecting commodities and investing in efficient markets.
Angular processes related to Cauchy random walks
Cammarota, Valemtina
2011-01-01
We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of non-linear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the $n$-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.
A Random Walk to Economic Freedom?
Witte, Mark David
2013-04-01
Full Text Available Given the wide use of economic freedom in economic literature it is imperative to understand how economic freedom evolves. Results suggest that levels of economic freedom are dominated by random shocks. Using a test for stationarity devised by Westerlund and Larsson (2012 we are unable to reject the null hypothesis of a random walk. The changes to economic freedom also are mostly driven by random shocks with only a minor role played by country specific characteristics. Additionally, changes to economic freedom are partially reversed as increases (decreases in one year are partially offset by decreases (increases in the next year.
Random Walk Picture of Basketball Scoring
Gabel, Alan
2011-01-01
We present evidence, based on play-by-play data from all 6087 games from the 2006/07--2009/10 seasons of the National Basketball Association (NBA), that basketball scoring is well described by a weakly-biased continuous-time random walk. The time between successive scoring events follows an exponential distribution, with little memory between different scoring intervals. Using this random-walk picture that is augmented by features idiosyncratic to basketball, we account for a wide variety of statistical properties of scoring, such as the distribution of the score difference between opponents and the fraction of game time that one team is in the lead. By further including the heterogeneity of team strengths, we build a computational model that accounts for essentially all statistical features of game scoring data and season win/loss records of each team.
Random walk centrality in interconnected multilayer networks
Solé-Ribalta, Albert; Gómez, Sergio; Arenas, Alex
2015-01-01
Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influential nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.
Delayed Random Walks: Modeling Human Posture Control
Ohira, Toru
1998-03-01
We consider a phenomenological description of a noisy trajectory which appears on a stabiliogram platform during human postural sway. We hypothesize that this trajectory arises due to a mixture of uncontrollable noise and a corrective delayed feedback to an upright position. Based on this hypothesis, we model the process with a biased random walk whose transition probability depends on its position at a fixed time delay in the past, which we call a delayed random walk. We first introduce a very simple model (T. Ohira and J. G. Milton, Phys.Rev.E. 52), 3277, (1995), which can nevertheless capture the rough qualitative features of the two--point mean square displacement of experimental data with reasonable estimation of delay time. Then, we discuss two approaches toward better capturing and understanding of the experimental data. The first approach is an extension of the model to include a spatial displacement threshold from the upright position below which no or only weak corrective feedback motion takes place. This can be incorporated into an extended delayed random walk model. Numerical simulations show that this extended model can better capture the three scaling region which appears in the two--point mean square displacement. The other approach studied the autocorrelation function of the experimental data, which shows oscillatory behavior. We recently investigated a delayed random walk model whose autocorrelation function has analytically tractable oscillatory behavior (T. Ohira, Phys.Rev.E. 55), R1255, (1997). We discuss how this analytical understanding and its application to delay estimation (T. Ohira and R. Sawatari, Phys.Rev.E. 55), R2077, (1997) could possibly be used to further understand the postural sway data.
Dynamic random walks theory and applications
Guillotin-Plantard, Nadine
2006-01-01
The aim of this book is to report on the progress realized in probability theory in the field of dynamic random walks and to present applications in computer science, mathematical physics and finance. Each chapter contains didactical material as well as more advanced technical sections. Few appendices will help refreshing memories (if necessary!).· New probabilistic model, new results in probability theory· Original applications in computer science· Applications in mathematical physics· Applications in finance
A Random Walk Picture of Basketball
Gabel, Alan; Redner, Sidney
2012-02-01
We analyze NBA basketball play-by-play data and found that scoring is well described by a weakly-biased, anti-persistent, continuous-time random walk. The time between successive scoring events follows an exponential distribution, with little memory between events. We account for a wide variety of statistical properties of scoring, such as the distribution of the score difference between opponents and the fraction of game time that one team is in the lead.
Deterministic Random Walks on Regular Trees
Cooper, Joshua; Friedrich, Tobias; Spencer, Joel; 10.1002/rsa.20314
2010-01-01
Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\\Z^d$ and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite $k$-ary tree ($k \\ge 3$), we show that for any deviation $D$ there is an initial configuration of chips such that after running the Propp model for a ...
FRACTAL DIMENSION RESULTS FOR CONTINUOUS TIME RANDOM WALKS.
Meerschaert, Mark M; Nane, Erkan; Xiao, Yimin
2013-04-01
Continuous time random walks impose random waiting times between particle jumps. This paper computes the fractal dimensions of their process limits, which represent particle traces in anomalous diffusion.
Random walk search in unstructured P2P
Jia Zhaoqing; You Jinyuan; Rao Ruonan; Li Minglu
2006-01-01
Unstructured P2P has power-law link distribution, and the random walk in power-law networks is analyzed. The analysis results show that the probability that a random walker walks through the high degree nodes is high in the power-law network, and the information on the high degree nodes can be easily found through random walk. Random walk spread and random walk search method (RWSS) is proposed based on the analysis result. Simulation results show that RWSS achieves high success rates at low cost and is robust to high degree node failure.
A simplified analytical random walk model for proton dose calculation
Yao, Weiguang; Merchant, Thomas E.; Farr, Jonathan B.
2016-10-01
We propose an analytical random walk model for proton dose calculation in a laterally homogeneous medium. A formula for the spatial fluence distribution of primary protons is derived. The variance of the spatial distribution is in the form of a distance-squared law of the angular distribution. To improve the accuracy of dose calculation in the Bragg peak region, the energy spectrum of the protons is used. The accuracy is validated against Monte Carlo simulation in water phantoms with either air gaps or a slab of bone inserted. The algorithm accurately reflects the dose dependence on the depth of the bone and can deal with small-field dosimetry. We further applied the algorithm to patients’ cases in the highly heterogeneous head and pelvis sites and used a gamma test to show the reasonable accuracy of the algorithm in these sites. Our algorithm is fast for clinical use.
Fragment formation in biased random walks
Ramola, Kabir
2008-10-01
We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability pc (Anteneodo and Morgado 2007 Phys. Rev. Lett. 99 180602). The behaviour of ρ(l), the probability of formation of fragments of length l, is analysed. An exact expression for the generating function of ρ(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form \\rho (l) \\simeq 3/[l(\\log \\ l)^2] .
Random walk of passive tracers among randomly moving obstacles
Gori, Matteo; Floriani, Elena; Nardecchia, Ilaria; Pettini, Marco
2016-01-01
Background: This study is mainly motivated by the need of understanding how the diffusion behaviour of a biomolecule (or even of a larger object) is affected by other moving macromolecules, organelles, and so on, inside a living cell, whence the possibility of understanding whether or not a randomly walking biomolecule is also subject to a long-range force field driving it to its target. Method: By means of the Continuous Time Random Walk (CTRW) technique the topic of random walk in random environment is here considered in the case of a passively diffusing particle in a crowded environment made of randomly moving and interacting obstacles. Results: The relevant physical quantity which is worked out is the diffusion cofficient of the passive tracer which is computed as a function of the average inter-obstacles distance. Coclusions: The results reported here suggest that if a biomolecule, let us call it a test molecule, moves towards its target in the presence of other independently interacting molecules, its m...
On Polya-Friedman random walks
Huillet, Thierry [Laboratoire de Physique Theorique et Modelisation, CNRS-UMR 8089 et Universite de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95032, Cergy-Pontoise (France)], E-mail: Thierry.Huillet@u-cergy.fr
2008-12-19
The Polya process is an urn scheme arising in the context of contagion spreading. It exhibits unstable persistence effects. The Friedman urn process is dual to the Polya one with antipersistent stabilizing effects. It appears in a safety campaign problem. A Polya-Friedman urn process is investigated with a tuning persistence parameter extrapolating the latter two extreme processes. The study includes the diffusion approximations of both the Polya-Friedman proportion process and the population gap random walk. The structure of the former is a generalized Wright-Fisher diffusion appearing in population genetics. The correlation structure of the latter presents an anomalous character at a critical value of the persistence parameter.
Randomized robot navigation algorithms
Berman, P. [Penn State Univ., University Park, PA (United States); Blum, A. [Carnegie Mellon Univ., Pittsburgh, PA (United States); Fiat, A. [Tel-Aviv Univ. (Israel)] [and others
1996-12-31
We consider the problem faced by a mobile robot that has to reach a given target by traveling through an unmapped region in the plane containing oriented rectangular obstacles. We assume the robot has no prior knowledge about the positions or sizes of the obstacles, and acquires such knowledge only when obstacles are encountered. Our goal is to minimize the distance the robot must travel, using the competitive ratio as our measure. We give a new randomized algorithm for this problem whose competitive ratio is O(n4/9 log n), beating the deterministic {Omega}({radical}n) lower bound of [PY], and answering in the affirmative an open question of [BRS] (which presented an optimal deterministic algorithm). We believe the techniques introduced here may prove useful in other on-line situations in which information gathering is part of the on-line process.
Random walk immunization strategy on scale-free networks
Weidong PEI; Zengqiang CHEN; Zhuzhi YUAN
2009-01-01
A novel immunization strategy called the random walk immunization strategy on scale-free networks is proposed. Different from other known immunization strategies, this strategy works as follows: a node is randomly chosen from the network. Starting from this node, randomly walk to one of its neighbor node; if the present node is not immunized, then immunize it and continue the random walk; otherwise go back to the previous node and randomly walk again. This process is repeated until a certain fraction of nodes is immunized. By theoretical analysis and numerical simulations, we found that this strategy is very effective in comparison with the other known immunization strategies.
Hitting times for random walks with restarts
Janson, Svante
2010-01-01
The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $T(x,0)$ is asymptotic to $2|x|^2\\log|x|$ as $|x| \\to \\infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.
Random Walk Routing in WSNs with Regular Topologies
Hui Tian; Hong Shen; Teruo Matsuzawa
2006-01-01
Topology is one of the most important characteristics for any type of networks because it represents the network's inherent properties and has great impact on the performance of the network. For wireless sensor networks (WSN),a well-deployed regular topology can help save more energy than what a random topology can do. WSNs with regular topologies can prolong network lifetime as studied in many previous work. However, little work has been done in developing effective routing algorithms for WSNs with regular topologies, except routing along a shortest path with the knowledge of global location information of sensor nodes. In this paper, a new routing protocol based on random walk is proposed. It does not require global location information. It also achieves load balancing property inherently for WSNs which is difficult to achieve by other routing protocols. In the scenarios where the message required to be sent to the base station is in comparatively small size with the inquiry message among neighboring nodes, it is proved that the random walk routing protocol can guarantee high probability of successful transmission from the source to the base station with the same amount of energy consumption as the shortest path routing. Since in many applications of WSNs, sensor nodes often send only beep-like small messages to the base station to report their status, our proposed random walk routing is thus a viable scheme and can work very efficiently especially in these application scenarios. The random walk routing provides load balancing in the WSN as mentioned, however, the nodes near to the base station are inevitably under heavier burden than those far away from the base station. Therefore, a density-aware deployment scheme is further proposed to guarantee that the heavy-load nodes do not affect the network lifetime even if their energy is exhausted. The main idea is deploying sensors with different densities according to their distance to the base station. It will be
On the non-equivalence of two standard random walks
Bénichou, O.; Lindenberg, K.; Oshanin, G.
2013-09-01
We focus on two models of nearest-neighbour random walks on d-dimensional regular hyper-cubic lattices that are usually assumed to be identical-the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll-Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.
许健才; 张良均; 余燕团
2016-01-01
在图像分割中，针对 FCM 算法存在聚类数目需要预先给定、收敛速度慢等缺点，本文把快速模糊 C 均值聚类算法和随机游走算法相结合，具体方法为先采用快速模糊 C 均值聚类算法对图像进行预分割，以便获得聚类中心的位置，然后将该中心作为随机游走的种子点，再进行图像分割，实验结果得到了较为满意的预期效果，证明该方法是可行的。本文的研究为快速 FCM 实现自适应性和开发图形图像预处理系统提供了技术支持与理论依据。%As far as image segmentation, the defeat of the number of clusters for FCM algorithm is reeded to be improued. In this paper, the fast fuzzy C-means clustering and random walk algorithm are combined to solve the problem of image segmentation. Firstly, the fast FCM for image pre-segmentation to obtain the number of clusters and cluster central location as the seed points of random walk firstly. Then, for image segmentation, experimental results show that this method is feasible, and get a more satisfactory desired purpose. Results of this study achieve self-adaptive and fast FCM develop graphical image preprocessing system provides technical support and theoretical basis.
Concave Majorants of Random Walks and Related Poisson Processes
Abramson, Josh
2010-01-01
We offer a unified approach to the theory of concave majorants of random walks by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant for a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.
Optimal paths as correlated random walks
Perlsman, E.; Havlin, S.
2006-01-01
A numerical study of optimal paths in the directed polymer model shows that the paths are similar to correlated random walks. It is shown that when a directed optimal path of length t is divided into 3 segments whose length is t/3, the correlation between the transversal movements along the first and last path segments is independent of the path length t. It is also shown that the transversal correlations along optimal paths decrease as the paths approach their endpoints. The numerical results obtained for optimal paths in 1+4 dimensions are qualitatively similar to those obtained for optimal paths in lower dimensions, and the data supplies a strong numerical indication that 1+4 is not the upper critical dimension of this model, and of the associated KPZ equation.
Information Filtering via Biased Random Walk on Coupled Social Network
Da-Cheng Nie
2014-01-01
Full Text Available The recommender systems have advanced a great deal in the past two decades. However, most researchers focus their attentions on mining the similarities among users or objects in recommender systems and overlook the social influence which plays an important role in users’ purchase process. In this paper, we design a biased random walk algorithm on coupled social networks which gives recommendation results based on both social interests and users’ preference. Numerical analyses on two real data sets, Epinions and Friendfeed, demonstrate the improvement of recommendation performance by taking social interests into account, and experimental results show that our algorithm can alleviate the user cold-start problem more effectively compared with the mass diffusion and user-based collaborative filtering methods.
The parabolic Anderson model random walk in random potential
König, Wolfgang
2016-01-01
This is a comprehensive survey on the research on the parabolic Anderson model – the heat equation with random potential or the random walk in random potential – of the years 1990 – 2015. The investigation of this model requires a combination of tools from probability (large deviations, extreme-value theory, e.g.) and analysis (spectral theory for the Laplace operator with potential, variational analysis, e.g.). We explain the background, the applications, the questions and the connections with other models and formulate the most relevant results on the long-time behavior of the solution, like quenched and annealed asymptotics for the total mass, intermittency, confinement and concentration properties and mass flow. Furthermore, we explain the most successful proof methods and give a list of open research problems. Proofs are not detailed, but concisely outlined and commented; the formulations of some theorems are slightly simplified for better comprehension.
The Simple Random Walk Snake on Z^4 is Recurrent
Benjamini, Itai
2011-01-01
Consider the branching simple random walk on Z^d indexed by a critical geometric Galton-Watson tree conditioned to survive. Using the concept of unimodular random graphs, we show that the walk is recurrent if and only if d is less than or equal to 4.
Asymptotic Properties of Multistate Random Walks. I. Theory
Roerdink, J.B.T.M.; Shuler, K.E.
1985-01-01
A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) for multistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of
Pseudo Memory Effects, Majorization and Entropy in Quantum Random Walks
Bracken, A J; Tsohantjis, I; Bracken, Anthony J.; Ellinas, Demosthenes; Tsohantjis, Ioannis
2004-01-01
A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.
The Random Walk Model Based on Bipartite Network
Zhang Man-Dun
2016-01-01
Full Text Available With the continuing development of the electronic commerce and growth of network information, there is a growing possibility for citizens to be confused by the information. Though the traditional technology of information retrieval have the ability to relieve the overload of information in some extent, it can not offer a targeted personality service based on user’s interests and activities. In this context, the recommendation algorithm arose. In this paper, on the basis of conventional recommendation, we studied the scheme of random walk based on bipartite network and the application of it. We put forward a similarity measurement based on implicit feedback. In this method, a uneven character vector is imported(the weight of item in the system. We put forward a improved random walk pattern which make use of partial or incomplete neighbor information to create recommendation information. In the end, there is an experiment in the real data set, the recommendation accuracy and practicality are improved. We promise the reality of the result of the experiment
Biased random walks on Kleinberg's spatial networks
Pan, Gui-Jun; Niu, Rui-Wu
2016-12-01
We investigate the problem of the particle or message that travels as a biased random walk toward a target node in Kleinberg's spatial network which is built from a d-dimensional (d = 2) regular lattice improved by adding long-range shortcuts with probability P(rij) ∼rij-α, where rij is the lattice distance between sites i and j, and α is a variable exponent. Bias is represented as a probability p of the packet to travel at every hop toward the node which has the smallest Manhattan distance to the target node. We study the mean first passage time (MFPT) for different exponent α and the scaling of the MFPT with the size of the network L. We find that there exists a threshold probability pth ≈ 0.5, for p ≥pth the optimal transportation condition is obtained with an optimal transport exponent αop = d, while for 0 pth, and increases with L less than a power law and get close to logarithmical law for 0 complex network with a highly efficient structure for navigation although nodes hold null local information with a relatively large probability, which gives a powerful evidence for the reason why many real networks' navigability have small world property.
Random walks in directed modular networks
Comin, Cesar H.; Viana, Mateus P.; Antiqueira, Lucas; Costa, Luciano da F.
2014-12-01
Because diffusion typically involves symmetric interactions, scant attention has been focused on studying asymmetric cases. However, important networked systems underlain by diffusion (e.g. cortical networks and WWW) are inherently directed. In the case of undirected diffusion, it can be shown that the steady-state probability of the random walk dynamics is fully correlated with the degree, which no longer holds for directed networks. We investigate the relationship between such probability and the inward node degree, which we call efficiency, in modular networks. Our findings show that the efficiency of a given community depends mostly on the balance between its ingoing and outgoing connections. In addition, we derive analytical expressions to show that the internal degree of the nodes does not play a crucial role in their efficiency, when considering the Erdős-Rényi and Barabási-Albert models. The results are illustrated with respect to the macaque cortical network, providing subsidies for improving transportation and communication systems.
Prostate Cancer Segmentation Using Multispectral Random Walks
Artan, Yusuf; Haider, Masoom A.; Yetik, Imam Samil
Several studies have shown the advantages of multispectral magnetic resonance imaging (MRI) as a noninvasive imaging technique for prostate cancer localization. However, a large proportion of these studies are with human readers. There is a significant inter and intra-observer variability for human readers, and it is substantially difficult for humans to analyze the large dataset of multispectral MRI. To solve these problems a few studies were proposed for fully automated cancer localization in the past. However, fully automated methods are highly sensitive to parameter selection and often may not produce desirable segmentation results. In this paper, we present a semi-supervised segmentation algorithm by extending a graph based semi-supervised random walker algorithm to perform prostate cancer segmentation with multispectral MRI. Unlike classical random walker which can be applied only to dataset of single type of MRI, we develop a new method that can be applied to multispectral images. We prove the effectiveness of the proposed method by presenting the qualitative and quantitative results of multispectral MRI datasets acquired from 10 biopsy-confirmed cancer patients. Our results demonstrate that the multispectral MRI noticeably increases the sensitivity and jakkard measures of prostate cancer localization compared to single MR images; 0.71 sensitivity and 0.56 jakkard for multispectral images compared to 0.51 sensitivity and 0.44 jakkard for single MR image based segmentation.
Variational data assimilation using targetted random walks
Cotter, S. L.
2011-02-15
The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis. In either of these scenarios, it can be important to assess uncertainties in the assimilated state. Ideally, it is desirable to have complete information concerning the Bayesian posterior distribution for unknown state given data. We show that complete computational probing of this posterior distribution is now within the reach in the offline situation. We introduce a Markov chain-Monte Carlo (MCMC) method which enables us to directly sample from the Bayesian posterior distribution on the unknown functions of interest given observations. Since we are aware that these methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however, more sophisticated MCMC methods are available which exploit derivative information. For simplicity of exposition, we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number flow in a two-dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces. © 2011 John Wiley & Sons, Ltd.
Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks
Zaburdaev, V.; Fouxon, I.; Denisov, S.; Barkai, E.
2016-12-01
It is recognized now that a variety of real-life phenomena ranging from diffusion of cold atoms to the motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.
Levandowski, William Brower; Boyd, Oliver; Briggs, Richard; Gold, Ryan D.
2015-01-01
This paper develops a Monte Carlo algorithm for extracting three-dimensional lithospheric density models from geophysical data. Empirical scaling relationships between velocity and density create a 3D starting density model, which is then iteratively refined until it reproduces observed gravity and topography. This approach permits deviations from uniform crustal velocity-density scaling, which provide insight into crustal lithology and prevent spurious mapping of crustal anomalies into the mantle.
Random walks of cytoskeletal motors in open and closed compartments
Lipowsky, R.; Klumpp, S.
2001-01-01
Random walks of molecular motors, which bind to and unbind from cytoskeletal filaments, are studied theoretically. The bound and unbound motors undergo directed and nondirected motion, respectively. Motors in open compartments exhibit anomalous drift velocities. Motors in closed compartments generat
Record statistics of financial time series and geometric random walks.
Sabir, Behlool; Santhanam, M S
2014-09-01
The study of record statistics of correlated series in physics, such as random walks, is gaining momentum, and several analytical results have been obtained in the past few years. In this work, we study the record statistics of correlated empirical data for which random walk models have relevance. We obtain results for the records statistics of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lying in the range 1.5≤α≤1.8. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data.
A scaling law for random walks on networks
Perkins, Theodore J.; Foxall, Eric; Glass, Leon; Edwards, Roderick
2014-10-01
The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.
Randomized Filtering Algorithms
Katriel, Irit; Van Hentenryck, Pascal
2008-01-01
of AllDifferent and is generalization, the Global Cardinality Constraint. The first delayed filtering scheme is a Monte Carlo algorithm: its running time is superior, in the worst case, to that of enforcing are consistency after every domain event, while its filtering effectiveness is analyzed...... in the expected sense. The second scheme is a Las Vegas algorithm using filtering triggers: Its effectiveness is the same as enforcing are consistency after every domain event, while in the expected case it is faster by a factor of m/n, where n and m are, respectively, the number of nodes and edges...
A random walk-based method for segmentation of intravascular ultrasound images
Yan, Jiayong; Liu, Hong; Cui, Yaoyao
2014-04-01
Intravascular ultrasound (IVUS) is an important imaging technique that is used to study vascular wall architecture for diagnosis and assessment of the vascular diseases. Segmentation of lumen and media-adventitia boundaries from IVUS images is a basic and necessary step for quantitative assessment of the vascular walls. Due to ultrasound speckles, artifacts and individual differences, automated segmentation of IVUS images represents a challenging task. In this paper, a random walk based method is proposed for fully automated segmentation of IVUS images. Robust and accurate determination of the seed points for different regions is the key to successful use of the random walk algorithm in segmentation of IVUS images and is the focus of our work. The presented method mainly comprises five steps: firstly, the seed points inside the lumen and outside the adventitia are roughly estimated with intensity information, respectively; secondly, the seed points outside the adventitia are refined, and those of the media are determined through the results of applying random walk to the IVUS image with the roughly estimated seed points; thirdly, the media-adventitia boundary is detected by using random walk with the seed points obtained in the second step and the image gradient; fourthly, the seed points for media and lumen are refined; finally, the lumen boundary is extracted by using random walk again with the seed points obtained in the fourth step and the image gradient. The tests of the proposed algorithm on the in vivo dataset demonstrate the effectiveness of the presented IVUS image segmentation approach.
Sampling the equilibrium: the j-walking algorithm revisited
Rimas, Zilvinas
2016-01-01
The j-walking Monte-Carlo algorithm is revisited and updated to study the equilibrium properties of a system exhibiting broken ergodicity. The updated algorithm is tested on the Ising model and applied to the lattice-gas model for sorption in aerogel at low temperatures, when dynamics of the system is critically slowed down. It is demonstrated that the updated j-walking simulations are able to produce equilibrium isotherm which are typically hidden by the hysteresis effect within the standard single-flip simulations.
On the random walk characteristics of stock returns in India
Hiremath, Gourishankar S; Bandi, Kamaiah
2009-01-01
An attempt is made in this paper to examine whether stock returns in two premier two exchanges in India namely, Bombay Stock Exchange (BSE), and National Stock Exchange (NSE) follow a random walk. Towards this end, data on major indices during the period 1997 to 2009 are analyzed by using non-parametric Runs and BDS tests. The findings of the study reveal that the stock returns do not follow random walk during the sample period.
Peer-to-Peer Topology Formation Using Random Walk
Kwong, Kin-Wah; Tsang, Danny H. K.
Peer-to-Peer (P2P) systems such as live video streaming and content sharing are usually composed of a huge number of users with heterogeneous capacities. As a result, designing a distributed algorithm to form such a giant-scale topology in a heterogeneous environment is a challenging question because, on the one hand, the algorithm should exploit the heterogeneity of users' capacities to achieve load-balancing and, on the other hand, the overhead of the algorithm should be kept as low as possible. To meet such requirements, we introduce a very simple protocol for building heterogeneous unstructured P2P networks. The basic idea behind our protocol is to exploit a simple, distributed nature of random walk sampling to assist the peers in selecting their suitable neighbors in terms of capacity and connectivity to achieve load-balancing. To gain more insights into our proposed protocol, we also develop a detailed analysis to investigate our protocol under any heterogeneous P2P environment. The analytical results are validated by the simulations. The ultimate goal of this chapter is to stimulate further research to explore the fundamental issues in heterogeneous P2P networks.
Random hypergraphs and algorithmics
Andriamampianina, Tsiriniaina
2008-01-01
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by mean of exponential generating functions. The number of hypergraph component is bounded, as a generalisation of Wright inequalities for graphs: the proof is a combinatorial understanding of the structure by inclusion exclusion. Asymptotic results are obtained, thanks to generating functions proofs are at the end very easy to read, through complex analysis by saddle point method. By this way, we characterized: - the components with a given number of vertices and of hyperedges by the expected size of a random hypermatching in these structures. - the random hypergraphs (evolving hyperedge by hyperedge) according to the expected number of hyperedges when the first cycle appears in the evolving structure. This work is an open road to further works on random hypergraphs such as threshold phenomenon, tools used here seem to be sufficien...
Eich, H-J; Mach, H; Werner, C; Hesse, S
2004-09-01
To evaluate the immediate and long-term effects of aerobic treadmill plus Bobath walking training in subacute stroke survivors compared with Bobath walking training alone. Randomized controlled trial. Rehabilitation unit. Fifty patients, first-time supratentorial stroke, stroke interval less than six weeks, Barthel Index (0-100) from 50 to 80, able to walk a minimum distance of 12 m with either intermittent help or stand-by while walking, cardiovascular stable, minimum 50 W in the bicycle ergometry, randomly allocated to two groups, A and B. Group A 30 min of treadmill training, harness secured and minimally supported according to patients' needs, and 30 min of physiotherapy, every workday for six weeks, speed and inclination of the treadmill were adjusted to achieve a heart rate of HR: (Hrmax-HRrest)*0.6+HRrest; in group B 60 min of daily physiotherapy for six weeks. Primary outcome variables were the absolute improvement of walking velocity (m/s) and capacity (m), secondary were gross motor function including walking ability (score out of 13) and walking quality (score out of 41), blindly assessed before and after the intervention, and at follow-up three months later. Patients tolerated the aerobic training well with no side-effects, significantly greater improvement of walking velocity and capacity both at study end (p =0.001 versus p =0.002) and at follow-up (p Bobath walking training in moderately affected stroke patients was better than Bobath walking training alone with respect to the improvement of walking velocity and capacity. The treatment approach is recommended in patients meeting the inclusion criteria. A multicentre trial should follow to strengthen the evidence.
Predicting genetic interactions with random walks on biological networks
Singh Ambuj K
2009-01-01
Full Text Available Abstract Background Several studies have demonstrated that synthetic lethal genetic interactions between gene mutations provide an indication of functional redundancy between molecular complexes and pathways. These observations help explain the finding that organisms are able to tolerate single gene deletions for a large majority of genes. For example, system-wide gene knockout/knockdown studies in S. cerevisiae and C. elegans revealed non-viable phenotypes for a mere 18% and 10% of the genome, respectively. It has been postulated that the low percentage of essential genes reflects the extensive amount of genetic buffering that occurs within genomes. Consistent with this hypothesis, systematic double-knockout screens in S. cerevisiae and C. elegans show that, on average, 0.5% of tested gene pairs are synthetic sick or synthetic lethal. While knowledge of synthetic lethal interactions provides valuable insight into molecular functionality, testing all combinations of gene pairs represents a daunting task for molecular biologists, as the combinatorial nature of these relationships imposes a large experimental burden. Still, the task of mapping pairwise interactions between genes is essential to discovering functional relationships between molecular complexes and pathways, as they form the basis of genetic robustness. Towards the goal of alleviating the experimental workload, computational techniques that accurately predict genetic interactions can potentially aid in targeting the most likely candidate interactions. Building on previous studies that analyzed properties of network topology to predict genetic interactions, we apply random walks on biological networks to accurately predict pairwise genetic interactions. Furthermore, we incorporate all published non-interactions into our algorithm for measuring the topological relatedness between two genes. We apply our method to S. cerevisiae and C. elegans datasets and, using a decision tree
Performance of redirected walking algorithms in a constrained virtual world.
Hodgson, Eric; Bachmann, Eric; Thrash, Tyler
2014-04-01
Redirected walking algorithms imperceptibly rotate a virtual scene about users of immersive virtual environment systems in order to guide them away from tracking area boundaries. Ideally, these distortions permit users to explore large unbounded virtual worlds while walking naturally within a physically limited space. Many potential virtual worlds are composed of corridors, passageways, or aisles. Assuming users are not expected to walk through walls or other objects within the virtual world, these constrained worlds limit the directions of travel and as well as the number of opportunities to change direction. The resulting differences in user movement characteristics within the physical world have an impact on redirected walking algorithm performance. This work presents a comparison of generalized RDW algorithm performance within a constrained virtual world. In contrast to previous studies involving unconstrained virtual worlds, experimental results indicate that the steer-to-orbit keeps users in a smaller area than the steer-to-center algorithm. Moreover, in comparison to steer-to-center, steer-to-orbit is shown to reduce potential wall contacts by over 29%.
Search for Directed Networks by Different Random Walk Strategies
ZHU Zi-Qi; JIN Xiao-Ling; HUANG Zhi-Long
2012-01-01
A comparative study is carried out on the effciency of five different random walk strategies searching on directed networks constructed based on several typical complex networks.Due to the difference in search effciency of the strategies rooted in network clustering,the clustering coeFfcient in a random walker's eye on directed networks is defined and computed to be half of the corresponding undirected networks.The search processes are performed on the directed networks based on Erd(o)s-Rényi model,Watts-Strogatz model,Barabási-Albert model and clustered scale-free network model.It is found that self-avoiding random walk strategy is the best search strategy for such directed networks.Compared to unrestricted random walk strategy,path-iteration-avoiding random walks can also make the search process much more effcient. However,no-triangle-loop and no-quadrangle-loop random walks do not improve the search effciency as expected,which is different from those on undirected networks since the clustering coefficient of directed networks are smaller than that of undirected networks.
Random walk models of worker sorting in ant colonies.
Sendova-Franks, Ana B; Van Lent, Jan
2002-07-21
Sorting can be an important mechanism for the transfer of information from one level of biological organization to another. Here we study the algorithm underlying worker sorting in Leptothorax ant colonies. Worker sorting is related to task allocation and therefore to the adaptive advantages associated with an efficient system for the division of labour in ant colonies. We considered four spatially explicit individual-based models founded on two-dimensional correlated random walk. Our aim was to establish whether sorting at the level of the worker population could occur with minimal assumptions about the behavioural algorithm of individual workers. The behaviour of an individual worker in the models could be summarized by the rule "move if you can, turn always". We assume that the turning angle of a worker is individually specific and negatively dependent on the magnitude of an internal parameter micro which could be regarded as a measure of individual experience or task specialization. All four models attained a level of worker sortedness that was compatible with results from experiments onLeptothorax ant colonies. We found that the presence of a sorting pivot, such as the nest wall or an attraction force towards the centre of the worker population, was crucial for sorting. We make a distinction between such pivots and templates and discuss the biological implications of their difference.
First steps in random walks from tools to applications
Klafter, J
2011-01-01
The name ""random walk"" for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of ""Nature"". The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier such a problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays the theory of random walks has proved useful in physics andchemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcel
A random walk down Main Street
David Matthew Levinson
2016-08-01
Full Text Available US suburbs have often been characterized by their relatively low walk accessibility compared to more urban environments, and US urban environments have been char- acterized by low walk accessibility compared to cities in other countries. Lower overall density in the suburbs implies that activities, if spread out, would have a greater distance between them. But why should activities be spread out instead of developed contiguously? This brief research note builds a positive model for the emergence of contiguous development along “Main Street” to illustrate the trade-offs that result in the built environment we observe. It then suggests some policy interventions to place a “thumb on the scale” to choose which parcels will develop in which sequence to achieve socially preferred outcomes.
Perturbing transient Random Walk in a Random Environment with cookies of maximal strength
Bauernschubert, Elisabeth
2011-01-01
We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.
Random walk of second class particles in product shock measures
Balazs, Marton; Kovacs, Peter; Rakos, Attila
2009-01-01
We consider shock measures in a class of conserving stochastic particle systems on Z. These shock measures have a product structure with a step-like density profile and include a second class particle at the shock position. We show for the asymmetric simple exclusion process, for the exponential bricklayers' process, and for a generalized zero range process, that under certain conditions these shocks, and therefore the second class particles, perform a simple random walk. Some previous results, including random walks of product shock measures and stationary shock measures seen from a second class particle, are direct consequences of our more general theorem. Multiple shocks can also be handled easily in this framework. Similar shock structure is also found in a nonconserving model, the branching coalescing random walk, where the role of the second class particle is played by the rightmost (or leftmost) particle.
A generalized model via random walks for information filtering
Ren, Zhuo-Ming, E-mail: zhuomingren@gmail.com [Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700, Fribourg (Switzerland); Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, ChongQing, 400714 (China); Kong, Yixiu [Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700, Fribourg (Switzerland); Shang, Ming-Sheng, E-mail: msshang@cigit.ac.cn [Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, ChongQing, 400714 (China); Zhang, Yi-Cheng [Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700, Fribourg (Switzerland)
2016-08-06
There could exist a simple general mechanism lurking beneath collaborative filtering and interdisciplinary physics approaches which have been successfully applied to online E-commerce platforms. Motivated by this idea, we propose a generalized model employing the dynamics of the random walk in the bipartite networks. Taking into account the degree information, the proposed generalized model could deduce the collaborative filtering, interdisciplinary physics approaches and even the enormous expansion of them. Furthermore, we analyze the generalized model with single and hybrid of degree information on the process of random walk in bipartite networks, and propose a possible strategy by using the hybrid degree information for different popular objects to toward promising precision of the recommendation. - Highlights: • We propose a generalized recommendation model employing the random walk dynamics. • The proposed model with single and hybrid of degree information is analyzed. • A strategy with the hybrid degree information improves precision of recommendation.
Optical variability of quasars: a damped random walk
Ivezic, Zeljko
2013-01-01
A damped random walk is a stochastic process, defined by an exponential covariance matrix that behaves as a random walk for short time scales and asymptotically achieves a finite variability amplitude at long time scales. Over the last few years, it has been demonstrated, mostly but not exclusively using SDSS data, that a damped random walk model provides a satisfactory statistical description of observed quasar variability in the optical wavelength range, for rest-frame timescales from 5 days to 2000 days. The best-fit characteristic timescale and asymptotic variability amplitude scale with the luminosity, black hole mass, and rest wavelength, and appear independent of redshift. In addition to providing insights into the physics of quasar variability, the best-fit model parameters can be used to efficiently separate quasars from stars in imaging surveys with adequate long-term multi-epoch data, such as expected from LSST.
Application of continuous-time random walk to statistical arbitrage
Sergey Osmekhin
2015-01-01
Full Text Available An analytical statistical arbitrage strategy is proposed, where the distribution of the spread is modelled as a continuous-time random walk. Optimal boundaries, computed as a function of the mean and variance of the firstpassage time ofthe spread,maximises an objective function. The predictability of the trading strategy is analysed and contrasted for two forms of continuous-time random walk processes. We found that the waiting-time distribution has a significant impact on the prediction of the expected profit for intraday trading
Quenched Point-to-Point Free Energy for Random Walks in Random Potentials
Rassoul-Agha, Firas
2012-01-01
We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of an entropy. We establish regularity properties of the point-to-point free energy, as a function of the potential and as a function on the convex hull of the admissible steps of the walk, and link it to the infinite volume free energy and quenched large deviations of the endpoint of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.
Mindful Walking in Psychologically Distressed Individuals: A Randomized Controlled Trial
M. Teut
2013-01-01
Full Text Available Background. The aim of this randomized, controlled study was to investigate the effectiveness of a mindful walking program in patients with high levels of perceived psychological distress. Methods. Participants aged between 18 and 65 years with moderate to high levels of perceived psychological distress were randomized to 8 sessions of mindful walking in 4 weeks (each 40 minutes walking, 10 minutes mindful walking, 10 minutes discussion or to no study intervention (waiting group. Primary outcome parameter was the difference to baseline on Cohen’s Perceived Stress Scale (CPSS after 4 weeks between intervention and control. Results. Seventy-four participants were randomized in the study; 36 (32 female, 52.3 ± 8.6 years were allocated to the intervention and 38 (35 female, 49.5 ± 8.8 years to the control group. Adjusted CPSS differences after 4 weeks were −8.8 [95% CI: −10.8; −6.8] (mean 24.2 [22.2; 26.2] in the intervention group and −1.0 [−2.9; 0.9] (mean 32.0 [30.1; 33.9] in the control group, resulting in a highly significant group difference (. Conclusion. Patients participating in a mindful walking program showed reduced psychological stress symptoms and improved quality of life compared to no study intervention. Further studies should include an active treatment group and a long-term follow-up.
Navigation by anomalous random walks on complex networks
Weng, Tongfeng; Zhang, Jie; Khajehnejad, Moein; Small, Michael; Zheng, Rui; Hui, Pan
2016-11-01
Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.
Navigation by anomalous random walks on complex networks
Weng, Tongfeng; Khajehnejad, Moein; Small, Michael; Zheng, Rui; Hui, Pan
2016-01-01
Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Levy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Levy walks and the underlying network structure. Interestingly, applying our framework to the famous PageRank search, we can explain why its damping factor empirically chosen to be around 0.85. The framework for analyzing anomalous random walks on complex networks offers a new us...
Riemann Hypothesis and Random Walks: the Zeta case
LeClair, André
2016-01-01
In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\\Re (s) > 1/2$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We use our results to argue that $ S_\\delta (t) \\equiv \\lim_{\\delta \\to 0^+} \\dfrac{1}{\\pi} \\arg \\zeta (\\tfrac{1}{2}+ \\delta + i t ) = O(1)$, and that it is nearly always on the principal branch. We conjecture that a 1-point correlation function of the Riemann zeros has a normal distribution. This leads to the construction of a probabilistic model for the zeros. Based on these results we describe a new algorithm for computing very high Riemann zeros as a kind of stochastic process, and we calculate the $10^{100}$-th zero to over 1...
Quenched Free Energy and Large Deviations for Random Walks in Random Potentials
Rassoul-Agha, Firas; Yilmaz, Atilla
2011-01-01
We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process-level quenched large deviation principle for the empirical measure. As a corollary we obtain LDPs for types of random walk in random environment not covered by earlier results.
A local limit theorem for random walks in random scenery and on randomly oriented lattices
Castell, Fabienne; Pène, Françoise; Schapira, Bruno
2010-01-01
Random walks in random scenery are processes defined by $Z_n:=\\sum_{k=1}^n\\xi_{X_1+...+X_k}$, where $(X_k,k\\ge 1)$ and $(\\xi_y,y\\in\\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\\alpha\\in (0,2]$ and $\\beta\\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\\alpha\
On a zero-drift nearest-neighbour random walk
Cohen, J.W.
1996-01-01
The present study concerns the analysis of the hitting point identity for a nearest-neighbour random walk of which the one-step transition to the $NE$, $SE$, $SW$ and $NW$ are the only transitions with nonzero probabilities. The one-step transition vector has a symmetrical probability distribution w
Adaptive importance sampling of random walks on continuous state spaces
Baggerly, K.; Cox, D.; Picard, R.
1998-11-01
The authors consider adaptive importance sampling for a random walk with scoring in a general state space. Conditions under which exponential convergence occurs to the zero-variance solution are reviewed. These results generalize previous work for finite, discrete state spaces in Kollman (1993) and in Kollman, Baggerly, Cox, and Picard (1996). This paper is intended for nonstatisticians and includes considerable explanatory material.
States recognition in random walk Markov chain via binary Entropy
Morteza Khodabin
2013-03-01
Full Text Available In this paper, a new method for specification of recurrence or transient of states in one and two dimensional simple random walk based on upper and lower bounds of {it r}-combinations from a set of m elements $(C^{m}_{r}$ via binary entropy is introduced.
Simulating intrafraction prostate motion with a random walk model
Tobias Pommer, PhD
2017-07-01
Conclusions: Random walk modeling is feasible and recreated the characteristics of the observed prostate motion. Introducing artificial transient motion did not improve the overall agreement, although the first 30 seconds of the traces were better reproduced. The model provides a simple estimate of prostate motion during delivery of radiation therapy.
Renewal theory for perturbed random walks and similar processes
Iksanov, Alexander
2016-01-01
This book offers a detailed review of perturbed random walks, perpetuities, and random processes with immigration. Being of major importance in modern probability theory, both theoretical and applied, these objects have been used to model various phenomena in the natural sciences as well as in insurance and finance. The book also presents the many significant results and efficient techniques and methods that have been worked out in the last decade. The first chapter is devoted to perturbed random walks and discusses their asymptotic behavior and various functionals pertaining to them, including supremum and first-passage time. The second chapter examines perpetuities, presenting results on continuity of their distributions and the existence of moments, as well as weak convergence of divergent perpetuities. Focusing on random processes with immigration, the third chapter investigates the existence of moments, describes long-time behavior and discusses limit theorems, both with and without scaling. Chapters fou...
Measuring the fractal dimension of an optical random walk
Savo, Romolo; Svensson, Tomas; Vynck, Kevin; Wiersma, Diederik S
2013-01-01
Random walks often grasp the essence of transport processes in complex systems, representing a model for a large variety of phenomena, from human travel, to molecular kinetics, to the propagation of light and sound in disordered media. Transport is generally driven by the topology of the system, which can range from a simply random distribution of scattering elements, to very rich self-similar structures like random fractals. In this context the fractal dimension of the random walk trajectory, $d_\\mathrm{w}$, crucially determines the nature of the resulting transport process and provides information on the way the spatial evolution scales with time. In living cells and turbulent flow it has been possible to study anomalous dynamics showing $d_\\mathrm{w}\
Law of large numbers for non-elliptic random walks in dynamic random environments
Hollander, Frank den; Sidoravicius, Vladas
2011-01-01
We prove a law of large numbers for a class of $\\Z^d$-valued random walks in dynamic random environments, including \\emph{non-elliptic} examples. We assume that the random environment has a mixing property called \\emph{conditional cone-mixing} and that the random walk tends to stay inside space-time cones. The proof is based on a generalization of the regeneration scheme developed by Comets and Zeitouni for static random environments, which was adapted by Avena, den Hollander and Redig to dynamic random environments. We exhibit some one-dimensional examples to which our result applies. In some cases, the sign of the speed can be determined.
Randomized approximate nearest neighbors algorithm.
Jones, Peter Wilcox; Osipov, Andrei; Rokhlin, Vladimir
2011-09-20
We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {x(j)} in R(d), the algorithm attempts to find k nearest neighbors for each of x(j), where k is a user-specified integer parameter. The algorithm is iterative, and its running time requirements are proportional to T·N·(d·(log d) + k·(d + log k)·(log N)) + N·k(2)·(d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d + k). A by-product of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {x(j)} for an arbitrary point x ∈ R(d). The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d + log k)), and the memory requirements for the requisite data structure are of the order N·(d + k) + T·(d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {x(j)} and illustrate its performance via several numerical examples.
Vibration driven random walk in a Chladni experiment
Grabec, Igor
2017-01-01
Drifting of sand particles bouncing on a vibrating membrane of a Chladni experiment is characterized statistically. Records of trajectories reveal that bounces are circularly distributed and random. The mean length of their horizontal displacement is approximately proportional to the vibration amplitude above the critical level and amounts about one fourth of the corresponding bounce height. For the description of horizontal drifting of particles a model of vibration driven random walk is proposed that yields a good agreement between experimental and numerically simulated data.
Avena, L
2012-01-01
We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamic setup too. Such anomalous fluctuations emerge in a new phase diagram. Further we discuss possible consequences for general static and dynamic random environments.
Random Walks and Diffusions on Graphs and Databases An Introduction
Blanchard, Philippe
2011-01-01
Most networks and databases that humans have to deal with contain large, albeit finite number of units. Their structure, for maintaining functional consistency of the components, is essentially not random and calls for a precise quantitative description of relations between nodes (or data units) and all network components. This book is an introduction, for both graduate students and newcomers to the field, to the theory of graphs and random walks on such graphs. The methods based on random walks and diffusions for exploring the structure of finite connected graphs and databases are reviewed (Markov chain analysis). This provides the necessary basis for consistently discussing a number of applications such diverse as electric resistance networks, estimation of land prices, urban planning, linguistic databases, music, and gene expression regulatory networks.
Periodic Walks on Large Regular Graphs and Random Matrix Theory
Oren, Idan
2011-01-01
We study the distribution of the number of (non-backtracking) periodic walks on large regular graphs. We propose a formula for the ratio between the variance of the number of $t$-periodic walks and its mean, when the cardinality of the vertex set $V$ and the period $t$ approach $\\infty$ with $t/V\\rightarrow \\tau$ for any $\\tau$. This formula is based on the conjecture that the spectral statistics of the adjacency eigenvalues is given by Random Matrix Theory (RMT). We provide numerical and theoretical evidence for the validity of this conjecture. The key tool used in this study is a trace formula which expresses the spectral density of $d$-regular graphs, in terms of periodic walks.
On a zero-one law for the norm process of transient random walk
Matsumoto, Ayako
2009-01-01
A zero-one law of Engelbert--Schmidt type is proven for the norm process of a transient random walk. An invariance principle for random walk local times and a limit version of Jeulin's lemma play key roles.
Integrated photonic 3D waveguide arrays for quantum random walks on a circle
Linjordet, Trond
2010-01-01
Quantum random walks (QRWs) can be used to perform both quantum simulations and quantum algorithms. In order to exploit this potential, quantum walks on different types of graphs must be physically implemented. To this end this we design, model and experimentally fabricate, using the femtosecond laser direct-write technique, a 3D tubular waveguide array within glass to implement a photonic quantum walk on a circle. The boundary conditions of a QRW on a circle naturally suggests a 3D waveguide implementation - allowing much simpler device design than what could be achieved using a 2D waveguide architecture. We show that, in some cases, three-dimensional photonic circuits can be more suited to the simulation of complex quantum phenomena.
An effective Hamiltonian approach to quantum random walk
Sarkar, Debajyoti; Paul, Niladri; Bhattacharya, Kaushik; Ghosh, Tarun Kanti
2017-03-01
In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, Sci. Rep. 3, 2829 (18)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.
An effective Hamiltonian approach to quantum random walk
DEBAJYOTI SARKAR; NILADRI PAUL; KAUSHIK BHATTACHARYA; TARUN KANTI GHOSH
2017-03-01
In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltoniansare generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, $\\it{Sci. Rep}$. 3, 2829 (2013)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.
a Random Walk in Theoretical Physics
Roberts, Bruce Wharton
1995-01-01
This thesis covers four diverse topics, representing a cross section of theoretical physics (and mathematical biology). The initial two subjects deal with nonequilibrium dynamics of spatially extended systems. These two topics are spatiotemporal chaos in the complex Ginzburg-Landau equation and a mathematical model for biological evolution and mass extinctions. The study of the complex Ginzburg -Landau equation provides results on the topological defects that exist in this system. The defect-defect correlation function is calculated numerically, and the result is discussed in the context of generic scale invariance and in terms of analytic bounds. The biological evolution work provides a simple, coarse-grained description of the evolutionary process and its interaction with the physical environment. It provides testable predictions about the distribution of extinction sizes and species lifetimes found in the fossil record. The third topic covered in this thesis is real -space renormalization of the random field Ising model. The results provide evidence about the nature of the phases of the system through calculation of renormalization flows in parameter space. The calculation also gives critical exponents for the transition between phases, as well as the heights of barriers between metastable states. The final topic deals with the fractional quantum Hall effect in double quantum wells. Analytic work is done which provides a framework for calculating collective excitations within the single-mode approximation. These excitations are then calculated using Monte Carlo techniques and numerical integration. These calculations yield information about density fluctuations and excitonic modes, the latter of which can be related to tunneling spectral moments.
Two-Dimensional Random Interlacements and Late Points for Random Walks
Comets, Francis; Popov, Serguei; Vachkovskaia, Marina
2016-04-01
We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for a random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.
Continuous Time Random Walks for the Evolution of Lagrangian Velocities
Dentz, Marco; Comolli, Alessandro; Borgne, Tanguy Le; Lester, Daniel R
2016-01-01
We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of...
On the Emergence of Shortest Paths by Reinforced Random Walks
Figueiredo, Daniel R
2016-01-01
The co-evolution between network structure and functional performance is a fundamental and challenging problem whose complexity emerges from the intrinsic interdependent nature of structure and function. Within this context, we investigate the interplay between the efficiency of network navigation (i.e., path lengths) and network structure (i.e., edge weights). We propose a simple and tractable model based on iterative biased random walks where edge weights increase over time as function of the traversed path length. Under mild assumptions, we prove that biased random walks will eventually only traverse shortest paths in their journey towards the destination. We further characterize the transient regime proving that the probability to traverse non-shortest paths decays according to a power-law. We also highlight various properties in this dynamic, such as the trade-off between exploration and convergence, and preservation of initial network plasticity. We believe the proposed model and results can be of inter...
Biasing the random walk of a molecular motor
Astumian, R Dean [Department of Physics, University of Maine, Orono, ME 04469-5709 (United States)
2005-11-30
Biomolecular motors are often described in mechanical terms, with analogy to cars, turbines, judo throws, levers, etc. It is important to remember however that because of their small size, and because of the aqueous environment in which molecular motors move, viscous drag and thermal noise dominate the inertial forces that drive macroscopic machines. The sequence of motions-conformational changes-by which a motor protein moves can best be described as a random walk, with transitions from one state to another occurring by thermal activation over energy barriers. In this paper I will address the question of how this random walk is biased by a non-equilibrium chemical reaction (ATP hydrolysis) so that the motor molecule moves preferentially (with almost unit certainty) in one direction, even when an external force is applied to drive it in the opposite direction. I will also discuss how these 'soft matter' motors can achieve thermodynamic efficiencies of nearly 100%.
Strong approximation of continuous local martingales by simple random walks
Szekely, Balazs
2010-01-01
The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or, equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone.
A generalized model via random walks for information filtering
Ren, Zhuo-Ming; Kong, Yixiu; Shang, Ming-Sheng; Zhang, Yi-Cheng
2016-08-01
There could exist a simple general mechanism lurking beneath collaborative filtering and interdisciplinary physics approaches which have been successfully applied to online E-commerce platforms. Motivated by this idea, we propose a generalized model employing the dynamics of the random walk in the bipartite networks. Taking into account the degree information, the proposed generalized model could deduce the collaborative filtering, interdisciplinary physics approaches and even the enormous expansion of them. Furthermore, we analyze the generalized model with single and hybrid of degree information on the process of random walk in bipartite networks, and propose a possible strategy by using the hybrid degree information for different popular objects to toward promising precision of the recommendation.
Visual Saliency and Attention as Random Walks on Complex Networks
Costa, L F
2006-01-01
The unmatched versatility of vision in mammals is totally dependent on purposive eye movements and selective attention guided by saliencies in the presented images. The current article shows how concepts and tools from the areas of random walks, Markov chains, complex networks and artificial image analysis can be naturally combined in order to provide a unified and biologically plausible model for saliency detection and visual attention, which become indistinguishable in the process. Images are converted into complex networks by considering pixels as nodes while connections are established in terms of fields of influence defined by visual features such as tangent fields induced by luminance contrasts, distance, and size. Random walks are performed on such networks in order to emulate attentional shifts and even eye movements in the case of large shapes, and the frequency of visits to each node is conveniently obtained from the eigenequation defined by the stochastic matrix associated to the respectively drive...
Convex minorants of random walks and L\\'evy processes
Abramson, Josh; Ross, Nathan; Bravo, Gerónimo Uribe
2011-01-01
This article provides an overview of recent work on descriptions and properties of the convex minorant of random walks and L\\'evy processes which summarize and extend the literature on these subjects. The results surveyed include point process descriptions of the convex minorant of random walks and L\\'evy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.
Statistical Modeling of Robotic Random Walks on Different Terrain
Naylor, Austin; Kinnaman, Laura
Issues of public safety, especially with crowd dynamics and pedestrian movement, have been modeled by physicists using methods from statistical mechanics over the last few years. Complex decision making of humans moving on different terrains can be modeled using random walks (RW) and correlated random walks (CRW). The effect of different terrains, such as a constant increasing slope, on RW and CRW was explored. LEGO robots were programmed to make RW and CRW with uniform step sizes. Level ground tests demonstrated that the robots had the expected step size distribution and correlation angles (for CRW). The mean square displacement was calculated for each RW and CRW on different terrains and matched expected trends. The step size distribution was determined to change based on the terrain; theoretical predictions for the step size distribution were made for various simple terrains. It's Dr. Laura Kinnaman, not sure where to put the Prefix.
Bose-Einstein Correlations from Random Walk Models
Tomasik, Boris; Pisút, J; Tomasik, Boris; Heinz, Ulrich; Pisut, Jan
1998-01-01
We argue that the recently suggested ``random walk models'' for the extrapolation of hadronic transverse mass spectra from pp or pA to AB collisions fail to describe existing data on Bose-Einstein correlations. In particular they are unable to reproduce the measured magnitude and K_\\perp-dependence of R_s in Pb+Pb collisions and the increase of R_l with increasing size of the collision system.
Tightness for Maxima of Generalized Branching Random Walks
Fang, Ming
2010-01-01
We study generalized branching random walks, which allow time dependence and local dependence between siblings. Under appropriate tail assumptions, we prove the tightness of $F_n(\\cdot-Med(F_n))$, where $F_n(\\cdot)$ is the maxima distribution at time $n$ and $Med(F_n)$ is the median of $F_n(\\cdot)$. The main component in the argument is a proof of exponential decay of the right tail $1-F_n(\\cdot-Med(F_n))$.
On the pertinence to Physics of random walks induced by random dynamical systems: a survey
Petritis, Dimitri
2016-08-01
Let be an abstract space and a denumerable (finite or infinite) alphabet. Suppose that is a family of functions such that for all we have and a family of transformations . The pair ((Sa)a , (pa)a ) is termed an iterated function system with place dependent probabilities. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given ; we pick then randomly, with probability pa (x), the transformation Sa and evolve to Sa (x). We are interested in the behaviour of the system when the iteration continues indefinitely. Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc. In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published.
Ant-inspired density estimation via random walks.
Musco, Cameron; Su, Hsin-Hao; Lynch, Nancy A
2017-09-19
Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density estimation for robot swarms, and random walk-based sampling for sensor networks.
Random walk theory and exchange rate dynamics in transition economies
Gradojević Nikola
2010-01-01
Full Text Available This paper investigates the validity of the random walk theory in the Euro-Serbian dinar exchange rate market. We apply Andrew Lo and Archie MacKinlay's (1988 conventional variance ratio test and Jonathan Wright's (2000 non-parametric ranks and signs based variance ratio tests to the daily Euro/Serbian dinar exchange rate returns using the data from January 2005 - December 2008. Both types of variance ratio tests overwhelmingly reject the random walk hypothesis over the data span. To assess the robustness of our findings, we examine the forecasting performance of a non-linear, nonparametric model in the spirit of Francis Diebold and James Nason (1990 and find that it is able to significantly improve upon the random walk model, thus confirming the existence of foreign exchange market imperfections in a small transition economy such as Serbia. In the last part of the paper, we conduct a comparative study on how our results relate to those of other transition economies in the region.
The random walk of a low-Reynolds-number swimmer
Rafaï, Salima; Garcia, Michaël; Berti, Stefano; Peyla, Philippe
2010-11-01
Swimming at a micrometer scale demands particular strategies. Indeed when inertia is negligible as compared to viscous forces (i.e. Reynolds number Re is lower than unity), hydrodynamics equations are reversible in time. To achieve propulsion a low Reynolds number, swimmers must then deform in a way that is not invariant under time reversal. Here we investigate the dispersal properties of self propelled organisms by means of microscopy and cell tracking. Our system of interest is the microalga Chlamydomonas Reinhardtii, a motile single celled green alga about 10 micrometers in diameter that swims with two flagellae. In the case of dilute suspensions, we show that tracked trajectories are well modelled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlations are lost and a standard random walk caracterizes the trajectories. Moreover, high speed imaging enables us to show how speed fluctuations at very short times affect the statistical description of the dynamics. Finally we show how drag forces modify the characteristics of this particular random walk.
Random walk of microswimmers: puller and pusher cases
Rafai, Salima; Peyla, Philippe; Dyfcom Team
2014-11-01
Swimming at a micrometer scale demands particular strategies. Indeed when inertia is negligible as compared to viscous forces (i.e. Reynolds number Re is lower than unity), hydrodynamics equations are reversible in time. To achieve propulsion a low Reynolds number, swimmers must then deform in a way that is not invariant under time reversal. Here we investigate the dispersal properties of self propelled organisms by means of microscopy and cell tracking. Our systems of interest are, on the one hand, the microalga Chlamydomonas Reinhardtii, a puller-type swimmer and on the other hand, Lingulodinium polyedrum, a pusher. Both are quasi-spherical single celled alga. In the case of dilute suspensions, we show that tracked trajectories are well modelled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlations are lost and a standard random walk characterizes the trajectories. Finally we show how drag forces modify the characteristics of this particular random walk.
Random walks on coset spaces with applications to Furstenberg entropy
Bowen, Lewis
2010-01-01
We study the Poisson boundary of a random walk on the coset space of a random subgroup of a locally compact group whose law is conjugation-invariant. Then we construct a path of ergodic stationary actions of the free group on which the Furstenberg entropy varies continuously, thereby solving the Furstenberg entropy realization problem for free groups. This result is motivated by the general problem of understanding the structure of stationary actions and more specifically by works of Nevo and Zimmer which proved the Furstenberg entropies of stationary actions of a higher rank semisimple Lie group satisfying a certain mixing condition are restricted to a finite set.
Subdiffusivity of a Random Walk Among a Poisson System of Moving Traps on {Z}
Athreya, Siva; Drewitz, Alexander; Sun, Rongfeng
2017-03-01
We consider a random walk among a Poisson system of moving traps on {Z}. In earlier work (Drewitz et al. Springer Proc. Math. 11, 119-158 2012), the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.
An invariance principle for random walk bridges conditioned to stay positive
Caravenna, Francesco
2012-01-01
We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and Wachtel [38] and Doney [21]. We review and extend these relations to the absolutely continuous setting.
A self-similar process arising from a random walk with random environment in random scenery
Franke, Brice; 10.3150/09-BEJ234
2011-01-01
In this article, we merge celebrated results of Kesten and Spitzer [Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25] and Kawazu and Kesten [J. Stat. Phys. 37 (1984) 561-575]. A random walk performs a motion in an i.i.d. environment and observes an i.i.d. scenery along its path. We assume that the scenery is in the domain of attraction of a stable distribution and prove that the resulting observations satisfy a limit theorem. The resulting limit process is a self-similar stochastic process with non-trivial dependencies.
Some Minorants and Majorants of Random Walks and Levy Processes
Abramson, Joshua Simon
This thesis consists of four chapters, all relating to some sort of minorant or majorant of random walks or Levy processes. In Chapter 1 we provide an overview of recent work on descriptions and properties of the convex minorant of random walks and Levy processes as detailed in Chapter 2, [72] and [73]. This work rejuvenated the field of minorants, and led to the work in all the subsequent chapters. The results surveyed include point process descriptions of the convex minorant of random walks and Levy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander. This chapter is based on [3], which was co-written with Jim Pitman, Nathan Ross and Geronimo Uribe Bravo. Chapter 1 serves as a long introduction to Chapter 2, in which we offer a unified approach to the theory of concave majorants of random walks. The reasons for the switch from convex minorants to concave majorants are discussed in Section 1.1, but the results are all equivalent. This unified theory is arrived at by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant - the path transformation is different from the one discussed in Chapter 1, but this is necessary to deal with a more general case than the standard one as done in Section 2.6. The path transformation of Chapter 1, which is discussed in detail in Section 2.8, is more relevant to the limiting results for Levy processes that are of interest in Chapter 1. Our results lead to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave
Influence of weight heterogeneity on random walks in scale-free networks
Li, Ling; Guan, Jihong; Qi, Zhaohui
2016-07-01
Many systems are best described by weighted networks, in which the weights of the edges are heterogeneous. In this paper, we focus on random walks in weighted network, investigating the impacts of weight heterogeneity on the behavior of random walks. We study random walks in a family of weighted scale-free tree-like networks with power-law weight distribution. We concentrate on three cases of random walk problems: with a trap located at a hub node, a leaf adjacent to a hub node, and a farthest leaf node from a hub. For all these cases, we calculate analytically the global mean first passage time (GMFPT) measuring the efficiency of random walk, as well as the leading scaling of GMFPT. We find a significant decrease in the dominating scaling of GMFPT compared with the corresponding binary networks in all three random walk problems, which implies that weight heterogeneity has a significant influence on random walks in scale-free networks.
Process-level quenched large deviations for random walk in random environment
Rassoul-Agha, Firas
2009-01-01
We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.
Almost Sure Invariance Principle for Continuous-Space Random Walk in Dynamic Random Environment
Joseph, Mathew
2010-01-01
We consider a random walk on $\\R^d$ in a polynomially mixing random environment that is refreshed at each time step. We use a martingale approach to give a necessary and sufficient condition for the almost-sure functional central limit theorem to hold.
Critical Random Walk in Random Environment on Trees of Exponential Growth
Pemantle, Robin
2004-01-01
This paper studies the behavior of RWRE on trees in the critical case left open in previous work. For trees of exponential growth, a random perturbation of the transition probabilities can change a transient random walk into a recurrent one. This is the opposite of what occurs on trees of sub-exponential growth.
Some Probability Properties of Random Walk in Time-Random Environment
Zhang Xiao-min; Li Bo
2004-01-01
A general formulation of the stochastic model for random walk in time-random environment and an equivalent definition is established in this paper. Moreover, some basic probability relations similar to the classical case which are very useful in the corresponding research of fractal properties are given. At the end, a typical example is provided to show the recurrence and transience.
Strong approximation for the general Kesten-Spitzer random walk in independent random scenery
无
2001-01-01
This paper is to prove that, if a one-dimensional random wa lkcan be approximated by a Brownian motion, then the related random walk in a g eneral independent scenery can be approximated by a Brownian motion in Brownian scenery.
Novel pseudo-random number generator based on quantum random walks
Yang, Yu-Guang; Zhao, Qian-Qian
2016-02-01
In this paper, we investigate the potential application of quantum computation for constructing pseudo-random number generators (PRNGs) and further construct a novel PRNG based on quantum random walks (QRWs), a famous quantum computation model. The PRNG merely relies on the equations used in the QRWs, and thus the generation algorithm is simple and the computation speed is fast. The proposed PRNG is subjected to statistical tests such as NIST and successfully passed the test. Compared with the representative PRNG based on quantum chaotic maps (QCM), the present QRWs-based PRNG has some advantages such as better statistical complexity and recurrence. For example, the normalized Shannon entropy and the statistical complexity of the QRWs-based PRNG are 0.999699456771172 and 1.799961178212329e-04 respectively given the number of 8 bits-words, say, 16Mbits. By contrast, the corresponding values of the QCM-based PRNG are 0.999448131481064 and 3.701210794388818e-04 respectively. Thus the statistical complexity and the normalized entropy of the QRWs-based PRNG are closer to 0 and 1 respectively than those of the QCM-based PRNG when the number of words of the analyzed sequence increases. It provides a new clue to construct PRNGs and also extends the applications of quantum computation.
Law of large numbers for a class of random walks in dynamic random environments
Avena, L; Redig, F
2009-01-01
In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.
Percon8 Algorithm for Random Number Generation
Dr. Mrs. Saylee Gharge
2014-05-01
Full Text Available In today’s technology savvy world, computer security holds a prime importance. Most computer security algorithms require some amount of random data for generating public and private keys, session keys or for other purposes. Random numbers are those numbers that occur in a sequence such that the future value of the sequence cannot be predicted based on present or past values. Random numbers find application in statistical analysis and probability theory. The many applications of randomness have led to the development of random number generating algorithms. These algorithms generate a sequence of random numbers either computationally or physically. In our proposed technique, we have implemented a random number generation algorithm combining two existing random number generation techniques viz. Mid square method and Linear Congruential Generator
Guex, Guillaume
2016-05-01
In recent articles about graphs, different models proposed a formalism to find a type of path between two nodes, the source and the target, at crossroads between the shortest-path and the random-walk path. These models include a freely adjustable parameter, allowing to tune the behavior of the path toward randomized movements or direct routes. This article presents a natural generalization of these models, namely a model with multiple sources and targets. In this context, source nodes can be viewed as locations with a supply of a certain good (e.g. people, money, information) and target nodes as locations with a demand of the same good. An algorithm is constructed to display the flow of goods in the network between sources and targets. With again a freely adjustable parameter, this flow can be tuned to follow routes of minimum cost, thus displaying the flow in the context of the optimal transportation problem or, by contrast, a random flow, known to be similar to the electrical current flow if the random-walk is reversible. Moreover, a source-targetcoupling can be retrieved from this flow, offering an optimal assignment to the transportation problem. This algorithm is described in the first part of this article and then illustrated with case studies.
Characteristic times of biased random walks on complex networks.
Bonaventura, Moreno; Nicosia, Vincenzo; Latora, Vito
2014-01-01
We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k(α), where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely (i) the time the walker needs to come back to the starting node, (ii) the time it takes to visit a given node for the first time, and (iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times differs from the value α(min)=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of α(min) in the range [-1,-0.5], while disassortative networks have α(min) in the range [-0.5,0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value α(min), which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks by means of an appropriate tuning of the motion bias.
Aging Renewal Theory and Application to Random Walks
Johannes H. P. Schulz
2014-02-01
Full Text Available We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_{a},t_{a}+t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.
Discrete random walk models for space-time fractional diffusion
Gorenflo, Rudolf; Mainardi, Francesco; Moretti, Daniele; Pagnini, Gianni; Paradisi, Paolo
2002-11-01
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order {alpha} is part of (0,2] and skewness {theta} (module{theta}{<=}{l_brace}{alpha},2-{alpha}{r_brace}), and the first-order time derivative with a Caputo derivative of order {beta} is part of (0,1]. Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.
A General Random Walk Model of Molecular Motor
WANG Xian-Ju; AI Bao-Quan; LIU Guo-Tao; LIU Liang-Gang
2003-01-01
A general random walk model framework is presented which can be used to statistically describe the internaldynamics and external mechanical movement of molecular motors along filament track. The motion of molecular motorin a periodic potential and a constant force is considered. We show that the molecular motor's movement becomesslower with the potential barrier increasing, but if the forceis increased, the molecular motor's movement becomesfaster. The relation between the effective rate constant and the potential barrier's height, and that between the effectiverate constant and the value of the force are discussed. Our results are consistent with the experiments and relevanttheoretical consideration, and can be used to explain some physiological phenomena.
Non-equilibrium Phase Transitions: Activated Random Walks at Criticality
Cabezas, M.; Rolla, L. T.; Sidoravicius, V.
2014-06-01
In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including , and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.
Nonlocal operators, parabolic-type equations, and ultrametric random walks
Chacón-Cortes, L. F., E-mail: fchaconc@math.cinvestav.edu.mx; Zúñiga-Galindo, W. A., E-mail: wazuniga@math.cinvestav.edu.mx [Centro de Investigacion y de Estudios Avanzados del I.P.N., Departamento de Matematicas, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, Mexico D.F., C.P. 07360 (Mexico)
2013-11-15
In this article, we introduce a new type of nonlocal operators and study the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated to these operators. Some of these equations are the p-adic master equations of certain models of complex systems introduced by Avetisov, V. A. and Bikulov, A. Kh., “On the ultrametricity of the fluctuation dynamicmobility of protein molecules,” Proc. Steklov Inst. Math. 265(1), 75–81 (2009) [Tr. Mat. Inst. Steklova 265, 82–89 (2009) (Izbrannye Voprosy Matematicheskoy Fiziki i p-adicheskogo Analiza) (in Russian)]; Avetisov, V. A., Bikulov, A. Kh., and Zubarev, A. P., “First passage time distribution and the number of returns for ultrametric random walks,” J. Phys. A 42(8), 085003 (2009); Avetisov, V. A., Bikulov, A. Kh., and Osipov, V. A., “p-adic models of ultrametric diffusion in the conformational dynamics of macromolecules,” Proc. Steklov Inst. Math. 245(2), 48–57 (2004) [Tr. Mat. Inst. Steklova 245, 55–64 (2004) (Izbrannye Voprosy Matematicheskoy Fiziki i p-adicheskogo Analiza) (in Russian)]; Avetisov, V. A., Bikulov, A. Kh., and Osipov, V. A., “p-adic description of characteristic relaxation in complex systems,” J. Phys. A 36(15), 4239–4246 (2003); Avetisov, V. A., Bikulov, A. H., Kozyrev, S. V., and Osipov, V. A., “p-adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” J. Phys. A 35(2), 177–189 (2002); Avetisov, V. A., Bikulov, A. Kh., and Kozyrev, S. V., “Description of logarithmic relaxation by a model of a hierarchical random walk,” Dokl. Akad. Nauk 368(2), 164–167 (1999) (in Russian). The fundamental solutions of these parabolic-type equations are transition functions of random walks on the n-dimensional vector space over the field of p-adic numbers. We study some properties of these random walks, including the first passage time.
Coupling limit order books and branching random walks
2014-01-01
We consider a model for a one-sided limit order book proposed by Lakner, Reed and Stoikov (2013). We show that it can be coupled with a branching random walk and use this coupling to answer a nontrivial question about the long-term behavior of the price. The coupling relies on a classical idea of enriching the state space by artificially creating a filiation, in this context between orders of the book, which we believe has the potential of being useful for a broader class of...
A note on the recurrence of edge reinforced random walks
Tournier, Laurent
2009-01-01
We give a short proof of Theorem 2.1 from [MR07], stating that the linearly edge reinforced random walk (ERRW) on a locally finite graph is recurrent if and only if it returns to its starting point almost surely. This result was proved in [MR07] by means of the much stronger property that the law of the ERRW is a mixture of Markov chains. Our proof only uses this latter property on finite graphs, in which case it is a consequence of De Finetti's theorem on exchangeability.
Anomalous diffusion in correlated continuous time random walks
Tejedor, Vincent; Metzler, Ralf, E-mail: metz@ph.tum.d [Physics Department T30 g, Technical University of Munich, 85747 Garching (Germany)
2010-02-26
We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous diffusion with the mean squared displacement (r{sup 2}(t)) {approx_equal} t{sup 2/3}. Long-ranged correlations of the waiting times with a power-law exponent alpha (0 < alpha <= 2) give rise to subdiffusion of the form (r{sup 2}(t)) {approx_equal} t{sup {alpha}/(1+{alpha})}. In contrast, correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations. (fast track communication)
Statistical shape model with random walks for inner ear segmentation
Pujadas, Esmeralda Ruiz; Kjer, Hans Martin; Piella, Gemma
2016-01-01
Cochlear implants can restore hearing to completely or partially deaf patients. The intervention planning can be aided by providing a patient-specific model of the inner ear. Such a model has to be built from high resolution images with accurate segmentations. Thus, a precise segmentation...... is required. We propose a new framework for segmentation of micro-CT cochlear images using random walks combined with a statistical shape model (SSM). The SSM allows us to constrain the less contrasted areas and ensures valid inner ear shape outputs. Additionally, a topology preservation method is proposed...
Turbulent pair dispersion as a continuous-time random walk
Thalabard, Simon; Bec, Jeremie
2014-01-01
The phenomenology of turbulent relative dispersion is revisited. A heuristic scenario is proposed, in which pairs of tracers undergo a succession of independent ballistic separations during time intervals whose lengths fluctuate. This approach suggests that the logarithm of the distance between tracers self-averages and performs a continuous-time random walk. This leads to specific predictions for the probability distribution of separations, that differ from those obtained using scale-dependent eddy-diffusivity models (e.g. in the framework of Richardson's approach). Such predictions are tested against high-resolution simulations and shed new lights on the explosive separation between tracers.
Random Walks and Boundaries of CAT(0) Cubical complexes
Fernós, Talia; Lécureux, Jean; Mathéus, Fréderic
2016-01-01
We show under weak hypotheses that the pushforward $\\{Z_no\\}$ of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either the Roller boundary or the visual boundary, with the appropriate hypotheses. This study allows us to show that any nonelementary action necessarily contains regular elements, that is, elements that act as rank-1 hyperbolic isometries in each irreducible fa...
Approximated maximum likelihood estimation in multifractal random walks
Løvsletten, Ola
2011-01-01
We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry et al., Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the R computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.
Record statistics of a strongly correlated time series: random walks and Lévy flights
Godrèche, Claude; Majumdar, Satya N.; Schehr, Grégory
2017-08-01
We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful ‘laboratory’ to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statistics of the number of records, as well as of the ages of the records, i.e. the lapses of time between two successive record breaking events. Then we review the results that were obtained for a wide variety of random walk models, including random walks with a linear drift, continuous time random walks, constrained random walks (like the random walk bridge) and the case of multiple independent random walkers. Finally, we discuss further observables related to records, like the record increments, as well as some questions raised by physical applications of record statistics, like the effects of measurement error and noise.
Cochlea segmentation using iterated random walks with shape prior
Ruiz Pujadas, Esmeralda; Kjer, Hans Martin; Vera, Sergio; Ceresa, Mario; González Ballester, Miguel Ángel
2016-03-01
Cochlear implants can restore hearing to deaf or partially deaf patients. In order to plan the intervention, a model from high resolution µCT images is to be built from accurate cochlea segmentations and then, adapted to a patient-specific model. Thus, a precise segmentation is required to build such a model. We propose a new framework for segmentation of µCT cochlear images using random walks where a region term is combined with a distance shape prior weighted by a confidence map to adjust its influence according to the strength of the image contour. Then, the region term can take advantage of the high contrast between the background and foreground and the distance prior guides the segmentation to the exterior of the cochlea as well as to less contrasted regions inside the cochlea. Finally, a refinement is performed preserving the topology using a topological method and an error control map to prevent boundary leakage. We tested the proposed approach with 10 datasets and compared it with the latest techniques with random walks and priors. The experiments suggest that this method gives promising results for cochlea segmentation.
Do MENA stock market returns follow a random walk process?
Salim Lahmiri
2013-01-01
Full Text Available In this research, three variance ratio tests: the standard variance ratio test, the wild bootstrap multiple variance ratio test, and the non-parametric rank scores test are adopted to test the random walk hypothesis (RWH of stock markets in Middle East and North Africa (MENA region using most recent data from January 2010 to September 2012. The empirical results obtained by all three econometric tests show that the RWH is strongly rejected for Kuwait, Tunisia, and Morocco. However, the standard variance ratio test and the wild bootstrap multiple variance ratio test reject the null hypothesis of random walk in Jordan and KSA, while non-parametric rank scores test do not. We may conclude that Jordan and KSA stock market are weak efficient. In sum, the empirical results suggest that return series in Kuwait, Tunisia, and Morocco are predictable. In other words, predictable patterns that can be exploited in these markets still exit. Therefore, investors may make profits in such less efficient markets.
Dynamic decoupling in the presence of 1D random walk
Chakrabarti, Arnab; Chakraborty, Ipsita; Bhattacharyya, Rangeet
2016-05-01
In the recent past, many dynamic decoupling sequences have been proposed for the suppression of decoherence of spins connected to thermal baths of various natures. Dynamic decoupling schemes for suppressing decoherence due to Gaussian diffusion have also been developed. In this work, we study the relative performances of dynamic decoupling schemes in the presence of a non-stationary Gaussian noise such as a 1D random walk. Frequency domain analysis is not suitable to determine the performances of various dynamic decoupling schemes in suppressing decoherence due to such a process. Thus, in this work, we follow a time domain calculation to arrive at the following conclusions: in the presence of such a noise, we show that (i) the traditional Carr-Purcell-Meiboom-Gill (CPMG) sequence outperforms Uhrig’s dynamic decoupling scheme, (ii) CPMG remains the optimal sequence for suppression of decoherence due to random walk in the presence of an external field gradient. Later, the theoretical predictions are experimentally verified by using nuclear magnetic resonance spectroscopy on spin 1/2 particles diffusing in a liquid medium.
Random-walk baryogenesis via primordial black holes
Semiz, İbrahim
2016-01-01
Gravitation violates baryon number $B$: A star has a huge amount of it, while a black hole forming from the star has none. Consider primordial black holes before the hadronic annihiliation in the early universe, encountering and absorbing baryons and antibaryons: Each such absorption changes $B$ of the universe by one unit, up or down. But the absorption events are $uncorrelated$ $and$ $random$, hence they amount to a random walk in $B$-space, leading to the expectation of a net $|B|$ at the end. While the scale of this effect is most uncertain, it must exist. We explore some ramifications, including the change of net $|B|$ with expansion, connection with universe topology, and possible observational signatures.
Quantum walks: a comprehensive review
Venegas-Andraca, Salvador E
2012-01-01
Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important resul...
Random walks in nonuniform environments with local dynamic interactions
Baker, Christopher M.; Hughes, Barry D.; Landman, Kerry A.
2013-10-01
We consider a class of lattice random walk models in which the random walker is initially confined to a finite connected set of allowed sites but has the opportunity to enlarge this set by colliding with its boundaries, each such collision having a given probability of breaking through. The model is motivated by an analogy to cell motility in tissue, where motile cells have the ability to remodel extracellular matrix, but is presented here as a generic model for stochastic erosion. For the one-dimensional case, we report some exact analytic results, some mean-field type analytic approximate results and simulations. We compute exactly the mean and variance of the time taken to enlarge the interval from a single site to a given size. The problem of determining the statistics of the interval length and the walker's position at a given time is more difficult and we report several interesting observations from simulations. Our simulations include the case in which the initial interval length is random and the case in which the initial state of the lattice is a random mixture of allowed and forbidden sites, with the walker placed at random on an allowed site. To illustrate the extension of these ideas to higher-dimensional systems, we consider the erosion of the simple cubic lattice commencing from a single site and report simulations of measures of cluster size and shape and the mean-square displacement of the walker.
Aslam, Muhammad Zaheer
2011-01-01
Mobile Adhoc Network is a kind of wireless ad hoc network where nodes are connected wirelessly and the network is self configuring. MANET may work in a standalone manner or may be a part of another network. In this paper we have compared Random Walk Mobility Model and Random Waypoint Mobility Model over two reactive routing protocols Dynamic Source Routing (DSR) and Adhoc On-Demand Distance Vector Routing (AODV) protocol and one Proactive routing protocol Distance Sequenced Distance Vector Routing (DSDV) Our analysis showed that DSR, AODV & DSDV under Random Walk and Random Way Point Mobility models have similar results for similar inputs however as the pause time increases so does the difference in performance rises. They show that their motion, direction, angle of direction, speed is same under both mobility models. We have made their analysis on packet delivery ratio, throughput and routing overhead. We have tested them with different criteria like different number of nodes, speed and different maximum...
Scaling limit of the recurrent biased random walk on a Galton-Watson tree
Aïdékon, Elie; de Raphélis, Loïc
2015-01-01
We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.
THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS
无
2006-01-01
Suppose {Xn} is a random walk in time-random environment with state space Zd, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α.Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.
Scaling analysis of random walks with persistence lengths: Application to self-avoiding walks
Granzotti, C. R. F.; Martinez, A. S.; da Silva, M. A. A.
2016-05-01
We develop an approach for performing scaling analysis of N -step random walks (RWs). The mean square end-to-end distance, , is written in terms of inner persistence lengths (IPLs), which we define by the ensemble averages of dot products between the walker's position and displacement vectors, at the j th step. For RW models statistically invariant under orthogonal transformations, we analytically introduce a relation between and the persistence length, λN, which is defined as the mean end-to-end vector projection in the first step direction. For self-avoiding walks (SAWs) on 2D and 3D lattices we introduce a series expansion for λN, and by Monte Carlo simulations we find that λ∞ is equal to a constant; the scaling corrections for λN can be second- and higher-order corrections to scaling for . Building SAWs with typically 100 steps, we estimate the exponents ν0 and Δ1 from the IPL behavior as function of j . The obtained results are in excellent agreement with those in the literature. This shows that only an ensemble of paths with the same length is sufficient for determining the scaling behavior of , being that the whole information needed is contained in the inner part of the paths.
When human walking becomes random walking: fractal analysis and modeling of gait rhythm fluctuations
Hausdorff, Jeffrey M.; Ashkenazy, Yosef; Peng, Chang-K.; Ivanov, Plamen Ch.; Stanley, H. Eugene; Goldberger, Ary L.
2001-12-01
We present a random walk, fractal analysis of the stride-to-stride fluctuations in the human gait rhythm. The gait of healthy young adults is scale-free with long-range correlations extending over hundreds of strides. This fractal scaling changes characteristically with maturation in children and older adults and becomes almost completely uncorrelated with certain neurologic diseases. Stochastic modeling of the gait rhythm dynamics, based on transitions between different “neural centers”, reproduces distinctive statistical properties of the gait pattern. By tuning one model parameter, the hopping (transition) range, the model can describe alterations in gait dynamics from childhood to adulthood - including a decrease in the correlation and volatility exponents with maturation.
The average inter-crossing number of equilateral random walks and polygons
Diao, Y.; Dobay, A.; Stasiak, A.
2005-09-01
In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is a=\\frac{3\\ln 2}{8}\\approx 0.2599 . In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance ρ apart and ρ is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of ρ. Our simulation result shows that the model in fact works very well for the entire range of ρ. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.
Large deviations for the local times of a random walk among random conductances
König, Wolfgang; Wolff, Tilman
2011-01-01
We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\\Z^d$ in the spirit of Donsker-Varadhan \\cite{DV75}. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.
Maps of random walks on complex networks reveal community structure.
Rosvall, Martin; Bergstrom, Carl T
2008-01-29
To comprehend the multipartite organization of large-scale biological and social systems, we introduce an information theoretic approach that reveals community structure in weighted and directed networks. We use the probability flow of random walks on a network as a proxy for information flows in the real system and decompose the network into modules by compressing a description of the probability flow. The result is a map that both simplifies and highlights the regularities in the structure and their relationships. We illustrate the method by making a map of scientific communication as captured in the citation patterns of >6,000 journals. We discover a multicentric organization with fields that vary dramatically in size and degree of integration into the network of science. Along the backbone of the network-including physics, chemistry, molecular biology, and medicine-information flows bidirectionally, but the map reveals a directional pattern of citation from the applied fields to the basic sciences.
A Branching Random Walk Seen from the Tip
Brunet, Éric; Derrida, Bernard
2011-05-01
We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.
Composition of many spins, random walks and statistics
Polychronakos, Alexios P
2016-01-01
The multiplicities of the decomposition of the product of an arbitrary number $n$ of spin $s$ states into irreducible $SU(2)$ representations are computed. Two complementary methods are presented, one based on random walks in representation space and another based on the partition function of the system in the presence of a magnetic field. The large-$n$ scaling limit of these multiplicities is derived, including nonperturbative corrections, and related to semiclassical features of the system. A physical application of these results to ferromagnetism is explicitly worked out. Generalizations involving several types of spins, as well as spin distributions, are also presented. The corresponding problem for (anti-)symmetric composition of spins is also considered and shown to obey remarkable duality and bosonization relations and exhibit qualitatively different large-$n$ scaling properties.
Random walks on activity-driven networks with attractiveness
Alessandretti, Laura; Sun, Kaiyuan; Baronchelli, Andrea; Perra, Nicola
2017-05-01
Virtually all real-world networks are dynamical entities. In social networks, the propensity of nodes to engage in social interactions (activity) and their chances to be selected by active nodes (attractiveness) are heterogeneously distributed. Here, we present a time-varying network model where each node and the dynamical formation of ties are characterized by these two features. We study how these properties affect random-walk processes unfolding on the network when the time scales describing the process and the network evolution are comparable. We derive analytical solutions for the stationary state and the mean first-passage time of the process, and we study cases informed by empirical observations of social networks. Our work shows that previously disregarded properties of real social systems, such as heterogeneous distributions of activity and attractiveness as well as the correlations between them, substantially affect the dynamical process unfolding on the network.
A General Random Walk Model of Molecular Motor
WANGXian-Ju; AIBao-Quan; LIUGuo-Tao; LIULiang-Gang
2003-01-01
A general random walk model framework is presented which can be used to statistically describe the internal dynamics and external mechanical movement of molecular motors along filament track. The motion of molecular motor in a periodic potential and a constant force is considered. We show that the molecular motor's movement becomes slower with the potential barrier increasing, but if the force is increased, the molecular motor''s movement becomes faster. The relation between the effective rate constant and the potential battler's height, and that between the effective rate constant and the value of the force are discussed. Our results are consistent with the experiments and relevant theoretical consideration, and can be used to explain some physiological phenomena.
Multifractal analysis and simulation of multifractal random walks
Schmitt, Francois G.; Huang, Yongxiang
2016-04-01
Multifractal time series, characterized by a scale invariance and large fluctuations at all scales, are found in many fields of natural and applied sciences. They are found i.e. in many geophysical fields, such as atmospheric and oceanic turbulence, hydrology, earth sciences. Here we consider a quite general type of multifractal time series, called multifractal random walk, as non stationary stochastic processes with intermittent stationary increments. We first quickly recall how such time series can be analyzed and characterized, using structure functions and arbitrary order Hilbert spectral analysis. We then discuss the simulation approach. The main object is to provide a stochastic process generating time series having the same multiscale properties We review recent works on this topic, and provide stochastic simulations in order to verify the theoretical predictions. In the lognormal framework we provide a h - μ plane expressing the scale invariant properties of these simulations. The theoretical plane is compared to simulation results.
Random walks on Sierpinski gaskets of different dimensions
Weber, Sebastian; Klafter, Joseph; Blumen, Alexander
2010-11-01
We study random walks (RWs) on classical and dual Sierpinski gaskets (SG and DSG), naturally embedded in d -dimensional Euclidian spaces (ESs). For large d the spectral dimension ds approaches 2, the marginal RW dimension. In contrast to RW over two-dimensional ES, RWs over SG and DSG show a very rich behavior. First, the time discrete scale invariance leads to logarithmic-periodic (log-periodic) oscillations in the RW properties monitored, which increase in amplitude with d . Second, the asymptotic approach to the theoretically predicted RW power laws is significantly altered depending on d and on the variant of the fractal (SG or DSG) under study. In addition, we discuss the suitability of standard RW properties to determine ds , a question of great practical relevance.
Correlated continuous time random walk and option pricing
Lv, Longjin; Xiao, Jianbin; Fan, Liangzhong; Ren, Fuyao
2016-04-01
In this paper, we study a correlated continuous time random walk (CCTRW) with averaged waiting time, whose probability density function (PDF) is proved to follow stretched Gaussian distribution. Then, we apply this process into option pricing problem. Supposing the price of the underlying is driven by this CCTRW, we find this model captures the subdiffusive characteristic of financial markets. By using the mean self-financing hedging strategy, we obtain the closed-form pricing formulas for a European option with and without transaction costs, respectively. At last, comparing the obtained model with the classical Black-Scholes model, we find the price obtained in this paper is higher than that obtained from the Black-Scholes model. A empirical analysis is also introduced to confirm the obtained results can fit the real data well.
Lee, Chul-Ho; Eun, Do Young
2012-01-01
Graph sampling via crawling has been actively considered as a generic and important tool for collecting uniform node samples so as to consistently estimate and uncover various characteristics of complex networks. The so-called simple random walk with re-weighting (SRW-rw) and Metropolis-Hastings (MH) algorithm have been popular in the literature for such unbiased graph sampling. However, an unavoidable downside of their core random walks -- slow diffusion over the space, can cause poor estimation accuracy. In this paper, we propose non-backtracking random walk with re-weighting (NBRW-rw) and MH algorithm with delayed acceptance (MHDA) which are theoretically guaranteed to achieve, at almost no additional cost, not only unbiased graph sampling but also higher efficiency (smaller asymptotic variance of the resulting unbiased estimators) than the SRW-rw and the MH algorithm, respectively. In particular, a remarkable feature of the MHDA is its applicability for any non-uniform node sampling like the MH algorithm,...
Limit theorems for one and two-dimensional random walks in random scenery
Castell, Fabienne; Pène, Françoise
2011-01-01
Random walks in random scenery are processes defined by $Z_n:=\\sum_{k=1}^n\\xi_{X_1+...+X_k}$, where $(X_k,k\\ge 1)$ and $(\\xi_y,y\\in{\\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\\mathbb Z}^d$ and $\\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $\\xi_0$ belong to the normal basin of attraction of stable distribution of index $\\alpha\\in(0,2]$ and $\\beta\\in(0,2]$. When $d=1$ and $\\alpha\
Fernández-Del-Olmo, Miguel Angel; Sanchez, Jose Andres; Bello, Olalla; Lopez-Alonso, Virginia; Márquez, Gonzalo; Morenilla, Luis; Castro, Xabier; Giraldez, Manolo; Santos-García, Diego
2014-01-01
Gait disturbances are one of the principal and most incapacitating symptoms of Parkinson's disease (PD). In addition, walking economy is impaired in PD patients and could contribute to excess fatigue in this population. An important number of studies have shown that treadmill training can improve kinematic parameters in PD patients. However, the effects of treadmill and overground walking on the walking economy remain unknown. The goal of this study was to explore the walking economy changes in response to a treadmill and an overground training program, as well as the differences in the walking economy during treadmill and overground walking. Twenty-two mild PD patients were randomly assigned to a treadmill or overground training group. The training program consisted of 5 weeks (3 sessions/week). We evaluated the energy expenditure of overground walking, before and after each of the training programs. The energy expenditure of treadmill walking (before the program) was also evaluated. The treadmill, but not the overground training program, lead to an improvement in the walking economy (the rate of oxygen consumed per distance during overground walking at a preferred speed) in PD patients. In addition, walking on a treadmill required more energy expenditure compared with overground walking at the same speed. This study provides evidence that in mild PD patients, treadmill training is more beneficial compared with that of walking overground, leading to a greater improvement in the walking economy. This finding is of clinical importance for the therapeutic administration of exercise in PD.
Maximal Displacement for Bridges of Random Walks in a Random Environment
Gantert, Nina
2009-01-01
It is well known that the distribution of simple random walks on $\\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\\{S_{2n}=0\\}$ the maximal displacement $\\max_{k\\leq 2n} |S_k|$ converges in distribution when scaled by $\\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\\bf{Z}$. We show that under the quenched law $P_\\omega$ (conditioned on the environment $\\omega$), the maximal displacement of the random walk when conditioned to return to the origin at time $2n$ is no longer necessarily of the order $\\sqrt{n}$. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time $2n$ is of order $n^{\\kappa/(\\kappa+1)}$, where the constant $\\kappa>0$ depends on the law on environment. On the other hand, if the environment is marginally nestli...
Law of large numbers for a transient random walk driven by a symmetric exclusion process
Avena, Luca; Völlering, Florian
2011-01-01
We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We prove that the random walk has an a.s. positive constant global speed by using a regeneration-time argument. This result is part of an ongoing project aiming to analyze the behavior of random walks in slowly mixing dynamic random environments. A brief discussion on this topic is presented.
Stretched Exponential Relaxation in Disordered Complex Systems: Fractal Time Random Walk Model
Ekrem Aydmer
2007-01-01
We have analytically derived the relaxation function for one-dimensional disordered complex systems in terms of autocorrelation function of fractal time random walk by using operator formalism. We have shown that the relaxation function has stretched exponential, i.e. the Kohlrausch-Williams-Watts character for a fractal time random walk process.
Random walk with a boundary line as a free massive boson with a defect line
Valleriani, A
1995-01-01
We show that the problem of Random Walk with boundary attractive potential may be mapped onto the free massive bosonic Quantum Field Theory with a line of defect. This mapping permits to recover the statistical properties of the Random Walks by using boundary S--matrix and Form Factor techniques.
An example of the difference between quantum and classical random walks
Childs, A M; Gutmann, S; Childs, Andrew M.; Farhi, Edward; Gutmann, Sam
2002-01-01
In this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analogue. In this graph, propagation between a particular pair of nodes is exponentially faster in the quantum case.
Singularly perturbed telegraph equations with applications in the random walk theory
Jacek Banasiak
1998-01-01
Full Text Available In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.
Ranking competitors using degree-neutralized random walks.
Shin, Seungkyu; Ahnert, Sebastian E; Park, Juyong
2014-01-01
Competition is ubiquitous in many complex biological, social, and technological systems, playing an integral role in the evolutionary dynamics of the systems. It is often useful to determine the dominance hierarchy or the rankings of the components of the system that compete for survival and success based on the outcomes of the competitions between them. Here we propose a ranking method based on the random walk on the network representing the competitors as nodes and competitions as directed edges with asymmetric weights. We use the edge weights and node degrees to define the gradient on each edge that guides the random walker towards the weaker (or the stronger) node, which enables us to interpret the steady-state occupancy as the measure of the node's weakness (or strength) that is free of unwarranted degree-induced bias. We apply our method to two real-world competition networks and explore the issues of ranking stabilization and prediction accuracy, finding that our method outperforms other methods including the baseline win-loss differential method in sparse networks.
Characteristic times of biased random walks on complex networks
Bonaventura, Moreno; Latora, Vito
2013-01-01
We consider degree-biased random walkers whose probability to move from a node to one of its neighbours of degree k is proportional to k^{\\alpha}, where \\alpha is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to pass from a given node, and iii) the time it takes to visit all the nodes of the network. We consider a large database of real-world networks and we show that the value of \\alpha which minimizes the three characteristic times is different from the value \\alpha_{min}=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, assortative networks have preferentially a value of \\alpha_{min} in the range [-1,-0.5], while disassortative networks have \\alpha_{min} in the range [-0.5, 0]. When only local information is available, degree-biased random walks can guarantee smaller characteristic times by means of an appropr...
Ranking Competitors Using Degree-Neutralized Random Walks
Shin, Seungkyu; Park, Juyong
2016-01-01
Competition is ubiquitous in many complex biological, social, and technological systems, playing an integral role in the evolutionary dynamics of the systems. It is often useful to determine the dominance hierarchy or the rankings of the components of the system that compete for survival and success based on the outcomes of the competitions between them. Here we propose a ranking method based on the random walk on the network representing the competitors as nodes and competitions as directed edges with asymmetric weights. We use the edge weights and node degrees to define the gradient on each edge that guides the random walker towards the weaker (or the stronger) node, which enables us to interpret the steady-state occupancy as the measure of the node's weakness (or strength) that is free of unwarranted degree-induced bias. We apply our method to two real-world competition networks and explore the issues of ranking stabilization and prediction accuracy, finding that our method outperforms other methods includ...
Ranking competitors using degree-neutralized random walks.
Seungkyu Shin
Full Text Available Competition is ubiquitous in many complex biological, social, and technological systems, playing an integral role in the evolutionary dynamics of the systems. It is often useful to determine the dominance hierarchy or the rankings of the components of the system that compete for survival and success based on the outcomes of the competitions between them. Here we propose a ranking method based on the random walk on the network representing the competitors as nodes and competitions as directed edges with asymmetric weights. We use the edge weights and node degrees to define the gradient on each edge that guides the random walker towards the weaker (or the stronger node, which enables us to interpret the steady-state occupancy as the measure of the node's weakness (or strength that is free of unwarranted degree-induced bias. We apply our method to two real-world competition networks and explore the issues of ranking stabilization and prediction accuracy, finding that our method outperforms other methods including the baseline win-loss differential method in sparse networks.
Randomized Algorithms for Matrices and Data
Mahoney, Michael W.
2012-03-01
This chapter reviews recent work on randomized matrix algorithms. By “randomized matrix algorithms,” we refer to a class of recently developed random sampling and random projection algorithms for ubiquitous linear algebra problems such as least-squares (LS) regression and low-rank matrix approximation. These developments have been driven by applications in large-scale data analysis—applications which place very different demands on matrices than traditional scientific computing applications. Thus, in this review, we will focus on highlighting the simplicity and generality of several core ideas that underlie the usefulness of these randomized algorithms in scientific applications such as genetics (where these algorithms have already been applied) and astronomy (where, hopefully, in part due to this review they will soon be applied). The work we will review here had its origins within theoretical computer science (TCS). An important feature in the use of randomized algorithms in TCS more generally is that one must identify and then algorithmically deal with relevant “nonuniformity structure” in the data. For the randomized matrix algorithms to be reviewed here and that have proven useful recently in numerical linear algebra (NLA) and large-scale data analysis applications, the relevant nonuniformity structure is defined by the so-called statistical leverage scores. Defined more precisely below, these leverage scores are basically the diagonal elements of the projection matrix onto the dominant part of the spectrum of the input matrix. As such, they have a long history in statistical data analysis, where they have been used for outlier detection in regression diagnostics. More generally, these scores often have a very natural interpretation in terms of the data and processes generating the data. For example, they can be interpreted in terms of the leverage or influence that a given data point has on, say, the best low-rank matrix approximation; and this
Dobkin, Bruce H.; Xu, Xiaoyu; Batalin, Maxim; Thomas, Seth; Kaiser, William
2015-01-01
Background and Purpose Outcome measures of mobility for large stroke trials are limited to timed walks for short distances in a laboratory, step counters and ordinal scales of disability and quality of life. Continuous monitoring and outcome measurements of the type and quantity of activity in the community would provide direct data about daily performance, including compliance with exercise and skills practice during routine care and clinical trials. Methods Twelve adults with impaired ambulation from hemiparetic stroke and 6 healthy controls wore triaxial accelerometers on their ankles. Walking speed for repeated outdoor walks was determined by machine-learning algorithms and compared to a stopwatch calculation of speed for distances not known to the algorithm. The reliability of recognizing walking, exercise, and cycling by the algorithms was compared to activity logs. Results A high correlation was found between stopwatch-measured outdoor walking speed and algorithm-calculated speed (Pearson coefficient, 0.98; P=0.001) and for repeated measures of algorithm-derived walking speed (P=0.01). Bouts of walking >5 steps, variations in walking speed, cycling, stair climbing, and leg exercises were correctly identified during a day in the community. Compared to healthy subjects, those with stroke were, as expected, more sedentary and slower, and their gait revealed high paretic-to-unaffected leg swing ratios. Conclusions Test–retest reliability and concurrent and construct validity are high for activity pattern-recognition Bayesian algorithms developed from inertial sensors. This ratio scale data can provide real-world monitoring and outcome measurements of lower extremity activities and walking speed for stroke and rehabilitation studies. PMID:21636815
Kardar-Parisi-Zhang Equation and Large Deviations for Random Walks in Weak Random Environments
Corwin, Ivan; Gu, Yu
2017-01-01
We consider the transition probabilities for random walks in 1+1 dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.
Random walk approach for dispersive transport in pipe networks
Sämann, Robert; Graf, Thomas; Neuweiler, Insa
2016-04-01
Keywords: particle transport, random walk, pipe, network, HYSTEM-EXTAN, OpenGeoSys After heavy pluvial events in urban areas the available drainage system may be undersized at peak flows (Fuchs, 2013). Consequently, rainwater in the pipe network is likely to spill out through manholes. The presence of hazardous contaminants in the pipe drainage system represents a potential risk to humans especially when the contaminated drainage water reaches the land surface. Real-time forecasting of contaminants in the drainage system needs a quick calculation. Numerical models to predict the fate of contaminants are usually based on finite volume methods. Those are not applicable here because of their volume averaging elements. Thus, a more efficient method is preferable, which is independent from spatial discretization. In the present study, a particle-based method is chosen to calculate transport paths and spatial distribution of contaminants within a pipe network. A random walk method for particles in turbulent flow in partially filled pipes has been developed. Different approaches for in-pipe-mixing and node-mixing with respect to the geometry in a drainage network are shown. A comparison of dispersive behavior and calculation time is given to find the fastest model. The HYSTEM-EXTRAN (itwh, 2002) model is used to provide hydrodynamic conditions in the pipe network according to surface runoff scenarios in order to real-time predict contaminant transport in an urban pipe network system. The newly developed particle-based model will later be coupled to the subsurface flow model OpenGeoSys (Kolditz et al., 2012). References: Fuchs, L. (2013). Gefährdungsanalyse zur Überflutungsvorsorge kommunaler Entwässerungssysteme. Sanierung und Anpassung von Entwässerungssystemen-Alternde Infrastruktur und Klimawandel, Österreichischer Wasser-und Abfallwirtschaftsverband, Wien, ISBN, 978-3. itwh (2002). Modellbeschreibung, Institut für technisch-wissenschaftliche Hydrologie Gmb
Stochastic calculus for uncoupled continuous-time random walks.
Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L
2009-06-01
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.
From elongated spanning trees to vicious random walks
Gorsky, A. [ITEP, B. Cheryomushkinskaya 25, 117218 Moscow (Russian Federation); Nechaev, S., E-mail: nechaev@lptms.u-psud.fr [LPTMS, Université Paris Sud, 91405 Orsay Cedex (France); P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow (Russian Federation); Poghosyan, V.S. [Institute for Informatics and Automation Problems NAS of Armenia, 375044 Yerevan (Armenia); Priezzhev, V.B. [Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna (Russian Federation)
2013-05-01
Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of k paths (k is odd) along branches of trees or, equivalently, k loop-erased random walks. Starting and ending points of the paths are grouped such that they form a k-leg watermelon. For large distance r between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as r{sup −ν}logr with ν=(k{sup 2}−1)/2. Considering the spanning forest stretched along the meridian of this watermelon, we show that the two-dimensional k-leg loop-erased watermelon exponent ν is converting into the scaling exponent for the reunion probability (at a given point) of k(1+1)-dimensional vicious walkers, ν{sup -tilde=}k{sup 2}/2. At the end, we express the conjectures about the possible relation to integrable systems.
Scale-free avalanches in the multifractal random walk
Bartolozzi, M
2007-01-01
Avalanches, or Avalanche-like, events are often observed in the dynamical behaviour of many complex systems which span from solar flaring to the Earth's crust dynamics and from traffic flows to financial markets. Self-organized criticality (SOC) is one of the most popular theories able to explain this intermittent charge/discharge behaviour. Despite a large amount of theoretical work, empirical tests for SOC are still in their infancy. In the present paper we address the common problem of revealing SOC from a simple time series without having much information about the underlying system. As a working example we use a modified version of the multifractal random walk originally proposed as a model for the stock market dynamics. The study reveals, despite the lack of the typical ingredients of SOC, an avalanche-like dynamics similar to that of many physical systems. While, on one hand, the results confirm the relevance of cascade models in representing turbulent-like phenomena, on the other, they also raise the ...
Scale-free avalanches in the multifractal random walk
Bartolozzi, M.
2007-06-01
Avalanches, or Avalanche-like, events are often observed in the dynamical behaviour of many complex systems which span from solar flaring to the Earth's crust dynamics and from traffic flows to financial markets. Self-organized criticality (SOC) is one of the most popular theories able to explain this intermittent charge/discharge behaviour. Despite a large amount of theoretical work, empirical tests for SOC are still in their infancy. In the present paper we address the common problem of revealing SOC from a simple time series without having much information about the underlying system. As a working example we use a modified version of the multifractal random walk originally proposed as a model for the stock market dynamics. The study reveals, despite the lack of the typical ingredients of SOC, an avalanche-like dynamics similar to that of many physical systems. While, on one hand, the results confirm the relevance of cascade models in representing turbulent-like phenomena, on the other, they also raise the question about the current state of reliability of SOC inference from time series analysis.
Radio Variability and Random Walk Noise Properties of Four Blazars
Park, Jong-Ho
2014-01-01
We present the results of a time series analysis of the long-term radio lightcurves of four blazars: 3C 279, 3C 345, 3C 446, and BL Lacertae. We exploit the data base of the University of Michigan Radio Astronomy Observatory (UMRAO) monitoring program which provides densely sampled lightcurves spanning 32 years in time in three frequency bands located at 4.8, 8, and 14.5 GHz. Our sources show mostly flat or inverted (spectral indices -0.5 < alpha < 0) spectra, in agreement with optically thick emission. All lightcurves show strong variability on all time scales. Analyzing the time lags between the lightcurves from different frequency bands, we find that we can distinguish high-peaking flares and low-peaking flares in accord with the classification of Valtaoja et al. (1992). The periodograms (temporal power spectra) of the observed lightcurves are consistent with random-walk powerlaw noise without any indication of (quasi-)periodic variability. The fact that all four sources studied are in agreement with...
Electron avalanche structure determined by random walk theory
Englert, G. W.
1973-01-01
A self-consistent avalanche solution which accounts for collective long range Coulomb interactions as well as short range elastic and inelastic collisions between electrons and background atoms is made possible by a random walk technique. Results show that the electric field patterns in the early formation stages of avalanches in helium are close to those obtained from theory based on constant transport coefficients. Regions of maximum and minimum induced electrostatic potential phi are located on the axis of symmetry and within the volume covered by the electron swarm. As formation time continues, however, the region of minimum phi moves to slightly higher radii and the electric field between the extrema becomes somewhat erratic. In the intermediate formation periods the avalanche growth is slightly retarded by the high concentration of ions in the tail which oppose the external electric field. Eventually the formation of ions and electrons in the localized regions of high field strength more than offset this effect causing a very abrupt increase in avalanche growth.
Estimates for the Tail Probability of the Supremum of a Random Walk with Independent Increments
Yang YANG; Kaiyong WANG
2011-01-01
The authors investigate the tail probability of the supremum of a random walk with independent increments and obtain some equivalent assertions in the case that the increments are independent and identically distributed random variables with Osubexponential integrated distributions.A uniform upper bound is derived for the distribution of the supremum of a random walk with independent but non-identically distributed increments,whose tail distributions are dominated by a common tail distribution with an O-subexponential integrated distribution.
The linking number and the writhe of uniform random walks and polygons in confined spaces
Panagiotou, E.; Millett, K. C.; Lambropoulou, S.
2010-01-01
Random walks and polygons are used to model polymers. In this paper we consider the extension of the writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(\\sqrt{n}) . Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.
On the Domination of Random Walk on a Discrete Cylinder by Random Interlacements
Sznitman, Alain-Sol
2009-01-01
We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order N, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite (d+1)-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when d is at least 17, in arXiv: math/0701414.
Kapadia, Naaz; Masani, Kei; Catharine Craven, B; Giangregorio, Lora M; Hitzig, Sander L; Richards, Kieva; Popovic, Milos R
2014-09-01
Multi-channel surface functional electrical stimulation (FES) for walking has been used to improve voluntary walking and balance in individuals with spinal cord injury (SCI). To investigate short- and long-term benefits of 16 weeks of thrice-weekly FES-assisted walking program, while ambulating on a body weight support treadmill and harness system, versus a non-FES exercise program, on improvements in gait and balance in individuals with chronic incomplete traumatic SCI, in a randomized controlled trial design. Individuals with traumatic and chronic (≥18 months) motor incomplete SCI (level C2 to T12, American Spinal Cord Injury Association Impairment Scale C or D) were recruited from an outpatient SCI rehabilitation hospital, and randomized to FES-assisted walking therapy (intervention group) or aerobic and resistance training program (control group). Outcomes were assessed at baseline, and after 4, 6, and 12 months. Gait, balance, spasticity, and functional measures were collected. Spinal cord independence measure (SCIM) mobility sub-score improved over time in the intervention group compared with the control group (baseline/12 months: 17.27/21.33 vs. 19.09/17.36, respectively). On all other outcome measures the intervention and control groups had similar improvements. Irrespective of group allocation walking speed, endurance, and balance during ambulation all improved upon completion of therapy, and majority of participants retained these gains at long-term follow-ups. Task-oriented training improves walking ability in individuals with incomplete SCI, even in the chronic stage. Further randomized controlled trials, involving a large number of participants are needed, to verify if FES-assisted treadmill training is superior to aerobic and strength training.
Yazdi, Ebrahim
2010-01-01
In this paper, a simple Neural controller has been used to achieve stable walking in a NAO biped robot, with 22 degrees of freedom that implemented in a virtual physics-based simulation environment of Robocup soccer simulation environment. The algorithm uses a Matsuoka base neural oscillator to generate control signal for the biped robot. To find the best angular trajectory and optimize network parameters, a new population-based search algorithm, called the Harmony Search (HS) algorithm, has been used. The algorithm conceptualized a group of musicians together trying to search for better state of harmony. Simulation results demonstrate that the modification of the step period and the walking motion due to the sensory feedback signals improves the stability of the walking motion.
Asymptotic normality of randomly truncated stochastic algorithms
Lelong, Jérôme
2010-01-01
We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins-Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard algorithms fail because the expected-value function grows too fast. In this work, we give a self contained proof of a central limit theorem for this algorithm under local assumptions on the expected-value function, which are fairly easy to check in practice.
Asymptotic normality of randomly truncated stochastic algorithms
Lelong, Jérôme
2010-01-01
We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins-Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard algorithms fail because the expected-value function grows too fast. In this work, we give a self contained proof of a central limit theorem for this algorithm under local assumptions on the expected-value function, which are fairly easy to check in practice.
Quantum Random Walks and their Convergence to Evans-Hudson Flows
Lingaraj Sahu
2008-08-01
Using coordinate-free basic operators on toy Fock spaces, quantum random walks are defined following the ideas of Attal and Pautrat. Extending the result for one dimensional noise, strong convergence of quantum random walks associated with bounded structure maps to Evans–Hudson flow is proved under suitable assumptions. Starting from the bounded generator of a given uniformly continuous quantum dynamical semigroup on a von Neumann algebra, we have constructed quantum random walks which converges strongly and the strong limit gives an Evans–Hudson dilation for the semigroup.
Universal adaptive self-stabilizing traversal scheme: random walk and reloading wave
Bernard, Thibault; Sohier, Devan
2011-01-01
In this paper, we investigate random walk based token circulation in dynamic environments subject to failures. We describe hypotheses on the dynamic environment that allow random walks to meet the important property that the token visits any node infinitely often. The randomness of this scheme allows it to work on any topology, and require no adaptation after a topological change, which is a desirable property for applications to dynamic systems. For random walks to be a traversal scheme and to answer the concurrence problem, one needs to guarantee that exactly one token circulates in the system. In the presence of transient failures, configurations with multiple tokens or with no token can occur. The meeting property of random walks solves the cases with multiple tokens. The reloading wave mechanism we propose, together with timeouts, allows to detect and solve cases with no token. This traversal scheme is self-stabilizing, and universal, meaning that it needs no assumption on the system topology. We describ...
Random walk-based similarity measure method for patterns in complex object
Liu Shihu
2017-04-01
Full Text Available This paper discusses the similarity of the patterns in complex objects. The complex object is composed both of the attribute information of patterns and the relational information between patterns. Bearing in mind the specificity of complex object, a random walk-based similarity measurement method for patterns is constructed. In this method, the reachability of any two patterns with respect to the relational information is fully studied, and in the case of similarity of patterns with respect to the relational information can be calculated. On this bases, an integrated similarity measurement method is proposed, and algorithms 1 and 2 show the performed calculation procedure. One can find that this method makes full use of the attribute information and relational information. Finally, a synthetic example shows that our proposed similarity measurement method is validated.
Random walks with shape prior for cochlea segmentation in ex vivo μCT
Ruiz Pujadas, Esmeralda; Kjer, Hans Martin; Piella, Gemma;
2016-01-01
Purpose Cochlear implantation is a safe and effective surgical procedure to restore hearing in deaf patients. However, the level of restoration achieved may vary due to differences in anatomy, implant type and surgical access. In order to reduce the variability of the surgical outcomes, we...... previously proposed the use of a high-resolution model built from μCT images and then adapted to patient-specific clinical CT scans. As the accuracy of the model is dependent on the precision of the original segmentation, it is extremely important to have accurate μCT segmentation algorithms. Methods We...... propose a new framework for cochlea segmentation in ex vivo μCT images using random walks where a distance-based shape prior is combined with a region term estimated by a Gaussian mixture model. The prior is also weighted by a confidence map to adjust its influence according to the strength of the image...
Biased Random-Walk Learning A Neurobiological Correlate to Trial-and-Error
Anderson, R W
1993-01-01
Neural network models offer a theoretical testbed for the study of learning at the cellular level. The only experimentally verified learning rule, Hebb's rule, is extremely limited in its ability to train networks to perform complex tasks. An identified cellular mechanism responsible for Hebbian-type long-term potentiation, the NMDA receptor, is highly versatile. Its function and efficacy are modulated by a wide variety of compounds and conditions and are likely to be directed by non-local phenomena. Furthermore, it has been demonstrated that NMDA receptors are not essential for some types of learning. We have shown that another neural network learning rule, the chemotaxis algorithm, is theoretically much more powerful than Hebb's rule and is consistent with experimental data. A biased random-walk in synaptic weight space is a learning rule immanent in nervous activity and may account for some types of learning -- notably the acquisition of skilled movement.
Walking Algorithm of Humanoid Robot on Uneven Terrain with Terrain Estimation
Jiang Yi
2016-02-01
Full Text Available Humanoid robots are expected to achieve stable walking on uneven terrains. In this paper, a control algorithm for humanoid robots walking on previously unknown terrains with terrain estimation is proposed, which requires only minimum modification to the original walking gait. The swing foot trajectory is redesigned to ensure that the foot lands at the desired horizontal positions under various terrain height. A compliant terrain adaptation method is applied to the landing foot to achieve a firm contact with the ground. Then a terrain estimation method that takes into account the deformations of the linkages is applied, providing the target for the following correction and adjustment. The algorithm was validated through walking experiments on uneven terrains with the full-size humanoid robot Kong.
Selecting materialized views using random algorithm
Zhou, Lijuan; Hao, Zhongxiao; Liu, Chi
2007-04-01
The data warehouse is a repository of information collected from multiple possibly heterogeneous autonomous distributed databases. The information stored at the data warehouse is in form of views referred to as materialized views. The selection of the materialized views is one of the most important decisions in designing a data warehouse. Materialized views are stored in the data warehouse for the purpose of efficiently implementing on-line analytical processing queries. The first issue for the user to consider is query response time. So in this paper, we develop algorithms to select a set of views to materialize in data warehouse in order to minimize the total view maintenance cost under the constraint of a given query response time. We call it query_cost view_ selection problem. First, cost graph and cost model of query_cost view_ selection problem are presented. Second, the methods for selecting materialized views by using random algorithms are presented. The genetic algorithm is applied to the materialized views selection problem. But with the development of genetic process, the legal solution produced become more and more difficult, so a lot of solutions are eliminated and producing time of the solutions is lengthened in genetic algorithm. Therefore, improved algorithm has been presented in this paper, which is the combination of simulated annealing algorithm and genetic algorithm for the purpose of solving the query cost view selection problem. Finally, in order to test the function and efficiency of our algorithms experiment simulation is adopted. The experiments show that the given methods can provide near-optimal solutions in limited time and works better in practical cases. Randomized algorithms will become invaluable tools for data warehouse evolution.
Survival, extinction and approximation of discrete-time branching random walks
Zucca, Fabio
2010-01-01
We consider a general discrete-time branching random walk on a countable set X. We relate local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.
Mathematical conversations multicolor problems, problems in the theory of numbers, and random walks
Dynkin, E B
2006-01-01
Comprises Multicolor Problems, dealing with map-coloring problems; Problems in the Theory of Numbers, an elementary introduction to algebraic number theory; Random Walks, addressing basic problems in probability theory. 1963 edition.
Position-space renormalization-group approach to the resistance of random walks
Sahimi, Muhammad; Jerauld, Gary R.; Scriven, L. E.; Davis, H. Ted
1984-06-01
We consider a Pólya random walk, i.e., an unbiased, nearest-neighbor walk, on a d-dimensional hypercubic lattice and study the scaling behavior of the mean end-to-end resistance of the walk as a function of the number of steps in the walk. The resistance of the walk is generated by assigning a constant conductance to each step of the walk. This problem was recently proposed by Banavar, Harris, and Koplik, and may be useful for understanding the physics of disordered systems. We develop a position-space renormalization-group approach, a generalization of the one developed for percolation conductivity, and study the problem and a modification of it proposed here in one, two, and three dimensions. Our results are in good agreement with the numerical estimates of Banavar et al.
A Realization of a Quasi-Random Walk for Atoms in Time-Dependent Optical Potentials
Torsten Hinkel
2015-09-01
Full Text Available We consider the time dependent dynamics of an atom in a two-color pumped cavity, longitudinally through a side mirror and transversally via direct driving of the atomic dipole. The beating of the two driving frequencies leads to a time dependent effective optical potential that forces the atom into a non-trivial motion, strongly resembling a discrete random walk behavior between lattice sites. We provide both numerical and analytical analysis of such a quasi-random walk behavior.
Quantum random walks with multiphoton interference and high order correlation functions
Gard, Bryan T; Anisimov, Petr M; Lee, Hwang; Dowling, Jonathan P
2011-01-01
We show a simulation of quantum random walks with multiple photons using a staggered array of 50/50 beam splitters with a bank of detectors at any desired level. We discuss the multiphoton interference effects that are inherent to this setup, and introduce one, two, and threefold coincidence detection schemes. The use of Feynman diagrams are used to intuitively explain the unique multiphoton interference effects of these quantum random walks.
Path statistics, memory, and coarse-graining of continuous-time random walks on networks.
Manhart, Michael; Kion-Crosby, Willow; Morozov, Alexandre V
2015-12-01
Continuous-time random walks (CTRWs) on discrete state spaces, ranging from regular lattices to complex networks, are ubiquitous across physics, chemistry, and biology. Models with coarse-grained states (for example, those employed in studies of molecular kinetics) or spatial disorder can give rise to memory and non-exponential distributions of waiting times and first-passage statistics. However, existing methods for analyzing CTRWs on complex energy landscapes do not address these effects. Here we use statistical mechanics of the nonequilibrium path ensemble to characterize first-passage CTRWs on networks with arbitrary connectivity, energy landscape, and waiting time distributions. Our approach can be applied to calculating higher moments (beyond the mean) of path length, time, and action, as well as statistics of any conservative or non-conservative force along a path. For homogeneous networks, we derive exact relations between length and time moments, quantifying the validity of approximating a continuous-time process with its discrete-time projection. For more general models, we obtain recursion relations, reminiscent of transfer matrix and exact enumeration techniques, to efficiently calculate path statistics numerically. We have implemented our algorithm in PathMAN (Path Matrix Algorithm for Networks), a Python script that users can apply to their model of choice. We demonstrate the algorithm on a few representative examples which underscore the importance of non-exponential distributions, memory, and coarse-graining in CTRWs.
State-independent importance sampling for random walks with regularly varying increments
Karthyek R. A. Murthy
2015-03-01
Full Text Available We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1 the large deviation probabilities, 2 the level crossing probabilities, and 3 the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.
Patching, Geoffrey R; Rahm, Johan; Jansson, Märit; Johansson, Maria
2017-01-01
Accurate assessment of people's preferences for different outdoor lighting applications is increasingly considered important in the development of new urban environments. Here a new method of random environmental walking is proposed to complement current methods of assessing urban lighting applications, such as self-report questionnaires. The procedure involves participants repeatedly walking between different lighting applications by random selection of a lighting application and preferred choice or by random selection of a lighting application alone. In this manner, participants are exposed to all lighting applications of interest more than once and participants' preferences for the different lighting applications are reflected in the number of times they walk to each lighting application. On the basis of an initial simulation study, to explore the feasibility of this approach, a comprehensive field test was undertaken. The field test included random environmental walking and collection of participants' subjective ratings of perceived pleasantness (PP), perceived quality, perceived strength, and perceived flicker of four lighting applications. The results indicate that random environmental walking can reveal participants' preferences for different lighting applications that, in the present study, conformed to participants' ratings of PP and perceived quality of the lighting applications. As a complement to subjectively stated environmental preferences, random environmental walking has the potential to expose behavioral preferences for different lighting applications.
On a random walk with memory and its relation with Markovian processes
Turban, Loic, E-mail: turban@lpm.u-nancy.f [Groupe de Physique Statistique, Departement Physique de la Matiere et des Materiaux, Institut Jean Lamour (Laboratoire associe au CNRS UMR 7198), CNRS-Nancy Universite-UPV Metz, BP 70239, F-54506 Vandoeuvre les Nancy Cedex (France)
2010-07-16
We study a one-dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker. The feedback is quite efficient and leads to a non-diffusive walk. The time evolution of the displacement is given by an equivalent Markovian dynamical process. The probability density for the position of the walker is the same at any time as for a random walk with shrinking steps, although the two-time correlation functions are quite different.
Novel Control Algorithm for the Foot Placement of a Walking Bipedal Robot
Wanli Liu
2013-04-01
Full Text Available A novel control algorithm for the foot placement of walking bipedal robots is proposed which can output the optimal step time and step location to obtain a desired walking gait from every feasible robot state. The step time and step location are determined by approximating the robot dynamics with the 3D linear inverted pendulum model and analytically solving the constraint equations. Intensive simulation studies are conducted to check the validity of the theoretical results. The results of this study show that the proposed control algorithm can get the system to a desired gait cycle from every feasible state within a finite number of steps.
Random walk study of electron motion in helium in crossed electromagnetic fields
Englert, G. W.
1972-01-01
Random walk theory, previously adapted to electron motion in the presence of an electric field, is extended to include a transverse magnetic field. In principle, the random walk approach avoids mathematical complexity and concomitant simplifying assumptions and permits determination of energy distributions and transport coefficients within the accuracy of available collisional cross section data. Application is made to a weakly ionized helium gas. Time of relaxation of electron energy distribution, determined by the random walk, is described by simple expressions based on energy exchange between the electron and an effective electric field. The restrictive effect of the magnetic field on electron motion, which increases the required number of collisions per walk to reach a terminal steady state condition, as well as the effect of the magnetic field on electron transport coefficients and mean energy can be quite adequately described by expressions involving only the Hall parameter.
Continuous-time random walk and parametric subordination in fractional diffusion
Gorenflo, Rudolf [Department of Mathematics and Informatics, Free University of Berlin, Arnimallee 3, D-14195 Berlin (Germany); Mainardi, Francesco [Department of Physics, University of Bologna and INFN, Via Irnerio 46, I-40126 Bologna (Italy)]. E-mail: mainardi@bo.infn.it; Vivoli, Alessandro [Department of Physics, University of Bologna and INFN, Via Irnerio 46, I-40126 Bologna (Italy)
2007-10-15
The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Levy process, we generate and display sample paths for some special cases.
肖杰斌; 张绍武
2013-01-01
动态网络社团结构挖掘有助于获取整体网络特性和发展规律。由于动态网络具有多个时刻，传统静态网络社团挖掘算法不仅容易在相邻时刻产生具有较大差异的社团划分结果，而且导致较高时间复杂度。虽然最近受到广泛关注的动态网络增量算法可以一定程度上降低算法时间复杂度，但普遍存在人工设定参数、可扩展性差等局限性。该文提出一种随机游走与增量相关节点相结合的社团挖掘算法(RWIV)进行动态网络社团挖掘。利用动态网络时间局部性即相邻采样时刻网络变化不大的特点，通过对增量相关节点进行随机游走聚类后社团划分，避免了对整个网络中的节点全部重新划分。实验结果和分析表明：RWIV算法可有效解决IC(Incremental algorithm for Community identification)和IDCM(Increment and Density based Community detection Method)判定参数难以选定、累积误差及网络突变等问题，其社团挖掘效率高于现有IC和IDCM算法。%Community mining in dynamic networks can help to obtain the whole network characteristics and the trend of network development. As dynamic networks usually consist of many consecutive static networks, traditional methods of identifying network communities will lead to significant variations between communities close in time and high time complexity. Although the general incremental methods (e.g. Incremental algorithm for Community identification (IC) and Increment and Density based Community detection Method (IDCM)) can reduce the time complexity at a certain extent, but they need to manually set the judgment parameter, and fail to identify large networks in acceptable time. In this paper, an algorithm of integrating Random Walk and Increment correction Vertexes (RWIV) is proposed to identify the dynamic network structure. RWIV algorithm first deals with increment correlative vertexes with random walk, and then adjusts
Randomized algorithms for matrices and data
Mahoney, Michael W
2011-01-01
Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis. Although this work had its origins within theoretical computer science, where researchers were interested in proving worst-case bounds, i.e., bounds without any assumptions at all on the input data, researchers from numerical linear algebra, statistics, applied mathematics, data analysis, and machine learning, as well as domain scientists have subsequently extended and applied these methods in important ways. Although this has been great for the development of the area and for the technology transfer of theoretical ideas into practical applications, this interdisciplinarity has thus far sometimes obscured the underlying simplicity and generality of the core ideas. This review will provide a detailed overview of recent work on randomized algorithms for matrix problems, with an emphasis on a few simple core ideas that underlie not...
Tail estimates for one-dimensional non-nearest-neighbor random walk in random environment
无
2010-01-01
Suppose that the integers are assigned i.i.d. random variables {(β gx , . . . , β 1x , α x )} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {X n } (called RWRE) which, when at x, moves one step of length 1 to the right with probability α x and one step of length k to the left with probability β kx for 1≤ k≤ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment.
NewGOA: predicting new GO annotations of proteins by bi-random walks on a hybrid graph.
Yu, Guoxian; Fu, Guangyuan; Wang, Jun; Zhao, Yingwen
2017-06-15
A remaining key challenge of modern biology is annotating the functional roles of proteins. Various computational models have been proposed for this challenge. Most of them assume the annotations of annotated proteins are complete. But in fact, many of them are incomplete. We proposed a method called NewGOA to predict new Gene Ontology (GO) annotations for incompletely annotated proteins and for completely un-annotated ones. NewGOA employs a hybrid graph, composed of two types of nodes (proteins and GO terms), to encode interactions between proteins, hierarchical relationships between terms and available annotations of proteins. To account for structural difference between the terms subgraph and the proteins subgraph, NewGOA applies a bi-random walks algorithm, which executes asynchronous random walks on the hybrid graph, to predict new GO annotations of proteins. Experimental study on archived GO annotations of two model species (H. Sapiens and S. cerevisiae) shows that NewGOA can more accurately and efficiently predict new annotations of proteins than other related methods. Experimental results also indicate the bi-random walks can explore and further exploit the structural difference between terms subgraph and proteins subgraph. The supplementary files and codes of NewGOA are available at: http://mlda.swu.edu.cn/codes.php?name=NewGO.
Sun, Min; Chen, Xinjian; Zhang, Zhiqiang; Ma, Chiyuan
2017-02-01
Accurate volume measurements of pituitary adenoma are important to the diagnosis and treatment for this kind of sellar tumor. The pituitary adenomas have different pathological representations and various shapes. Particularly, in the case of infiltrating to surrounding soft tissues, they present similar intensities and indistinct boundary in T1-weighted (T1W) magnetic resonance (MR) images. Then the extraction of pituitary adenoma from MR images is still a challenging task. In this paper, we propose an interactive method to segment the pituitary adenoma from brain MR data, by combining graph cuts based active contour model (GCACM) and random walk algorithm. By using the GCACM method, the segmentation task is formulated as an energy minimization problem by a hybrid active contour model (ACM), and then the problem is solved by the graph cuts method. The region-based term in the hybrid ACM considers the local image intensities as described by Gaussian distributions with different means and variances, expressed as maximum a posteriori probability (MAP). Random walk is utilized as an initialization tool to provide initialized surface for GCACM. The proposed method is evaluated on the three-dimensional (3-D) T1W MR data of 23 patients and compared with the standard graph cuts method, the random walk method, the hybrid ACM method, a GCACM method which considers global mean intensity in region forces, and a competitive region-growing based GrowCut method planted in 3D Slicer. Based on the experimental results, the proposed method is superior to those methods.
A path integral formula with applications to quantum random walks in Z{sup d}
Yang Weishih [Department of Mathematics, Temple University, Philadelphia, PA 19122 (United States); Liu, Chaobin [Department of Mathematics, Bowie State University, Bowie, MD 20715 (United States); Zhang Kai [Department of Mathematics, Temple University, Philadelphia, PA 19122 (United States)
2007-07-20
We consider general quantum random walks in a d-dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d-dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O(1) if the lattice spacing is set to be 1/n. We also show that the rate of convergence of the total hitting probability has lower bound n{sup -2} and upper bound n{sup -2+{epsilon}} for any {epsilon} > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.
Human mammary epithelial cells exhibit a bimodal correlated random walk pattern.
Alka A Potdar
Full Text Available BACKGROUND: Organisms, at scales ranging from unicellular to mammals, have been known to exhibit foraging behavior described by random walks whose segments confirm to Lévy or exponential distributions. For the first time, we present evidence that single cells (mammary epithelial cells that exist in multi-cellular organisms (humans follow a bimodal correlated random walk (BCRW. METHODOLOGY/PRINCIPAL FINDINGS: Cellular tracks of MCF-10A pBabe, neuN and neuT random migration on 2-D plastic substrates, analyzed using bimodal analysis, were found to reveal the BCRW pattern. We find two types of exponentially distributed correlated flights (corresponding to what we refer to as the directional and re-orientation phases each having its own correlation between move step-lengths within flights. The exponential distribution of flight lengths was confirmed using different analysis methods (logarithmic binning with normalization, survival frequency plots and maximum likelihood estimation. CONCLUSIONS/SIGNIFICANCE: Because of the presence of non-uniform turn angle distribution of move step-lengths within a flight and two different types of flights, we propose that the epithelial random walk is a BCRW comprising of two alternating modes with varying degree of correlations, rather than a simple persistent random walk. A BCRW model rather than a simple persistent random walk correctly matches the super-diffusivity in the cell migration paths as indicated by simulations based on the BCRW model.
RRW: repeated random walks on genome-scale protein networks for local cluster discovery
Can Tolga
2009-09-01
Full Text Available Abstract Background We propose an efficient and biologically sensitive algorithm based on repeated random walks (RRW for discovering functional modules, e.g., complexes and pathways, within large-scale protein networks. Compared to existing cluster identification techniques, RRW implicitly makes use of network topology, edge weights, and long range interactions between proteins. Results We apply the proposed technique on a functional network of yeast genes and accurately identify statistically significant clusters of proteins. We validate the biological significance of the results using known complexes in the MIPS complex catalogue database and well-characterized biological processes. We find that 90% of the created clusters have the majority of their catalogued proteins belonging to the same MIPS complex, and about 80% have the majority of their proteins involved in the same biological process. We compare our method to various other clustering techniques, such as the Markov Clustering Algorithm (MCL, and find a significant improvement in the RRW clusters' precision and accuracy values. Conclusion RRW, which is a technique that exploits the topology of the network, is more precise and robust in finding local clusters. In addition, it has the added flexibility of being able to find multi-functional proteins by allowing overlapping clusters.
Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field
Biskup, Marek; Ding, Jian; Goswami, Subhajit
2016-01-01
Given any $\\gamma>0$ and for $\\eta=\\{\\eta_v\\}_{v\\in \\mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\\mathbb Z^2$ pinned at the origin, we consider the random walk on $\\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\\mathrm{e}^{\\gamma(\\eta_u + \\eta_v)}$. We show that, for almost every $\\eta$, this random walk is recurrent and that, with probability tending to 1 as $T\\to \\infty$, the return probability at time $2...
Su, Yansen; Wang, Bangju; Zhang, Xingyi
2017-02-01
Community detection has received a great deal of attention, since it could help to reveal the useful information hidden in complex networks. Although most previous modularity-based and local modularity-based community detection algorithms could detect strong communities, they may fail to exactly detect several weak communities. In this work, we define a network with clear or ambiguous community structures based on the types of its communities. A seed-expanding method based on random walks is proposed to detect communities for networks, especially for the networks with ambiguous community structures. We identify local maximum degree nodes, and detect seed communities in a network. Then, the probability of a node belonging to each community is calculated based on the total probability model and random walks, and each community is expanded by repeatedly adding the node which is most likely to belong to it. Finally, we use the community optimization method to ensure that each node is in a community. Experimental results on both computer-generated and real-world networks demonstrate that the quality of the communities detected by the proposed algorithm is superior to the- state-of-the-art algorithms in the networks with ambiguous community structures.
Einstein relation for biased random walk on Galton--Watson trees
Arous, Gerard Ben; Olla, Stefano; Zeitouni, Ofer
2011-01-01
We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.
Genetic Analysis of Daily Maximum Milking Speed by a Random Walk Model in Dairy Cows
Karacaören, Burak; Janss, Luc; Kadarmideen, Haja
Data were obtained from dairy cows stationed at research farm ETH Zurich for maximum milking speed. The main aims of this paper are a) to evaluate if the Wood curve is suitable to model mean lactation curve b) to predict longitudinal breeding values by random regression and random walk models...
Testing the imprint of nonstandard cosmologies on void profiles using Monte Carlo random walks
Achitouv, Ixandra
2016-11-01
Using Monte Carlo random walks of a log-normal distribution, we show how to qualitatively study void properties for nonstandard cosmologies. We apply this method to an f (R ) modified gravity model and recover the N -body simulation results of [1 I. Achitouv, M. Baldi, E. Puchwein, and J. Weller, Phys. Rev. D 93, 103522 (2016).] for the void profiles and their deviation from GR. This method can potentially be extended to study other properties of the large scale structures such as the abundance of voids or overdense environments. We also introduce a new way to identify voids in the cosmic web, using only a few measurements of the density fluctuations around random positions. This algorithm allows us to select voids with specific profiles and radii. As a consequence, we can target classes of voids with higher differences between f (R ) and standard gravity void profiles. Finally, we apply our void criteria to galaxy mock catalogues and discuss how the flexibility of our void finder can be used to reduce systematic errors when probing the growth rate in the galaxy-void correlation function.
Testing the imprint of non-standard cosmologies on void profiles using Monte Carlo random walks
Achitouv, Ixandra
2016-01-01
Using a Monte Carlo random walks of a log-normal distribution, we show how to qualitatively study void properties for non-standard cosmologies. We apply this method to an f(R) modified gravity model and recover the N-body simulation results of (Achitouv et al. 2016) for the void profiles and their deviation from GR. This method can potentially be extended to study other properties of the large scale structures such as the abundance of voids or overdense environments. We also introduce a new way to identify voids in the cosmic web, using only a few measurements of the density fluctuations around random positions. This algorithm allows to select voids with specific profiles and radii. As a consequence, we can target classes of voids with higher differences between f(R) and standard gravity void profiles. Finally we apply our void criteria to galaxy mock catalogues and discuss how the flexibility of our void finder can be used to reduce systematics errors when probing the growth rate in the galaxy-void correlati...
Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications
Zhongtuan Zheng
2017-01-01
Full Text Available This paper investigates, both theoretically and numerically, preferential random walks (PRW on weighted complex networks. By using two different analytical methods, two exact expressions are derived for the mean first passage time (MFPT between two nodes. On one hand, the MFPT is got explicitly in terms of the eigenvalues and eigenvectors of a matrix associated with the transition matrix of PRW. On the other hand, the center-product-degree (CPD is introduced as one measure of node strength and it plays a main role in determining the scaling of the MFPT for the PRW. Comparative studies are also performed on PRW and simple random walks (SRW. Numerical simulations of random walks on paradigmatic network models confirm analytical predictions and deepen discussions in different aspects. The work may provide a comprehensive approach for exploring random walks on complex networks, especially biased random walks, which may also help to better understand and tackle some practical problems such as search and routing on networks.
Real-time Walking Pattern Generation for a Biped Robot with Hybrid CPG-ZMP Algorithm
Bin He
2014-10-01
Full Text Available Biped robots have better mobility than conventional wheeled robots. The bio-inspired method based on a central pattern generator (CPG can be used to control biped robot walking in a manner like human beings. However, to achieve stable locomotion, it is difficult to modulate the parameters for the neural networks to coordinate every degree of freedom of the walking robot. The zero moment point (ZMP method is very popular for the stability control of biped robot walking. However, the reference trajectories have low energy efficiency, lack naturalness and need significant offline calculation. This paper presents a new method for biped real-time walking generation using a hybrid CPG-ZMP control algorithm. The method can realize a stable walking pattern by combining the ZMP criterion with rhythmic motion control. The CPG component is designed to generate the desired motion for each robot joint, which is modulated by phase resetting according to foot contact information. By introducing the ZMP location, the activity of the CPG output signal is adjusted to coordinate the limbs’ motion and allow the robot to maintain balance during the process of locomotion. The numerical simulation results show that, compared with the CPG method, the new hybrid CPG-ZMP algorithm can enhance the robustness of the CPG parameters and improve the stability of the robot. In addition, the proposed algorithm is more energy efficient than the ZMP method. The results also demonstrate that the control system can generate an adaptive walking pattern through interactions between the robot, the CPG and the environment.
Random Walks on Directed Networks: Inference and Respondent-driven Sampling
Malmros, Jens; Britton, Tom
2013-01-01
Respondent driven sampling (RDS) is a method often used to estimate population properties (e.g. sexual risk behavior) in hard-to-reach populations. It combines an effective modified snowball sampling methodology with an estimation procedure that yields unbiased population estimates under the assumption that the sampling process behaves like a random walk on the social network of the population. Current RDS estimation methodology assumes that the social network is undirected, i.e. that all edges are reciprocal. However, empirical social networks in general also have non-reciprocated edges. To account for this fact, we develop a new estimation method for RDS in the presence of directed edges on the basis of random walks on directed networks. We distinguish directed and undirected edges and consider the possibility that the random walk returns to its current position in two steps through an undirected edge. We derive estimators of the selection probabilities of individuals as a function of the number of outgoing...
Random walk of a swimmer in a low-Reynolds-number medium
Garcia, Michaël; Berti, Stefano; Peyla, Philippe; Rafaï, Salima
2011-03-01
Swimming at a micrometer scale demands particular strategies. When inertia is negligible compared to viscous forces, hydrodynamics equations are reversible in time. To achieve propulsion, microswimmers must therefore deform in a way that is not invariant under time reversal. Here, we investigate dispersal properties of the microalga Chlamydomonas reinhardtii by means of microscopy and cell tracking. We show that tracked trajectories are well modeled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlation is lost and a standard random walk characterizes the trajectories. Moreover, high-speed imaging enables us to show how the back-and-forth motion of flagella at very short times affects the statistical description of the dynamics. Finally, we show how drag forces modify the characteristics of this particular random walk.
Enumeration of closed random walks in the square lattice according to their areas
Mohammad-Noori, Morteza
2010-01-01
We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple argument. It is concluded that the number of walks of length $n$ and area $s$ equals to the coefficient of $z^s$ in the expression $(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a special group ring $R[x,y,z]$. A polynomial time algorithm for calculating these values, is then concluded. Finally, the provided algorithm and the results of implementation are compared with previous works.
Kim, Chang-Yong; Lee, Jung-Sun; Kim, Hyeong-Dong
2017-02-01
The purposes of the present study were to compare the effects of backward and lateral walking training and to identify whether additional backward or lateral walking training would be more effective in increasing the walking function of poststroke patients. Fifty-one subjects with hemiplegic stroke were randomly allocated to 3 groups, each containing 17 subjects: the control group, the backward walking training group, and the lateral walking training group. The walking abilities of each group were assessed using a 10-m walk test and the GAITRite system for spatiotemporal gait. The results show that there were significantly greater posttest increases in gait velocity (F = -12.09, P = 0.02) and stride length (F = -11.50, P = 0.02), decreases in the values of the 10-m walk test (F = -7.10, P = 0.03) (P training group compared with those in the other 2 groups. These findings demonstrate that asymmetric gait patterns in poststroke patients could be improved by receiving additional lateral walking training therapy rather than backward walking training. Complete the self-assessment activity and evaluation online at http://www.physiatry.org/JournalCME CME OBJECTIVES: Upon completion of this article, the reader should be able to: (1) understand the potential benefits of backward walking (BW) and lateral walking (LW) training on improving muscle strength and gait; (2) appreciate the potential value of backward and lateral walking gait training in the treatment of hemiplegic stroke patients; and (3) appropriately incorporate backward and lateral walking gait training into the treatment plan of hemiplegic stroke patients. Advanced ACCREDITATION: The Association of Academic Physiatrists is accredited by the Accreditation Council for Continuing Medical Education to provide continuing medical education for physicians.The Association of Academic Physiatrists designates this activity for a maximum of 1.5 AMA PRA Category 1 Credit(s)™. Physicians should only claim credit
Analysis and Development of Walking Algorithm Kinematic Model for 5-Degree of Freedom Bipedal Robot
Gerald Wahyudi Setiono
2012-12-01
Full Text Available A design of walking diagram and the calculation of a bipedal robot have been developed. The bipedal robot was designed and constructed with several kinds of servo bracket for the legs, two feet and a hip. Each of the bipedal robot leg was 5-degrees of freedom, three pitches (hip joint, knee joint and ankle joint and two rolls (hip joint and ankle joint. The walking algorithm of this bipedal robot was based on the triangle formulation of cosine law to get the angle value at each joint. The hip height, height of the swinging leg and the step distance are derived based on linear equation. This paper discussed the kinematic model analysis and the development of the walking diagram of the bipedal robot. Kinematics equations were derived, the joint angles were simulated and coded into Arduino board to be executed to the robot.
Hitting time for Bessel processes - walk on moving spheres algorithm
Deaconu, Madalina
2011-01-01
In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation is coming from mathematical finance when dealing with volatility models but the results can also be used in optimal control problems. The aim is here to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side an adaptation of the method of images to this particular situation and on the other side the connection existing between Cox-Ingersoll-Ross processes and Bessel processes.
Randomized Controlled Theory-Based, E-Mail-Mediated Walking Intervention.
Richards, Elizabeth A; Ogata, Niwako; Cheng, Ching-Wei
2017-02-01
The purpose of this study was to evaluate the ability of two concurrent randomized controlled interventions based on social cognitive theory to increase walking. A second purpose was to compare the efficacy of the intervention between two distinct groups: dog owners and non-dog owners. Adult dog owners ( n = 40) and non-dog owners ( n = 65) were randomized into control or intervention groups. Intervention groups received bi-weekly emails for first 4 weeks and then weekly email for the next 8 weeks targeting self-efficacy, social support, goal setting, and benefits/barriers to walking. Dog owner messages focused on dog walking while non-dog owners received general walking messages. Control groups received a 1-time email reviewing current physical activity guidelines. At 6 months, both intervention groups reported greater increases in walking and maintained these increases at 12 months. The greatest increases were seen in the dog owner intervention group. In conclusion, dog owners accumulated more walking, which may be attributed to the dog-owner relationship.
Random walks on finite lattices with multiple traps: Application to particle-cluster aggregation
Evans, J.W.; Nord, R.S.
1985-11-01
For random walks on finite lattices with multiple (completely adsorbing) traps, one is interested in the mean walk length until trapping and in the probability of capture for the various traps (either for a walk with a specific starting site, or for an average over all nontrap sites). We develop the formulation of Montroll to enable determination of the large-lattice-size asymptotic behavior of these quantities. (Only the case of a single trap has been analyzed in detail previously.) Explicit results are given for the case of symmetric nearest-neighbor random walks on two-dimensional (2D) square and triangular lattices. Procedures for exact calculation of walk lengths on a finite lattice with a single trap are extended to the multiple-trap case to determine all the above quantities. We examine convergence to asymptotic behavior as the lattice size increases. Connection with Witten-Sander irreversible particle-cluster aggregation is made by noting that this process corresponds to designating all sites adjacent to the cluster as traps. Thus capture probabilities for different traps determine the proportions of the various shaped clusters formed. (Reciprocals of) associated average walk lengths relate to rates for various irreversible aggregation processes involving a gas of walkers and clusters. Results are also presented for some of these quantities.
Eigenvalue vs perimeter in a shape theorem for self-interacting random walks
Biskup, Marek; Procaccia, Eviatar B.
2016-01-01
We study paths of time-length $t$ of a continuous-time random walk on $\\mathbb Z^2$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\\beta$; the "energy" is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range ...
Random-Walk Type Model with Fat Tails for Financial Markets
Matuttis, Hans-Geors
Starting from the random-walk model, practices of financial markets are included into the random-walk so that fat tail distributions like those in the high frequency data of the SP500 index are reproduced, though the individual mechanisms are modeled by normally distributed data. The incorporation of local correlation narrows the distribution for "frequent" events, whereas global correlations due to technical analysis leads to fat tails. Delay of market transactions in the trading process shifts the fat tail probabilities downwards. Such an inclusion of reactions to market fluctuations leads to mini-trends which are distributed with unit variance.
Bias phase and light power dependence of the random walk coefficient of fiber optic gyroscope
Jian Mi; Chunxi Zhang; Zheng Li; Zhanjun Wu
2006-01-01
@@ Taking account of shot noise, thermal noise, dark current noise, and intensity noise that come from broad band light source, the dependence of the random walk coefficient of fiber optic gyroscope (FOG) on bias phase and light power is studied theoretically and experimentally. It is shown that with different optical and electronic parameters, the optimal bias phase is different and should be adjusted accordingly to improve the FOG precision. By choosing appropriate bias phase, the random walk coefficient of the aim FOG is reduced from 0.0026 to 0.0019 deg./h1/2.
Daily Quantity of Infant Leg Movement: Wearable Sensor Algorithm and Relationship to Walking Onset
Beth A. Smith
2015-08-01
Full Text Available Background: Normative values are lacking for daily quantity of infant leg movements. This is critical for understanding the relationship between the quantity of leg movements and onset of independent walking, and will begin to inform early therapy intervention for infants at risk for developmental delay. Methods: We used wearable inertial movement sensors to record full-day leg movement activity from 12 infants with typical development, ages 1–12 months. Each infant was tested three times across 5 months, and followed until the onset of independent walking. We developed and validated an algorithm to identify infant-produced leg movements. Results: Infants moved their legs tens of thousands of times per day. There was a significant effect of leg movement quantity on walking onset. Infants who moved their legs more walked later than infants who moved their legs less, even when adjusting for age, developmental level or percentile length. We will need a much larger sample to adequately capture and describe the effect of movement experience on developmental rate. Our algorithm defines a leg movement in a specific way (each pause or change in direction is counted as a new movement, and further assessment of movement characteristics are necessary before we can fully understand and interpret our finding that infants who moved their legs more walked later than infants who moved their legs less. Conclusions: We have shown that typically-developing infants produce thousands of leg movements in a typical day, and that this can be accurately captured in the home environment using wearable sensors. In our small sample we can identify there is an effect of leg movement quantity on walking onset, however we cannot fully explain it.
Finding passwords by random walks: How long does it take?
Kabatiansky, G
2009-01-01
We compare an efficiency of a deterministic "lawnmower" and random search strategies for finding a prescribed sequence of letters (a password) of length M in which all letters are taken from the same Q-ary alphabet. We show that at best a random search takes two times longer than a "lawnmower" search.
A Non-Random Walk Down Hollywood Boulevard
Lepori, Gabriele
affect (i.e. grief, proxied by the death of Hollywood Walk of Fame celebrities) on people’s willingness to invest in risky assets (proxied by the daily performance of the U.S. stock market). Using a sample of 1,374 celebrity deaths over the period 1926-2009 and controlling for seasonalities, economic....../environmental factors, and market liquidity, I find that the death of popular and beloved celebrities is immediately followed by a 16 basis point increase in stock returns, which is consistent with a rise in the net demand for risky instruments. I also find evidence that the size of this celebrity-death effect...... is increasing in the popularity/media coverage of the celebrity in question, and is larger for stocks that are more affected by investor sentiment. Overall, my findings are consistent with the lab research on the affect management model, which maintains that incidental negative affect promotes risk...
Is walking a random walk? Evidence for long-range correlations in stride interval of human gait
Hausdorff, Jeffrey M.; Peng, C.-K.; Ladin, Zvi; Wei, Jeanne Y.; Goldberger, Ary L.
1995-01-01
Complex fluctuation of unknown origin appear in the normal gait pattern. These fluctuations might be described as being (1) uncorrelated white noise, (2) short-range correlations, or (3) long-range correlations with power-law scaling. To test these possibilities, the stride interval of 10 healthy young men was measured as they walked for 9 min at their usual rate. From these time series we calculated scaling indexes by using a modified random walk analysis and power spectral analysis. Both indexes indicated the presence of long-range self-similar correlations extending over hundreds of steps; the stride interval at any time depended on the stride interval at remote previous times, and this dependence decayed in a scale-free (fractallike) power-law fashion. These scaling indexes were significantly different from those obtained after random shuffling of the original time series, indicating the importance of the sequential ordering of the stride interval. We demonstrate that conventional models of gait generation fail to reproduce the observed scaling behavior and introduce a new type of central pattern generator model that sucessfully accounts for the experimentally observed long-range correlations.
Avena, Luca; Blondel, Oriane; Faggionato, Alessandra
2016-10-01
We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in Avena et al. (L^2-Perturbed Markov processes and applications to random walks in dynamic random environments, Preprint, 2016). As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical interfaces between empty and occupied regions. We study the asymptotic velocity and density profile for the environment seen from the walker. In particular, we determine the sign of the velocity when the density of the underlying East process is not 1 / 2, and we discuss the appearance of a drift in the balanced setting given by density 1 / 2.
Return probability for random walks on scale-free complex trees
Chelminiak, Przemyslaw, E-mail: geronimo@amu.edu.pl [Faculty of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan (Poland)
2011-08-15
The time course of random processes usually differs depending on the topology of complex networks which are a substrate for the process. However, as this Letter demonstrates, the first-return as well as the survival probabilities for random walks on the scale-free (SF) trees decay in time according to the same invariant power-law behavior. This means that both quantities are independent of the node power-law degree distributions which are distinguished by different scaling exponents. It is also shown here that the crucial property of the networks, affecting the dynamics of random walks, is their tree-like topology and not SF architecture. All analytical results quantifying these predictions have been verified through extensive computer simulations. -- Highlights: → We show that the first return of random walks on scale-free trees is recurrent. → As a consequence the survival probability of random walks is smaller than one. → This behavior is independent of the degree distribution of the scale-free trees. → It strongly depends on the network tree-like topology with a mean degree two.
The worst visibility walk in a random Delaunay triangulation is $O(\\sqrt{n}$
Olivier Devillers
2016-07-01
Full Text Available We show that the memoryless routing algorithms Greedy Walk, Compass Walk, and all variants of visibility walk based on orientation predicates are asymptotically optimal in the average case on the Delaunay triangulation. More specifically, we consider the Delaunay triangulation of an unbounded Poisson point process of unit rate and demonstrate that, for any pair of vertices $(s,t$ inside $[0,n]^2$, the ratio between the longest and shortest visibility walks between $s$ and $t$ is bounded by a constant with probability converging to one (as long as the vertices are sufficiently far apart. As a corollary, it follows that the worst-case path has $O(\\sqrt{n}\\,$ steps in the limiting case, under the same conditions. Our results have applications in routing in mobile networks and also settle a long-standing conjecture in point location using walking algorithms. Our proofs use techniques from percolation theory and stochastic geometry.
Solving Electromagnetism Differential Equations Based on Random Walk
邱尧峰; 曹毅; 李征帆
2004-01-01
An approach of solving the finite difference equations with Monte Carlo method was presented. The accuracy of algorithm is guaranteed by the Central Limit Theorem. The computation of the value on each single node is independent of each other, which makes it easy to realize the algorithm in parallel processing and greatly improves the efficiency while dealing with local area computation. As long as different kinds of boundary conditions are statistics modeled ,wide applications can the be made where the finite difference method is of competence.
Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs
S. Salimi; M.A. Jafarizadeh
2009-01-01
In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on complicated graphs. Using this method, we calculate the probability of continuous-time classical and quantum random walks on many of finite direct product Cayley graphs (complete cycle, complete Kn, charter and n-cube). Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.
Recurrence and Polya Number of General One-Dimensional Random Walks
张晓琨; 万晶; 陆静菊; 徐新平
2011-01-01
The recurrence properties of random walks can be characterized by P61ya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities l and r, or remain at the same position with probability o （l ＋ r ＋ o = 1）. We calculate Polya number P of this model and find a simple expression for P as, P = 1 - △, where △ is the absolute difference of l and r （△= ｜l - r｜）. We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability l equals to the right-moving probability r.
Simulation of the diffusion process in composite porous media by random walks
ZHANG Yong
2005-01-01
A new random-walk interpolation scheme was developed to simulate solute transport through composite porous media with different porosities as well as different diffusivities. The significant influences of the abrupt variations of porosity and diffusivity on solute transport were simulated by tracking random walkers through a linear interpolation domain across the heterogeneity interface. The displacements of the random walkers within the interpolation region were obtained explicitly by establishing the equivalence between the Fokker-Planck equation and the advection-dispersion equation. Applications indicate that the random-walk interpolation method can simulate one- and two-dimensional, 2nd-order diffusion processes in composite media without local mass conservation errors. In addition, both the theoretical derivations and the numerical simulations show that the drift and dispersion of particles depend on the type of Markov process selected to reflect the dynamics of random walkers. If the nonlinear Langevin equation is used, the gradient of porosity and the gradient of diffusivity strongly affect the drift displacement of particles. Therefore, random-walking particles driven by the gradient of porosity,the gradient of diffusivity, and the random diffusion, can imitate the transport of solute under only pure diffusion in composite porous media containing abrupt variations of porosity and diffusivity.
Nordic Walking and chronic low back pain: design of a randomized clinical trial
Hartvigsen Jan
2006-10-01
Full Text Available Abstract Background Low Back Pain is a major public health problem all over the western world. Active approaches including exercise in the treatment of low back pain results in better outcomes for patients, but it is not known exactly which types of back exercises are most beneficial or whether general physical activity provide similar benefits. Nordic Walking is a popular and fast growing type of exercise in Northern Europe. Initial studies have demonstrated that persons performing Nordic Walking are able to exercise longer and harder compared to normal walking thereby increasing their cardiovascular metabolism. Until now no studies have been performed to investigate whether Nordic Walking has beneficial effects in relation to low back pain. The primary aim of this study is to investigate whether supervised Nordic Walking can reduce pain and improve function in a population of chronic low back pain patients when compared to unsupervised Nordic Walking and advice to stay active. In addition we investigate whether there is an increase in the cardiovascular metabolism in persons performing supervised Nordic Walking compared to persons who are advised to stay active. Finally, we investigate whether there is a difference in compliance between persons receiving supervised Nordic Walking and persons doing unsupervised Nordic Walking. Methods One hundred and fifty patients with low back pain for at least eight weeks and referred to a specialized secondary sector outpatient back pain clinic are included in the study. After completion of the standard back centre treatment patients are randomized into one of three groups: A Nordic Walking twice a week for eight weeks under supervision of a specially trained instructor; B Unsupervised Nordic Walking for eight weeks after one training session with an instructor; C A one hour motivational talk including advice to stay active. Outcome measures are pain, function, overall health, cardiovascular ability and
Functional equation for the crossover in the model of one-dimensional Weierstrass random walks
Rudoi, Yu. G.; Kotel'nikova, O. A.
2016-12-01
We consider the problem of one-dimensional symmetric diffusion in the framework of Markov random walks of the Weierstrass type using two-parameter scaling for the transition probability. We construct a solution for the characteristic Lyapunov function as a sum of regular (homogeneous) and singular (nonhomogeneous) solutions and find the conditions for the crossover from normal to anomalous diffusion.
Tail asymptotics for the total progeny of the critical killed branching random walk
Aidekon, Elie
2009-01-01
We consider a branching random walk on $\\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.
Visser, Andre
1997-01-01
Random walk simulation has the potential to be an extremely powerful tool in the investigation of turbulence in environmental processes. However, care must be taken in applying such simulations to the motion of particles in turbulent marine systems where turbulent diffusivity is commonly spatially...
Elliptic random-walk equation for suspension and tracer transport in porous media
Shapiro, Alexander; Bedrikovetsky, P. G.
2008-01-01
We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time...
Return Probability of the Open Quantum Random Walk with Time-Dependence
Clement Ampadu
2013-01-01
We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution.As an application we study the return probability of the OQRW.We also ask,"What is the average time for the return probability of the OQRW?"
Wang Kaiyong; Wang Yuebao; Yin Chuancun
2011-01-01
This article gives the equivalent conditions of the local asymptotics for the overshoot of a random walk with heavy-tailed increments, from which we find that the above asymptotics are different from the local asymptoties for the supremum of the random walk. To do this, the article first extends and improves some existing results about the solutions of renewal equations.
Uphill and Downhill Walking in Multiple Sclerosis: A Randomized Controlled Trial.
Samaei, Afshin; Bakhtiary, Amir Hoshang; Hajihasani, Abdolhamid; Fatemi, Elham; Motaharinezhad, Fatemeh
2016-01-01
Various exercise protocols have been recommended for patients with multiple sclerosis (MS). We investigated the effects of uphill and downhill walking exercise on mobility, functional activities, and muscle strength in MS patients. Thirty-four MS patients were randomly allocated to either the downhill or uphill treadmill walking group for 12 sessions (3 times/wk) of 30 minutes' walking on a 10% negative slope (n = 17) or a 10% positive slope (n = 17), respectively. Measurements were taken before and after the intervention and after 4-week follow-up and included fatigue by Modified Fatigue Impact Scale; mobility by Modified Rivermead Mobility Index; disability by Guy's Neurological Disability Scale; functional activities by 2-Minute Walk Test, Timed 25-Foot Walk test, and Timed Up and Go test; balance indices by Biodex Balance System; and quadriceps and hamstring isometric muscles by torque of left and right knee joints. Analysis of variance with repeated measures was used to investigate the intervention effects on the measurements. After the intervention, significant improvement was found in the downhill group versus the uphill group in terms of fatigue, mobility, and disability indices; functional activities; balance indices; and quadriceps isometric torque (P < .05). The results were stable at 4-week follow-up. Downhill walking on a treadmill may improve muscle performance, functional activity, and balance control in MS patients. These findings support the idea of using eccentric exercise training in MS rehabilitation protocols.
Elavsky, Steriani; McAuley, Edward
2007-01-01
To examine the effects of walking and yoga on multidimensional self-esteem and roles played by self-efficacy, body composition, and physical activity (PA) in changes in esteem. Four-month randomized controlled exercise trial with three arms: walking, yoga, and control. Previously low-active middle-aged women (n=164; M age = 49.9; SD = 3.6). Structured and supervised walking program meeting three times per week for I hour and supervised yoga program meeting twice per week for 90 minutes. Body composition, fitness assessment, and battery of psychologic measures. Panel analysis within a structural equation modeling framework using Mplus 3.0. The walking and yoga interventions failed to enhance global or physical self-esteem but improved subdomain esteem relative to physical condition and strength (for walking) and body attractiveness (for both walking and yoga). Over time the effects of PA, self-efficacy, and body fat on changes in physical self-esteem and global esteem were mediated by changes in physical condition and body attractiveness subdomain esteem. Women reporting greater levels of self-efficacy and PA with lower body fat also reported greater enhancements in subdomain esteem. These results provide support for the hierarchic and multidimensional nature of self-esteem and indicate that middle-aged women may enhance certain aspects of physical self-esteem by participating in PA.
Random walks in weighted networks with a perfect trap: an application of Laplacian spectra.
Lin, Yuan; Zhang, Zhongzhi
2013-06-01
Trapping processes constitute a primary problem of random walks, which characterize various other dynamical processes taking place on networks. Most previous works focused on the case of binary networks, while there is much less related research about weighted networks. In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and Lévy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For Lévy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing Lévy random walks, which have the minimal average of ATTs taken over all
Bounding the Edge Cover Time of Random Walks on Graphs
2011-07-21
34. The Annals of Probability, Vol 16, No. 1, pp. 189-199, 1988. [21] Niels Erik N6rlund. Vorlesungen Uber Diffcrcnzenrechnung. New York, Chelsea, 1954...16, No. 1, pp. 189-199, 1988. [21] Niels Erik N6rlund. Voriesungen Uber Differenzenrcchnung. New York, Chelsea, 1954. [22] Prasad Tetali. "Random
On the genealogy of branching random walks and of directed polymers
Derrida, Bernard; Mottishaw, Peter
2016-08-01
It is well known that the mean-field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite-size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.
Burton Nicola W
2009-07-01
Full Text Available Abstract Background Interventions designed to increase workplace physical activity may not automatically reduce high volumes of sitting, a behaviour independently linked to chronic diseases such as obesity and type II diabetes. This study compared the impact two different walking strategies had on step counts and reported sitting times. Methods Participants were white-collar university employees (n = 179; age 41.3 ± 10.1 years; 141 women, who volunteered and undertook a standardised ten-week intervention at three sites. Pre-intervention step counts (Yamax SW-200 and self-reported sitting times were measured over five consecutive workdays. Using pre-intervention step counts, employees at each site were randomly allocated to a control group (n = 60; maintain normal behaviour, a route-based walking group (n = 60; at least 10 minutes sustained walking each workday or an incidental walking group (n = 59; walking in workday tasks. Workday step counts and reported sitting times were re-assessed at the beginning, mid- and endpoint of intervention and group mean± SD steps/day and reported sitting times for pre-intervention and intervention measurement points compared using a mixed factorial ANOVA; paired sample-t-tests were used for follow-up, simple effect analyses. Results A significant interactive effect (F = 3.5; p t = 3.9, p t = 2.5, p Conclusion Compared to controls, both route and incidental walking increased physical activity in white-collar employees. Our data suggests that workplace walking, particularly through incidental movement, also has the potential to decrease employee sitting times, but there is a need for on-going research using concurrent and objective measures of sitting, standing and walking.
Harmonic maps on amenable groups and a diffusive lower bound for random walks
Lee, James R
2009-01-01
We prove that on any infinite, connected, locally finite, transitive graph G, the probability of the random walk being within $\\eps \\sqrt{t}$ of the origin after t steps is at most $O(\\eps)$. A similar statement holds for finite graphs, up to the relaxation time of the walk. Our approach uses non-constant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok-Korevaar-Schoen theorem on existence of such harmonic maps by constructing them from the heat flow on a Folner set.
Deterministic walks in quenched random environments of chaotic maps
Simula, Tapio [Mathematical Physics Laboratory, Department of Physics, Okayama University, Okayama 700-8530 (Japan); Stenlund, Mikko [Courant Institute of Mathematical Sciences, New York, NY 10012 (United States)], E-mail: mikko@cims.nyu.edu
2009-06-19
This paper concerns the propagation of particles through a quenched random medium. In the one- and two-dimensional models considered, the local dynamics is given by expanding circle maps and hyperbolic toral automorphisms, respectively. The particle motion in both models is chaotic and found to fluctuate about a linear drift. In the proper scaling limit, the cumulative distribution function of the fluctuations converges to a Gaussian one with system-dependent variance while the density function shows no convergence to any function. We have verified our analytical results using extreme precision numerical computations.
Test of Random Walk Behavior in Karachi Stock Exchange
Muhammad Mudassar
2013-05-01
Full Text Available Study was carried out to check the random behavior of the Karachi Stock Exchange (KSE 100 Index during the period of past three financial years to know whether investors could generate abnormal profits during the period or otherwise. Tests used were Runs Test, ADF Test, PP Test and Autocorrelation Function Test. During the study it was found that the performance of KSE 100 Index remained in weak form of inefficiency and investors have been able to generate excessive returns on their investment most of the times.
Kosmidis, Kosmas; Hütt, Marc-Thorsten
2015-01-01
Random walks are one of the best investigated dynamical processes on graphs. A particularly fascinating phenomenon is the scaling relationship of fluctuations $\\sigma $ with the average flux $\\langle f \\rangle $. Here we analyze how network topology and nodes with finite capacity lead to deviations from a simple scaling law $\\sigma \\sim \\langle f \\rangle ^\\alpha$. Sources of randomness are the random walk itself (internal noise) and the fluctuation of the number of walkers (external noise). We obtained exact results for the extreme case of a star network which are indicative of the behavior of large scale systems with a broad degree distribution.The latter are subsequently studied using Monte Carlo simulations. We find that the network heterogeneity amplifies the effects of external noise. By computing the `effective' scaling of each node we show that multiple scaling relationships can coexist in a graph with a heterogeneous degree distribution at an intermediate level of external noise. Finally, we analyze t...
δ-exceedance records and random adaptive walks
Park, Su-Chan; Krug, Joachim
2016-08-01
We study a modified record process where the kth record in a series of independent and identically distributed random variables is defined recursively through the condition {Y}k\\gt {Y}k-1-{δ }k-1 with a deterministic sequence {δ }k\\gt 0 called the handicap. For constant {δ }k\\equiv δ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences {δ }k, focusing in particular on the case when {δ }k matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when {δ }k is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where {δ }k corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.
Algorithmic Randomness as Foundation of Inductive Reasoning and Artificial Intelligence
Hutter, Marcus
2011-01-01
This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and philosophy. Intuitively, randomness is a lack of order or predictability. If randomness is the opposite of determinism, then algorithmic randomness is the opposite of computability. Besides many other things, these concepts have been used to quantify Ockham's razor, solve the induction problem, and define intelligence.
Interval-walking training for the treatment of type 2 diabetes: a randomized, controlled trial
Karstoft, Kristian; Winding, Kamilla; Knudsen, Sine H.
.05) and maximum (Ptraining is feasible in type 2 diabetes patients. CWT offsets the deterioration in glycemia seen in the control group, and IWT is superior to energy......Formål: To evaluate the feasibility of free-living walking training in type 2 diabetes patients, and to investigate the effects of interval-walking training (IWT) versus continuous-walking training (CWT) upon self reported health, physical fitness, body composition and glycemic control. Metoder......: Subjects with type 2 diabetes were randomized to a control (n = 8), CWT (n = 12), or IWT group (n = 12). Training groups were prescribed five sessions per week (60 min/session) and were controlled with an accelerometer and a heart-rate monitor. CWT performed all training at moderate intensity, whereas IWT...
A random walk on water (Henry Darcy Medal Lecture)
Koutsoyiannis, D.
2009-04-01
Randomness and uncertainty had been well appreciated in hydrology and water resources engineering in their initial steps as scientific disciplines. However, this changed through the years and, following other geosciences, hydrology adopted a naïve view of randomness in natural processes. Such a view separates natural phenomena into two mutually exclusive types, random or stochastic, and deterministic. When a classification of a specific process into one of these two types fails, then a separation of the process into two different, usually additive, parts is typically devised, each of which may be further subdivided into subparts (e.g., deterministic subparts such as periodic and aperiodic or trends). This dichotomous logic is typically combined with a manichean perception, in which the deterministic part supposedly represents cause-effect relationships and thus is physics and science (the "good"), whereas randomness has little relationship with science and no relationship with understanding (the "evil"). Probability theory and statistics, which traditionally provided the tools for dealing with randomness and uncertainty, have been regarded by some as the "necessary evil" but not as an essential part of hydrology and geophysics. Some took a step further to banish them from hydrology, replacing them with deterministic sensitivity analysis and fuzzy-logic representations. Others attempted to demonstrate that irregular fluctuations observed in natural processes are au fond manifestations of underlying chaotic deterministic dynamics with low dimensionality, thus attempting to render probabilistic descriptions unnecessary. Some of the above recent developments are simply flawed because they make erroneous use of probability and statistics (which, remarkably, provide the tools for such analyses), whereas the entire underlying logic is just a false dichotomy. To see this, it suffices to recall that Pierre Simon Laplace, perhaps the most famous proponent of determinism in
Efficient quantum walk on a quantum processor
Qiang, Xiaogang; Loke, Thomas; Montanaro, Ashley; Aungskunsiri, Kanin; Zhou, Xiao-Qi; O'Brien, Jeremy; Wang, Jingbo; Matthews, Jonathan
2016-01-01
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise quantum walks have shown much potential as a frame- work for developing new quantum algorithms. In this paper, we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs ef...
A Directed Continuous Time Random Walk Model with Jump Length Depending on Waiting Time
Long Shi
2014-01-01
Full Text Available In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density function P(x,t of finding the walker at position x at time t is completely determined by the Laplace transform of the probability density function φ(t of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.
Random walks across the sea: the origin of rogue waves?
Birkholz, Simon; Veselić, Ivan; Demircan, Ayhan; Steinmeyer, Günter
2015-01-01
Ocean rogue waves are large and suddenly appearing surface gravity waves, which may cause severe damage to ships and other maritime structures. Despite years of research, the exact origin of rogue waves is still disputed. Linear interference of waves with random phase has often been cited as one possible explanation, but apparently does not satisfactorily explain the probability of extreme events in the ocean. Other explanations therefore suggested a decisive role of a nonlinearity in the system. Here we show that linear interference of a finite and variable number of waves may very well explain the heavy tail in the wave height distribution. Our model can explain all prototypical ocean rogue waves reported so far, including the "three sisters" as well as rogue holes. We further suggest nonlinear time series analysis for estimation of the characteristic number of interfering waves for a given sea state. If ocean dynamics is ruled by interference of less than ten waves, rogue waves cannot appear as a matter of...
Mikosch, Thomas Valentin; Moser, Martin
2013-01-01
We investigate the maximum increment of a random walk with heavy-tailed jump size distribution. Here heavy-tailedness is understood as regular variation of the finite-dimensional distributions. The jump sizes constitute a strictly stationary sequence. Using a continuous mapping argument acting on...... on the point processes of the normalized jump sizes, we prove that the maximum increment of the random walk converges in distribution to a Fréchet distributed random variable....
Random and Directed Walk-Based Top- Queries in Wireless Sensor Networks
Jun-Song Fu
2015-05-01
Full Text Available In wireless sensor networks, filter-based top- query approaches are the state-of-the-art solutions and have been extensively researched in the literature, however, they are very sensitive to the network parameters, including the size of the network, dynamics of the sensors’ readings and declines in the overall range of all the readings. In this work, a random walk-based top- query approach called RWTQ and a directed walk-based top- query approach called DWTQ are proposed. At the beginning of a top- query, one or several tokens are sent to the specific node(s in the network by the base station. Then, each token walks in the network independently to record and process the readings in a random or directed way. A strategy of choosing the “right” way in DWTQ is carefully designed for the token(s to arrive at the high-value regions as soon as possible. When designing the walking strategy for DWTQ, the spatial correlations of the readings are also considered. Theoretical analysis and simulation results indicate that RWTQ and DWTQ both are very robust against these parameters discussed previously. In addition, DWTQ outperforms TAG, FILA and EXTOK in transmission cost, energy consumption and network lifetime.
Central limit theorem for biased random walk on multi-type Galton-Watson trees
Dembo, Amir
2010-01-01
Let T be a rooted multi-type Galton-Watson (MGW) tree of finitely many types with at least one offspring at each vertex, and an offspring distribution with exponential tails. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/(lambda+d(v)), and to each of the offspring with probability 1/(lambda+d(v)). This walk is recurrent for lambda >= rho and transient for 0 <= lambda < rho, with rho the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. We prove the following quenched CLT for the critical value lambda = rho: for almost every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to infinity to a deterministic positive multiple of a reflected Brownian motion. Following the approach of Peres and Zeitouni (2008) for Galton-Watson trees, our proof is based on a new explicit description of a reversing measure for the walk...
Musho, M.K.; Kozak, J.J.
1984-10-01
A method is presented for calculating exactly the relative width (sigma/sup 2/)/sup 1/2//
Schauer, Michael; Mauritz, Karl-Heinz
2003-11-01
To demonstrate the effect of rhythmical auditory stimulation in a musical context for gait therapy in hemiparetic stroke patients, when the stimulation is played back measure by measure initiated by the patient's heel-strikes (musical motor feedback). Does this type of musical feedback improve walking more than a less specific gait therapy? The randomized controlled trial considered 23 registered stroke patients. Two groups were created by randomization: the control group received 15 sessions of conventional gait therapy and the test group received 15 therapy sessions with musical motor feedback. Inpatient rehabilitation hospital. Median post-stroke interval was 44 days and the patients were able to walk without technical aids with a speed of approximately 0.71 m/s. Gait velocity, step duration, gait symmetry, stride length and foot rollover path length (heel-on-toe-off distance). The test group showed more mean improvement than the control group: stride length increased by 18% versus 0%, symmetry deviation decreased by 58% versus 20%, walking speed increased by 27% versus 4% and rollover path length increased by 28% versus 11%. Musical motor feedback improves the stroke patient's walk in selected parameters more than conventional gait therapy. A fixed memory in the patient's mind about the song and its timing may stimulate the improvement of gait even without the presence of an external pacemaker.
Kittas, Aristotelis; Delobelle, Aurélien; Schmitt, Sabrina; Breuhahn, Kai; Guziolowski, Carito; Grabe, Niels
2016-01-01
An effective means to analyze mRNA expression data is to take advantage of established knowledge from pathway databases, using methods such as pathway-enrichment analyses. However, pathway databases are not case-specific and expression data could be used to infer gene-regulation patterns in the context of specific pathways. In addition, canonical pathways may not always describe the signaling mechanisms properly, because interactions can frequently occur between genes in different pathways. Relatively few methods have been proposed to date for generating and analyzing such networks, preserving the causality between gene interactions and reasoning over the qualitative logic of regulatory effects. We present an algorithm (MCWalk) integrated with a logic programming approach, to discover subgraphs in large-scale signaling networks by random walks in a fully automated pipeline. As an exemplary application, we uncover the signal transduction mechanisms in a gene interaction network describing hepatocyte growth factor-stimulated cell migration and proliferation from gene-expression measured with microarray and RT-qPCR using in-house perturbation experiments in a keratinocyte-fibroblast co-culture. The resulting subgraphs illustrate possible associations of hepatocyte growth factor receptor c-Met nodes, differentially expressed genes and cellular states. Using perturbation experiments and Answer Set programming, we are able to select those which are more consistent with the experimental data. We discover key regulator nodes by measuring the frequency with which they are traversed when connecting signaling between receptors and significantly regulated genes and predict their expression-shift consistently with the measured data. The Java implementation of MCWalk is publicly available under the MIT license at: https://bitbucket.org/akittas/biosubg.
Identifying co-targets to fight drug resistance based on a random walk model
Chen Liang-Chun
2012-01-01
Full Text Available Abstract Background Drug resistance has now posed more severe and emergent threats to human health and infectious disease treatment. However, wet-lab approaches alone to counter drug resistance have so far still achieved limited success due to less knowledge about the underlying mechanisms of drug resistance. Our approach apply a heuristic search algorithm in order to extract active network under drug treatment and use a random walk model to identify potential co-targets for effective antibacterial drugs. Results We use interactome network of Mycobacterium tuberculosis and gene expression data which are treated with two kinds of antibiotic, Isoniazid and Ethionamide as our test data. Our analysis shows that the active drug-treated networks are associated with the trigger of fatty acid metabolism and synthesis and nicotinamide adenine dinucleotide (NADH-related processes and those results are consistent with the recent experimental findings. Efflux pumps processes appear to be the major mechanisms of resistance but SOS response is significantly up-regulation under Isoniazid treatment. We also successfully identify the potential co-targets with literature confirmed evidences which are related to the glycine-rich membrane, adenosine triphosphate energy and cell wall processes. Conclusions With gene expression and interactome data supported, our study points out possible pathways leading to the emergence of drug resistance under drug treatment. We develop a computational workflow for giving new insights to bacterial drug resistance which can be gained by a systematic and global analysis of the bacterial regulation network. Our study also discovers the potential co-targets with good properties in biological and graph theory aspects to overcome the problem of drug resistance.
Limit distributions of random walks on stochastic matrices
Santanu Chakraborty; Arunava Mukherjea
2014-11-01
Problems similar to Ann. Prob. 22 (1994) 424–430 and J. Appl. Prob. 23 (1986) 1019–1024 are considered here. The limit distribution of the sequence $X_{n}X_{n−1}\\ldots X_{1}$, where $(X_{n})_{n≥ 1}$ is a sequence of i.i.d. 2 × 2 stochastic matrices with each $X_{n}$ distributed as , is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components $C_{n}$ and $D_{n}$ (not necessarily independent), $(C_{n}, D_{n})$ being the first column of $X_{n}$, have the same (or different) Bernoulli distributions. Thus $(C_{n}, D_{n})$ is valued in $\\{0, r\\}^{2}$, where is a positive real number. If for a given positive real , with $0 < r ≤ \\frac{1}{2}$, $r^{-1}C_{n}$ and $r^{-1}D_{n}$ are each Bernoulli with parameters $p_{1}$ and $p_{2}$ respectively, $0 < p_{1}$, $p_{2} < 1$ (which means $C_{n}\\sim p_{1}_{\\{r\\}} + (1 - p_{1})_{\\{0\\}}$ and $D_{n} \\sim p_{2}_{\\{r\\}} + (1 - p_{2})_{\\{0\\}}$), then it is well known that the weak limit of the sequence $^{n}$ exists whose support is contained in the set of all 2 × 2 rank one stochastic matrices. We show that $S()$, the support of , consists of the end points of a countable number of disjoint open intervals and we have calculated the -measure of each such point. To the best of our knowledge, these results are new.
Randomized Speedup of the Bellman-Ford Algorithm
Bannister, Michael J
2011-01-01
We describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.
MATRIX ALGEBRA ALGORITHM OF STRUCTURE RANDOM RESPONSE NUMERICAL CHARACTERISTICS
无
2003-01-01
A new algorithm of structure random response numerical characteristics, named as matrix algebra algorithm of structure analysis is presented.Using the algorithm, structure random response numerical characteristics can easily be got by directly solving linear matrix equations rather than structure motion differential equations.Moreover, in order to solve the corresponding linear matrix equations, the numerical integration fast algorithm is presented.Then according to the results, dynamic design and life-span estimation can be done.Besides, the new algorithm can solve non-proportion damp structure response.
On Origin of Power-Law Distributions in Self-Organized Criticality from Random Walk Treatment
CHANG Zhe; CAO Xiao-Feng; GUAN Cheng-Bo; DENG Zong-Wei; HUANG Chao-Guang; YANG Chun-Bin; LI Xin
2008-01-01
The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random walk process in a one-dimensional lattice. We assume that the variation of the number of active sites has three possibilities in each update: to increase by 1 with probability fl, to decrease by 1 with probability f2, or remain unchanged with probability 1-f1-f2. This mimics the dynamics in the system. Power-law distributions of the lifetime are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in self-organized criticality may be caused by the balance of competitive interactions.
Ingo, Carson; Sui, Yi; Chen, Yufen; Parrish, Todd; Webb, Andrew; Ronen, Itamar
2015-03-01
In this paper, we provide a context for the modeling approaches that have been developed to describe non-Gaussian diffusion behavior, which is ubiquitous in diffusion weighted magnetic resonance imaging of water in biological tissue. Subsequently, we focus on the formalism of the continuous time random walk theory to extract properties of subdiffusion and superdiffusion through novel simplifications of the Mittag-Leffler function. For the case of time-fractional subdiffusion, we compute the kurtosis for the Mittag-Leffler function, which provides both a connection and physical context to the much-used approach of diffusional kurtosis imaging. We provide Monte Carlo simulations to illustrate the concepts of anomalous diffusion as stochastic processes of the random walk. Finally, we demonstrate the clinical utility of the Mittag-Leffler function as a model to describe tissue microstructure through estimations of subdiffusion and kurtosis with diffusion MRI measurements in the brain of a chronic ischemic stroke patient.
Stationary Probability and First-Passage Time of Biased Random Walk
Li, Jing-Wen; Tang, Shen-Li; Xu, Xin-Ping
2016-09-01
In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively (0 ⩽ p, q ⩽ 1, p + q = 1). We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F ˜ (N - 1)γ, where the exponent γ = 2 for N |p - q|-1. Our study sheds useful insights into the biased random-walk process. Supported by the National Natural Science Foundation of China under Grant No. 11205110, Shanghai Key Laboratory of Intelligent Information Processing (IIPL-2011-009), and Innovative Training Program for College Students under Grant No. 2015xj070
Castell, Fabienne; Mélot, Clothilde
2012-01-01
Let $(X_t, t \\geq 0)$ be an $\\alpha$-stable random walk with values in $\\Z^d$. Let $l_t(x) = \\int_0^t \\delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \\sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \\geq r_t)$ for all scales $r_t \\gg \\E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and K\\"onig 2010, and Laurent 2012.
Elephant random walks and their connection to Pólya-type urns
Baur, Erich; Bertoin, Jean
2016-11-01
In this paper, we explain the connection between the elephant random walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced in [Phys. Rev. E 70, 045101 (2004), 10.1103/PhysRevE.70.045101] to study memory effects in a highly non-Markovian setting. More specifically, the ERW is a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a memory parameter p between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of p . Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.
Observing random walks of atoms in buffer gas through resonant light absorption
Aoki, Kenichiro
2016-01-01
Using resonant light absorption, random walk motions of rubidium atoms in nitrogen buffer gas are observed directly. The transmitted light intensity through atomic vapor is measured and its spectrum is obtained, down to orders of magnitude below the shot noise level to detect fluctuations caused by atomic motions. To understand the measured spectra, the spectrum for atoms performing random walks in a gaussian light beam is computed and its analytical form is obtained. The spectrum has $1/f^2$ ($f$: frequency) behavior at higher frequencies, crossing over to a different, but well defined behavior at lower frequencies. The properties of this theoretical spectrum agree excellently with the measured spectrum. This understanding also enables us to obtain the diffusion constant, the photon cross section of atoms in buffer gas and the atomic number density, from a single spectral measurement. We further discuss other possible applications of our experimental method and analysis.
Lagging/Leading Coupled Continuous Time Random Walks, Renewal Times and their Joint Limits
Straka, Peter
2010-01-01
Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW) model for diffusion, including the possibility of anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of sequences of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. We identify two different limit processes $X_t$ and $Y_t$ when waiting times precede or follow jumps, respectively. In the limiting procedure, we keep track of the renewal times of the CTRWs and hence find two more limit processes. Finally, we calculate the joint law of all four limit processes evaluated at a fixed time $t$.
The continuous time random walk, still trendy: fifty-year history, state of art and outlook
Kutner, Ryszard; Masoliver, Jaume
2017-03-01
In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on">http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on] containing articles which show incredible possibilities of the CTRWs. Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver.
Continuous Time Random Walks for Non-Local Radial Solute Transport
Dentz, Marco; Borgne, Tanguy le
2016-01-01
This paper derives and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile-immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection-dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneou...
CHI Bing; LI Hong; FANG Dong
2007-01-01
Plume concentration prediction is one of the main contents of radioactive consequence assessment for early emergency response to nuclear accidents. Random characteristics of atmospheric diffusion itself was described, a random walk model of atmospheric diffusion (Random Walk) was introduced and compared with the Lagrangian puff model (RIMPUFF) in the nuclear emergency decision support system (RODOS) developed by the European Community for verification. The results show the concentrations calculated by the two models are quite close except that the plume area calculated by Random Walk is a little smaller than that by RIMPUFF. The random walk model for atmospheric diffusion can simulate the atmospheric diffusion in case of nuclear accidents, and provide more actual information for early emergency and consequence assessment as one of the atmospheric diffusion module of the nuclear emergency decision support system.
α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND L(E)VY PROCESSES
ZHANG HUIZENG; ZHAO MINZHI; YING JIANGANG
2005-01-01
The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasisymmetric if and only if X is not α-recurrent for all α＜ 0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.
Step-Step Random Walk Network with Power-Law Clique-Degree Distribution
YANG Han-Xin; WANG Bing-Hong; LIU Jian-Guo; HAN Xiao-Pu; ZHOU Tao
2008-01-01
We propose a simple mechanism for generating scale-free networks with degree exponent γ=3, where the new node is connected to the existing nodes by step-by-step random walk. It is found that the clique-degree distribution based on our model obeys a power-law form, which is in agreement with the recently empirical evidences. In addition, our model displays the small-world effect and the hierarchical structure.
Yan Xia REN
2008-01-01
The global supports of super-Poisson processes and super-random walks with a branching mechanism ψ(z)=z2 and constant branching rate are known to be noncompact. It turns out that, for any spatially dependent branching rate, this property remains true. However, the asymptotic extinction property for these two kinds of superprocesses depends on the decay rate of the branching-rate function at infinity.
Kullgren, Jeffrey T.; Harkins, Kristin A.; Bellamy, Scarlett L.; Gonzales, Amy; Tao, Yuanyuan; Zhu, Jingsan; Volpp, Kevin G.; Asch, David A.; Heisler, Michele; Karlawish, Jason
2014-01-01
Background: Financial incentives and peer networks could be delivered through eHealth technologies to encourage older adults to walk more. Methods: We conducted a 24-week randomized trial in which 92 older adults with a computer and Internet access received a pedometer, daily walking goals, and weekly feedback on goal achievement. Participants…
Kullgren, Jeffrey T.; Harkins, Kristin A.; Bellamy, Scarlett L.; Gonzales, Amy; Tao, Yuanyuan; Zhu, Jingsan; Volpp, Kevin G.; Asch, David A.; Heisler, Michele; Karlawish, Jason
2014-01-01
Background: Financial incentives and peer networks could be delivered through eHealth technologies to encourage older adults to walk more. Methods: We conducted a 24-week randomized trial in which 92 older adults with a computer and Internet access received a pedometer, daily walking goals, and weekly feedback on goal achievement. Participants…
Distributed clone detection in static wireless sensor networks: random walk with network division.
Wazir Zada Khan
Full Text Available Wireless Sensor Networks (WSNs are vulnerable to clone attacks or node replication attacks as they are deployed in hostile and unattended environments where they are deprived of physical protection, lacking physical tamper-resistance of sensor nodes. As a result, an adversary can easily capture and compromise sensor nodes and after replicating them, he inserts arbitrary number of clones/replicas into the network. If these clones are not efficiently detected, an adversary can be further capable to mount a wide variety of internal attacks which can emasculate the various protocols and sensor applications. Several solutions have been proposed in the literature to address the crucial problem of clone detection, which are not satisfactory as they suffer from some serious drawbacks. In this paper we propose a novel distributed solution called Random Walk with Network Division (RWND for the detection of node replication attack in static WSNs which is based on claimer-reporter-witness framework and combines a simple random walk with network division. RWND detects clone(s by following a claimer-reporter-witness framework and a random walk is employed within each area for the selection of witness nodes. Splitting the network into levels and areas makes clone detection more efficient and the high security of witness nodes is ensured with moderate communication and memory overheads. Our simulation results show that RWND outperforms the existing witness node based strategies with moderate communication and memory overheads.
Drug-target interaction prediction by random walk on the heterogeneous network.
Chen, Xing; Liu, Ming-Xi; Yan, Gui-Ying
2012-07-01
Predicting potential drug-target interactions from heterogeneous biological data is critical not only for better understanding of the various interactions and biological processes, but also for the development of novel drugs and the improvement of human medicines. In this paper, the method of Network-based Random Walk with Restart on the Heterogeneous network (NRWRH) is developed to predict potential drug-target interactions on a large scale under the hypothesis that similar drugs often target similar target proteins and the framework of Random Walk. Compared with traditional supervised or semi-supervised methods, NRWRH makes full use of the tool of the network for data integration to predict drug-target associations. It integrates three different networks (protein-protein similarity network, drug-drug similarity network, and known drug-target interaction networks) into a heterogeneous network by known drug-target interactions and implements the random walk on this heterogeneous network. When applied to four classes of important drug-target interactions including enzymes, ion channels, GPCRs and nuclear receptors, NRWRH significantly improves previous methods in terms of cross-validation and potential drug-target interaction prediction. Excellent performance enables us to suggest a number of new potential drug-target interactions for drug development.
Forecasting Performance of Random Walk with Drift and Feed Forward Neural Network Models
Augustine D. Pwasong
2015-08-01
Full Text Available In this study, linear and nonlinear methods were used to model forecasting performances on the daily crude oil production data of the Nigerian National Petroleum Corporation (NNPC. The linear model considered here is the random walk with drift, while the nonlinear model is the feed forward neural network model. The results indicate that nonlinear methods have better forecasting performance greater than linear methods based on the mean error square sense. The root mean square error (RMSE and the mean absolute error (MAE were applied to ascertain the assertion that nonlinear methods have better forecasting performance greater than linear methods. Autocorrelation functions emerging from the increment series, that is, log difference series and difference series of the daily crude oil production data of the NNPC indicates significant autocorrelations. As a result of the foregoing assertion we deduced that the daily crude oil production series of the NNPC is not firmly a random walk process. However, the original daily crude oil production series of the NNPC was considered to be a random walk with drift when we are not trying to forecast immediate values. The analysis for this study was simulated using MATLAB software, version 8.03
Hierarchical random walks in trace fossils and the origin of optimal search behavior.
Sims, David W; Reynolds, Andrew M; Humphries, Nicolas E; Southall, Emily J; Wearmouth, Victoria J; Metcalfe, Brett; Twitchett, Richard J
2014-07-29
Efficient searching is crucial for timely location of food and other resources. Recent studies show that diverse living animals use a theoretically optimal scale-free random search for sparse resources known as a Lévy walk, but little is known of the origins and evolution of foraging behavior and the search strategies of extinct organisms. Here, using simulations of self-avoiding trace fossil trails, we show that randomly introduced strophotaxis (U-turns)--initiated by obstructions such as self-trail avoidance or innate cueing--leads to random looping patterns with clustering across increasing scales that is consistent with the presence of Lévy walks. This predicts that optimal Lévy searches may emerge from simple behaviors observed in fossil trails. We then analyzed fossilized trails of benthic marine organisms by using a novel path analysis technique and find the first evidence, to our knowledge, of Lévy-like search strategies in extinct animals. Our results show that simple search behaviors of extinct animals in heterogeneous environments give rise to hierarchically nested Brownian walk clusters that converge to optimal Lévy patterns. Primary productivity collapse and large-scale food scarcity characterizing mass extinctions evident in the fossil record may have triggered adaptation of optimal Lévy-like searches. The findings suggest that Lévy-like behavior has been used by foragers since at least the Eocene but may have a more ancient origin, which might explain recent widespread observations of such patterns among modern taxa.
Large deviations for self-intersection local times of stable random walks
Laurent, Clément
2010-01-01
Let $(X_t,t\\geq 0)$ be a random walk on $\\mathbb{Z}^d$. Let $ l_T(x)= \\int_0^T \\delta_x(X_s)ds$ the local time at the state $x$ and $ I_T= \\sum\\limits_{x\\in\\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \\cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and M\\"orters obtain in \\cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \\cite{Asselah5} a large deviations principle for the SILT with $q=2$. We extend these results to an $\\alpha$-stable process (i.e. $\\alpha\\in]0,2]$) in the case where $q(d-\\alpha)\\geq d$.
Zhuo Qi Lee
Full Text Available Biased random walk has been studied extensively over the past decade especially in the transport and communication networks communities. The mean first passage time (MFPT of a biased random walk is an important performance indicator in those domains. While the fundamental matrix approach gives precise solution to MFPT, the computation is expensive and the solution lacks interpretability. Other approaches based on the Mean Field Theory relate MFPT to the node degree alone. However, nodes with the same degree may have very different local weight distribution, which may result in vastly different MFPT. We derive an approximate bound to the MFPT of biased random walk with short relaxation time on complex network where the biases are controlled by arbitrarily assigned node weights. We show that the MFPT of a node in this general case is closely related to not only its node degree, but also its local weight distribution. The MFPTs obtained from computer simulations also agree with the new theoretical analysis. Our result enables fast estimation of MFPT, which is useful especially to differentiate between nodes that have very different local node weight distribution even though they share the same node degrees.
Empirical scaling of the length of the longest increasing subsequences of random walks
Mendonça, J. Ricardo G.
2017-02-01
We provide Monte Carlo estimates of the scaling of the length L n of the longest increasing subsequences of n-step random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, {{L}n}∼ {{n}θ} with the leading scaling exponent 0.60≲ θ ≲ 0.69 for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and θ ≃ 0.57 for distributions of finite variance, irrespective of the particular distribution. The results are consistent with existing rigorous bounds for θ, although in a somewhat surprising manner. For random walks with step lengths of finite variance, we conjecture that the correct asymptotic behavior of L n is given by \\sqrt{n}\\ln n , and also propose the form for the subleading asymptotics. The distribution of L n was found to follow a simple scaling form with scaling functions that vary with θ. Accordingly, when the step lengths are of finite variance they seem to be universal. The nature of this scaling remains unclear, since we lack a working model, microscopic or hydrodynamic, for the behavior of the length of the longest increasing subsequences of random walks.
Irreversible Markov chain Monte Carlo algorithm for self-avoiding walk
Hu, Hao; Chen, Xiaosong; Deng, Youjin
2017-02-01
We formulate an irreversible Markov chain Monte Carlo algorithm for the self-avoiding walk (SAW), which violates the detailed balance condition and satisfies the balance condition. Its performance improves significantly compared to that of the Berretti-Sokal algorithm, which is a variant of the Metropolis-Hastings method. The gained efficiency increases with spatial dimension (D), from approximately 10 times in 2D to approximately 40 times in 5D. We simulate the SAW on a 5D hypercubic lattice with periodic boundary conditions, for a linear system with a size up to L = 128, and confirm that as for the 5D Ising model, the finite-size scaling of the SAW is governed by renormalized exponents, v* = 2/ d and γ/ v* = d/2. The critical point is determined, which is approximately 8 times more precise than the best available estimate.
Statistics at the tip of a branching random walk and the delay of traveling waves
Brunet, É.; Derrida, B.
2009-09-01
We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses.
Topics in Randomized Algorithms for Numerical Linear Algebra
Holodnak, John T.
In this dissertation, we present results for three topics in randomized algorithms. Each topic is related to random sampling. We begin by studying a randomized algorithm for matrix multiplication that randomly samples outer products. We show that if a set of deterministic conditions is satisfied, then the algorithm can compute the exact product. In addition, we show probabilistic bounds on the two norm relative error of the algorithm. two norm relative error of the algorithm. In the second part, we discuss the sensitivity of leverage scores to perturbations. Leverage scores are scalar quantities that give a notion of importance to the rows of a matrix. They are used as sampling probabilities in many randomized algorithms. We show bounds on the difference between the leverage scores of a matrix and a perturbation of the matrix. In the last part, we approximate functions over an active subspace of parameters. To identify the active subspace, we apply an algorithm that relies on a random sampling scheme. We show bounds on the accuracy of the active subspace identification algorithm and construct an approximation to a function with 3556 parameters using a ten-dimensional active subspace.
Phase Transitions in Sampling Algorithms and the Underlying Random Structures
Randall, Dana
Sampling algorithms based on Markov chains arise in many areas of computing, engineering and science. The idea is to perform a random walk among the elements of a large state space so that samples chosen from the stationary distribution are useful for the application. In order to get reliable results, we require the chain to be rapidly mixing, or quickly converging to equilibrium. For example, to sample independent sets in a given graph G, the so-called hard-core lattice gas model, we can start at any independent set and repeatedly add or remove a single vertex (if allowed). By defining the transition probabilities of these moves appropriately, we can ensure that the chain will converge to a use- ful distribution over the state space Ω. For instance, the Gibbs (or Boltzmann) distribution, parameterized by Λ> 0, is defined so that p(Λ) = π(I) = Λ|I| /Z, where Z = sum_{J in Ω} Λ^{|J|} is the normalizing constant known as the partition function. An interesting phenomenon occurs as Λ is varied. For small values of Λ, local Markov chains converge quickly to stationarity, while for large values, they are prohibitively slow. To see why, imagine the underlying graph G is a region of the Cartesian lattice. Large independent sets will dominate the stationary distribution π when Λ is sufficiently large, and yet it will take a very long time to move from an independent set lying mostly on the odd sublattice to one that is mostly even. This phenomenon is well known in the statistical physics community, and characterizes by a phase transition in the underlying model.
Clonal Selection Algorithm Based Iterative Learning Control with Random Disturbance
Yuanyuan Ju
2013-01-01
Full Text Available Clonal selection algorithm is improved and proposed as a method to solve optimization problems in iterative learning control. And a clonal selection algorithm based optimal iterative learning control algorithm with random disturbance is proposed. In the algorithm, at the same time, the size of the search space is decreased and the convergence speed of the algorithm is increased. In addition a model modifying device is used in the algorithm to cope with the uncertainty in the plant model. In addition a model is used in the algorithm cope with the uncertainty in the plant model. Simulations show that the convergence speed is satisfactory regardless of whether or not the plant model is precise nonlinear plants. The simulation test verify the controlled system with random disturbance can reached to stability by using improved iterative learning control law but not the traditional control law.
Continuous-time random walks with reset events. Historical background and new perspectives
Montero, Miquel; Masó-Puigdellosas, Axel; Villarroel, Javier
2017-09-01
In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.
Symmetry in stochasticity: Random walk models of large-scale structure
Ravi K Sheth
2011-07-01
This paper describes the insights gained from the excursion set approach, in which various questions about the phenomenology of large-scale structure formation can be mapped to problems associated with the ﬁrst crossing distribution of appropriately deﬁned barriers by random walks. Much of this is summarized in R K Sheth, AIP Conf. Proc. 1132, 158 (2009). So only a summary is given here, and instead a few new excursion set related ideas and results which are not published elsewhere are presented. One is a generalization of the formation time distribution to the case in which formation corresponds to the time when half the mass was ﬁrst assembled in pieces, each of which was at least 1/ times the ﬁnal mass, and where ≥ 2; another is an analysis of the ﬁrst crossing distribution of the Ornstein–Uhlenbeck process. The ﬁrst derives from the mirror-image symmetry argument for random walks which Chandrasekhar described so elegantly in 1943; the second corrects a misuse of this argument. Finally, some discussion of the correlated steps and correlated walks assumptions associated with the excursion set approach, and the relation between these and peaks theory are also included. These are problems in which Chandra’s mirror-image symmetry is broken.
Solvable continuous-time random walk model of the motion of tracer particles through porous media.
Fouxon, Itzhak; Holzner, Markus
2016-08-01
We consider the continuous-time random walk (CTRW) model of tracer motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported by Holzner et al. [M. Holzner et al., Phys. Rev. E 92, 013015 (2015)PLEEE81539-375510.1103/PhysRevE.92.013015]. The particle's passing through one channel is modeled as one step of the walk. The step (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability, mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step duration τ, length l, and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter α. Depending on that parameter, all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds, which was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer diffusion in the porous medium is considered.
Genetic algorithms as global random search methods
Peck, Charles C.; Dhawan, Atam P.
1995-01-01
Genetic algorithm behavior is described in terms of the construction and evolution of the sampling distributions over the space of candidate solutions. This novel perspective is motivated by analysis indicating that the schema theory is inadequate for completely and properly explaining genetic algorithm behavior. Based on the proposed theory, it is argued that the similarities of candidate solutions should be exploited directly, rather than encoding candidate solutions and then exploiting their similarities. Proportional selection is characterized as a global search operator, and recombination is characterized as the search process that exploits similarities. Sequential algorithms and many deletion methods are also analyzed. It is shown that by properly constraining the search breadth of recombination operators, convergence of genetic algorithms to a global optimum can be ensured.
A novel compound biped locomotion algorithm for humanoid robots to realize biped walking
Ruiwu XIN; Nanfeng XIAO
2009-01-01
In this paper,a compound biped locomotion algorithm for a humanoid robot under development is pre-sented.This paper is organized in two main parts.In the first part,it mainly focuses on the structural design for the humanoid.In the second part.the compound biped locomotion algorithm is presented based on the reference motion and reference Zero Moment Point(ZMP).This novel algorithm includes calculation of the upper body motion and trajectory of the Center of Gravity(COG) of the robot.First,disturbances from the environment are eliminated by the compensational movement of the upper body; then based on the error between a reference ZMP and the real ZMP as well as the relation between ZMP and CoG,the CoG error is calculated,thus leading to the CoG trajectory.Then,the motion of the robot converges to its reference motion,generating stable biped walking.Because the calculation of upper body motion and tra-jectory of CoG both depend on the reference motion,they can work in parallel,thus providing double insurances against the robot's collapse.Finally,the algorithm is validated by different kinds of simulation experiments.
Powder Mixing Simulation Using Random Walk Model in Eco-material Preparation
ZHANG Ji-ru; HU Zai-liang; LIU Zu-de
2004-01-01
The eco-material composition is not well-distributed in preparation. The eco-material samples were taken for computer image analysis, and its particle numbers and appearance parameters were measured. Based on the mechanism of connective mixing and diffusion, the particles distribution was simulated by a computer using the random walk with Levy flight. The results show that the eco-material microstructure simulated by a computer has an idealized porous structure. The particles distribution has a cluster characteristic that changes with the different size and number of particles in Levy flight trajectory. Each cluster consists of a collection of clusters and shows a structure of self-similar cluster,hence presents a well-defined fractal property. The results obtained from SEM observation are in good agreement with the numerical simulations, and show that the convective mixing presents in the Levy flight walk.
A New Type of Limit Theorems for the One-Dimensional Quantum Random Walk
Konno, N
2002-01-01
In this paper we consider the one-dimensional quantum random walk X^{\\phi}_n at time n starting from initial qubit state \\phi determined by 2 \\times 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X^{\\phi}_n. The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state \\phi. As a consequence of the above results, we present a new type of limit theorems for the Hadamard walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X^{\\phi}_n/n \\Rightarrow Z^{\\phi} where Z^{\\phi} has a density 1 / \\pi (1-x^2) \\sqrt{1-2x^2} for x \\in (- \\sqrt{2}/2, \\sqrt{2}/2). Moreover we discuss some known simulation results based on our limit theorems.
Excursions and local limit theorems for Bessel-like random walks
Alexander, Kenneth S
2009-01-01
We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -\\delta/2x + o(1/x) with \\delta > -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}\\phi(n), where c = (3+\\delta)/2 and \\phi is a slowly varying function given explicitly in terms of the o(1/x) terms.
Unbinding of mutually avoiding random walks and two-dimensional quantum gravity
Carlon, Enrico; Baiesi, Marco
2004-12-01
We analyze the unbinding transition for a two-dimensional lattice polymer in which the constituent strands are mutually avoiding random walks. At low temperatures the strands are bound and form a single self-avoiding walk. We show that unbinding in this model is a strong first order transition. The entropic exponents associated with denaturated loops and end-segment distributions show sharp differences at the transition point and in the high temperature phase. Their values can be deduced from some exact arguments relying on a conformal mapping of copolymer networks into a fluctuating geometry, i.e., in the presence of quantum gravity. An excellent agreement between analytical and numerical estimates is observed for all cases analyzed.
Physical implementation of quantum walks
Manouchehri, Kia
2013-01-01
Given the extensive application of random walks in virtually every science related discipline, we may be at the threshold of yet another problem solving paradigm with the advent of quantum walks. Over the past decade, quantum walks have been explored for their non-intuitive dynamics, which may hold the key to radically new quantum algorithms. This growing interest has been paralleled by a flurry of research into how one can implement quantum walks in laboratories. This book presents numerous proposals as well as actual experiments for such a physical realization, underpinned by a wide range of
A model for a correlated random walk based on the ordered extension of pseudopodia.
Peter J M Van Haastert
Full Text Available Cell migration in the absence of external cues is well described by a correlated random walk. Most single cells move by extending protrusions called pseudopodia. To deduce how cells walk, we have analyzed the formation of pseudopodia by Dictyostelium cells. We have observed that the formation of pseudopodia is highly ordered with two types of pseudopodia: First, de novo formation of pseudopodia at random positions on the cell body, and therefore in random directions. Second, pseudopod splitting near the tip of the current pseudopod in alternating right/left directions, leading to a persistent zig-zag trajectory. Here we analyzed the probability frequency distributions of the angles between pseudopodia and used this information to design a stochastic model for cell movement. Monte Carlo simulations show that the critical elements are the ratio of persistent splitting pseudopodia relative to random de novo pseudopodia, the Left/Right alternation, the angle between pseudopodia and the variance of this angle. Experiments confirm predictions of the model, showing reduced persistence in mutants that are defective in pseudopod splitting and in mutants with an irregular cell surface.
Weak convergence of stochastic integrals driven by continuous-time random walks
Burr, Meredith N
2011-01-01
Brownian motion is a well-known model for normal diffusion, but not all physical phenomena behave according to a Brownian motion. Many phenomena exhibit irregular diffusive behavior, called anomalous diffusion. Examples of anomalous diffusion have been observed in physics, hydrology, biology, and finance, among many other fields. Continuous-time random walks (CTRWs), introduced by Montroll and Weiss, serve as models for anomalous diffusion. CTRWs generalize the usual random walk model by allowing random waiting times between successive random jumps. Under certain conditions on the jumps and waiting times, scaled CTRWs can be shown to converge in distribution to a limit process M(t) in the cadlag space D[0,infinity) with the Skorohod J_1 or M_1 topology. An interesting question is whether stochastic integrals driven by the scaled CTRWs X^n(t) converge in distribution to a stochastic integral driven by the CTRW limit process M(t). We prove weak convergence of the stochastic integrals driven by CTRWs for certain...
Yang, Mingliang; Li, Jianjun; Guan, Xinyu; Gao, Lianjun; Gao, Feng; Du, Liangjie; Zhao, Hongmei; Yang, Degang; Yu, Yan; Wang, Qimin; Wang, Rencheng; Ji, Linhong
2017-09-01
The high energy cost of paraplegic walking using a reciprocating gait orthosis (RGO) is attributed to limited hip motion and excessive upper limb loading for support. To address the limitation, we designed the hip energy storage walking orthosis (HESWO) which uses a spring assembly on the pelvic shell to store energy from the movements of the healthy upper limbs and flexion-extension of the lumbar spine and hip and returns this energy to lift the pelvis and lower limb to assist with the swing and stance components of a stride. Our aim was to evaluate gait and energy cost indices for the HESWO compared to the RGO in patients with paraplegia. The cross-over design was used in the pilot study. Twelve patients with a complete T4-L5 chronic spinal cord injury underwent gait training using the HESWO and RGO. Gait performance (continuous walking distance, as well as the maximum and comfortable walking speeds) and energy expenditure (at a walking speed of 3.3m/min on a treadmill) were measured at the end of the 4-week training session. Compared to the RGO, the HESWO increased continuous walking distance by 24.7% (Penergy expenditure by 13.9% (P<0.05). Our preliminary results provide support for the use of the HESWO as an alternative support for paraplegic walking. Copyright © 2017. Published by Elsevier B.V.
Randomized Algorithms for Systems and Control: Theory and Applications
2008-05-01
IEIIT-CNR Randomized Algorithms for Systems and Control: Theory and Applications NATO LS Glasgow, Pamplona , Cleveland @RT 2008 Roberto Tempo IEIIT...Glasgow, Pamplona , Cleveland @RT 2008 roberto.tempo@polito.it IEIIT-CNR References R. Tempo, G. Calafiore and F. Dabbene, “Randomized Algorithms for...Analysis and Control of Uncertain Systems,” Springer-Verlag, London, 2005 R Tempo and H Ishii “Monte Carlo and Las Vegas NATO LS Glasgow, Pamplona , Cleveland
Quantitative characterisation of an engineering write-up using random walk analysis
Sunday A. Oke
2008-02-01
Full Text Available This contribution reports on the investigation of correlation properties in an English scientific text (engineering write-up by means of a random walk. Though the idea to use a random walk to characterise correlations is not new (it was used e.g. in the genome analysis and in the analysis of texts, a random walk approach to the analysis of an English scientific text is still far from being exploited in its full strength as demonstrated in this paper. A method of high-dimensional embedding is proposed. Case examples were drawn arbitrarily from four engineering write-ups (Ph.D. synopsis of three engineering departments in the Faculty of Technology, University of Ibadan, Nigeria. Thirteen additional analyses of non-engineering English texts were made and the results compared to the engineering English texts. Thus, a total of seventeen write-ups of eight Faculties and sixteen Departments of the University of Ibadan were considered. The characterising exponents which relate the average distance of random walkers away from a known starting position to the elapsed time steps were estimated for the seventeen cases according to the power law and in three different dimensional spaces. The average characteristic exponent obtained for the seventeen cases and over three different dimensional spaces studied was 1.42 to 2-decimal with a minimum and a maximum coefficient of determination (R2 of 0.9495 and 0.9994 respectively. This is found to be 284% of the average characterising exponent value (0.5, as supported by the literature for random walkers based on the pseudo-random number generator. The average characteristic exponent obtained for the four cases that were engineering-based and over the three different dimensional studied spaces was 1.41 to 2-decimal (closer by 99.3% to 1.42 with a minimum and a maximum coefficient of determination (R2 of 0.9507 and 0.9974 respectively. This is found to be 282% of the average characterising exponent value (0.5, as
IMPROVED RANDOMIZED ALGORITHM FOR THE EQUIVALENT 2-CATALOG SEGMENTATION PROBLEM
无
2005-01-01
An improved randomized algorithm of the equivalent 2-catalog segmentation problem is presented. The result obtained in this paper makes some progress to answer the open problem by analyze this algorithm with performance guarantee. A 0.6378-approximation for the equivalent 2-catalog segmentation problem is obtained.
Self-avoiding walks on random networks of resistors and diodes
Marković, D.; Milošević, S.; Stanley, H. E.
1987-07-01
We study the self-avoiding walks (SAW) on a square lattice whose various degrees of randomness encompasses many different random networks, including the incipient clusters of the directed, mixed and isotropic bond percolation. We apply the position-space renormalization group (PSRG) method and demonstrate that within the framework of this method one is bound to find that the critical exponent v of the mean end-to-end distance of SAW on various two-dimensional random networks should be equal to the critical exponent of SAW on the ordinary square lattice. A detailed analysis of this finding, and similar findings of other authors, lead us to conclude that a debatable opposite finding, which has been predicted on the basis of different approaches, could be attained after a substantial refinement of the method applied.
Decoding Algorithms for Random Linear Network Codes
Heide, Janus; Pedersen, Morten Videbæk; Fitzek, Frank
2011-01-01
achieve a high coding throughput, and reduce energy consumption.We use an on-the-fly version of the Gauss-Jordan algorithm as a baseline, and provide several simple improvements to reduce the number of operations needed to perform decoding. Our tests show that the improvements can reduce the number...
Algorithmic learning in a random world
Vovk, Vladimir; Shafer, Glenn
2005-01-01
A new scientific monograph developing significant new algorithmic foundations in machine learning theory. Researchers and postgraduates in CS, statistics, and A.I. will find the book an authoritative and formal presentation of some of the most promising theoretical developments in machine learning.
An Efficient Algorithm For Chinese Postman Walk on Bi-directed de Bruijn Graphs
Kundeti, Vamsi; Dinh, Hieu
2010-01-01
Sequence assembly from short reads is an important problem in biology. It is known that solving the sequence assembly problem exactly on a bi-directed de Bruijn graph or a string graph is intractable. However finding a Shortest Double stranded DNA string (SDDNA) containing all the k-long words in the reads seems to be a good heuristic to get close to the original genome. This problem is equivalent to finding a cyclic Chinese Postman (CP) walk on the underlying un-weighted bi-directed de Bruijn graph built from the reads. The Chinese Postman walk Problem (CPP) is solved by reducing it to a general bi-directed flow on this graph which runs in O(|E|2 log2(|V |)) time. In this paper we show that the cyclic CPP on bi-directed graphs can be solved without reducing it to bi-directed flow. We present a ?(p(|V | + |E|) log(|V |) + (dmaxp)3) time algorithm to solve the cyclic CPP on a weighted bi-directed de Bruijn graph, where p = max{|{v|din(v) - dout(v) > 0}|, |{v|din(v) - dout(v) < 0}|} and dmax = max{|din(v) - ...
GPU-accelerated algorithms for many-particle continuous-time quantum walks
Piccinini, Enrico; Benedetti, Claudia; Siloi, Ilaria; Paris, Matteo G. A.; Bordone, Paolo
2017-06-01
Many-particle continuous-time quantum walks (CTQWs) represent a resource for several tasks in quantum technology, including quantum search algorithms and universal quantum computation. In order to design and implement CTQWs in a realistic scenario, one needs effective simulation tools for Hamiltonians that take into account static noise and fluctuations in the lattice, i.e. Hamiltonians containing stochastic terms. To this aim, we suggest a parallel algorithm based on the Taylor series expansion of the evolution operator, and compare its performances with those of algorithms based on the exact diagonalization of the Hamiltonian or a 4th order Runge-Kutta integration. We prove that both Taylor-series expansion and Runge-Kutta algorithms are reliable and have a low computational cost, the Taylor-series expansion showing the additional advantage of a memory allocation not depending on the precision of calculation. Both algorithms are also highly parallelizable within the SIMT paradigm, and are thus suitable for GPGPU computing. In turn, we have benchmarked 4 NVIDIA GPUs and 3 quad-core Intel CPUs for a 2-particle system over lattices of increasing dimension, showing that the speedup provided by GPU computing, with respect to the OPENMP parallelization, lies in the range between 8x and (more than) 20x, depending on the frequency of post-processing. GPU-accelerated codes thus allow one to overcome concerns about the execution time, and make it possible simulations with many interacting particles on large lattices, with the only limit of the memory available on the device.
Lechman, Jeremy; Pierce, Flint
2012-02-01
Diffusive transport is a ubiquitous process that is typically understood in terms of a classical random walk of non-interacting particles. Here we present the results for a model of hard-sphere colloids in a Newtonian incompressible solvent at various volume fractions below the ordering transition (˜50%). We numerically simulate the colloidal systems via Fast Lubrication Dynamics -- a Brownian Dynamics approach with corrected mean-field hydrodynamic interactions. Colloid-colloid interactions are also included so that we effectively solve a system of interacting Langevin equations. The results of the simulations are analyzed in terms of the diffusion coefficient as a function of time with the early and late time diffusion coefficients comparing well with experimental results. An interpretation of the full time dependent behavior of the diffusion coefficient and mean-squared displacement is given in terms of a continuous time random walk. Therefore, the deterministic, continuum diffusion equation which arises from the discrete, interacting random walkers is presented. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Random walk model of subdiffusion in a system with a thin membrane.
Kosztołowicz, Tadeusz
2015-02-01
We consider in this paper subdiffusion in a system with a thin membrane. The subdiffusion parameters are the same in both parts of the system separated by the membrane. Using the random walk model with discrete time and space variables the probabilities (Green's functions) P(x,t) describing a particle's random walk are found. The membrane, which can be asymmetrical, is characterized by the two probabilities of stopping a random walker by the membrane when it tries to pass through the membrane in both opposite directions. Green's functions are transformed to the system in which the variables are continuous, and then the membrane permeability coefficients are given by special formulas which involve the probabilities mentioned above. From the obtained Green's functions, we derive boundary conditions at the membrane. One of the conditions demands the continuity of a flux at the membrane, but the other one is rather unexpected and contains the Riemann-Liouville fractional time derivative P(x(N)(-),t)=λ(1)P(x(N)(+),t)+λ(2)∂(α/2)P(x(N)(+),t)/∂t(α/2), where λ(1),λ(2) depending on membrane permeability coefficients (λ(1)=1 for a symmetrical membrane), α is a subdiffusion parameter, and x(N) is the position of the membrane. This boundary condition shows that the additional "memory effect," represented by the fractional derivative, is created by the membrane. This effect is also created by the membrane for a normal diffusion case in which α=1.
Multiple random walks on complex networks: A harmonic law predicts search time
Weng, Tongfeng; Zhang, Jie; Small, Michael; Hui, Pan
2017-05-01
We investigate multiple random walks traversing independently and concurrently on complex networks and introduce the concept of mean first parallel passage time (MFPPT) to quantify their search efficiency. The mean first parallel passage time represents the expected time required to find a given target by one or some of the multiple walkers. We develop a general theory that allows us to calculate the MFPPT analytically. Interestingly, we find that the global MFPPT follows a harmonic law with respect to the global mean first passage times of the associated walkers. Remarkably, when the properties of multiple walkers are identical, the global MFPPT decays in a power law manner with an exponent of unity, irrespective of network structure. These findings are confirmed by numerical and theoretical results on various synthetic and real networks. The harmonic law reveals a universal principle governing multiple random walks on networks that uncovers the contribution and role of the combined walkers in a target search. Our paradigm is also applicable to a broad range of random search processes.
Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics
Plyukhin, Dan; Plyukhin, Alex V.
2016-10-01
We consider the survival probability f (t ) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension ds≤2 , with spatially correlated traps. The traps form a sublattice with fractal dimension dawa which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (wa≪w ), we find that f (t ) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α =1 -(d -da) /dw , where dw is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f (t ) ˜t-α with the same exponent α as for the stretched exponential regime. For strong absorption wa≳w , including the limit of perfect traps wa→∞ , the stretched exponential regime is absent and the decay of f (t ) follows, after a short transient, the aforementioned power law for all times.
Elliptic equation for random walks. Application to transport in microporous media
Shapiro, Alexander
2007-01-01
We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes...... into account the distribution of the residence times of molecules ill pores. The new elliptic diffusion equation is strictly derived by the operator approach. A criterion showing where the new equation should be applied instead of the standard diffusion equation is obtained. Boundary conditions are studied...
Localization on 4 sites for Vertex-reinforced random walks on $\\mathbb Z$
Basdevant, Anne-Laure; Singh, Arvind
2012-01-01
We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\\log \\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\\log \\log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.
Slower deviations of the branching Brownian motion and of branching random walks
Derrida, Bernard; Shi, Zhan
2017-08-01
We have shown recently how to calculate the large deviation function of the position X\\max(t) of the rightmost particle of a branching Brownian motion at time t. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of X\\max(t) has, asymptotically in time, a prefactor characterized by a non trivial power law. Dedicated to John Cardy on the occasion of his 70th birthday.
Testing The Random Walk Hypothesis: An Application in the BRIC Countries and Turkey
Halime Temel Nalın
2015-03-01
Full Text Available This paper investigates the weak form efficiency in the BRIC countries and Turkey with use of autocorrelation analysis, unit root tests, Johansen cointegration and Granger causality test. Monthly data covers the period from July 1997 to December 2013. Our findings indicate the efficiency among the stock markets in the weak form. The empirical findings indicate monthly closing prices of indices follow the random walk procedure. According to Granger causality and Johansen cointegration tests we found the long-run relationship between China and India, also China and Turkey.
Exact Partition Function for the Random Walk of an Electrostatic Field
Gabriel González
2017-01-01
Full Text Available The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either ±σ is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example.
Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts
Fedotov, Sergei
2010-01-01
The main aim of the paper is to incorporate the nonlinear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with nonexponential waiting time distributions. We consider three different CTRW models with reactions. We derive nonlinear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov kinetics and anomalous diffusion. We have found an explicit expression for the speed of a propagating front in the case of subdiffusive transport.
Continuous Time Random Walk and Migration-Proliferation Dichotomy of Brain Cancer
Iomin, A.
A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed and explained in the framework of a continuous time random walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension Dfr < 3.
Random walk in degree space and the time-dependent Watts-Strogatz model
Grande, H L Casa; Hase, M O
2016-01-01
In this work, we propose a scheme that provides an analytical estimate for the time-dependent degree distribution of some networks. This scheme maps the problem into a random walk in degree space, and then we choose the paths that are responsible for the dominant contributions. The method is illustrated on the dynamical versions of the Erd\\"os-R\\'enyi and Watts-Strogatz graphs, which were introduced as static models in the original formulation. We have succeeded in obtaining an analytical form for the dynamics Watts-Strogatz model, which is asymptotically exact for some regimes.
Scaling Law for Photon Transmission through Optically Turbid Slabs Based on Random Walk Theory
Xuesong Li
2012-03-01
Full Text Available Past work has demonstrated the value of a random walk theory (RWT to solve multiple-scattering problems arising in numerous contexts. This paper’s goal is to investigate the application range of the RWT using Monte Carlo simulations and extending it to anisotropic media using scaling laws. Meanwhile, this paper also reiterates rules for converting RWT formulas to real physical dimensions, and corrects some errors which appear in an earlier publication. The RWT theory, validated by the Monte Carlo simulations and combined with the scaling law, is expected to be useful to study multiple scattering and to greatly reduce the computation cost.
Zhang, Zhongzhi; Sheng, Yibin
2015-01-01
Random walks including non-nearest-neighbor jumps appear in many real situations such as the diffusion of adatoms and have found numerous applications including PageRank search algorithm, however, related theoretical results are much less for this dynamical process. In this paper, we present a study of mixed random walks in a family of fractal scale-free networks, where both nearest-neighbor and next-nearest-neighbor jumps are included. We focus on trapping problem in the network family, which is a particular case of random walks with a perfect trap fixed at the central high-degree node. We derive analytical expressions for the average trapping time (ATT), a quantitative indicator measuring the efficiency of the trapping process, by using two different methods, the results of which are consistent with each other. Furthermore, we analytically determine all the eigenvalues and their multiplicities for the fundamental matrix characterizing the dynamical process. Our results show that although next-nearest-neighb...
OnlineMin: A Fast Strongly Competitive Randomized Paging Algorithm
Brodal, Gerth Stølting; Moruz, Gabriel; Negoescu, Andrei
2012-01-01
n the field of online algorithms paging is one of the most studied problems. For randomized paging algorithms a tight bound of H k on the competitive ratio has been known for decades, yet existing algorithms matching this bound have high running times. We present the first randomized paging...... approach that both has optimal competitiveness and selects victim pages in subquadratic time. In fact, if k pages fit in internal memory the best previous solution required O(k 2) time per request and O(k) space, whereas our approach takes also O(k) space, but only O(logk) time in the worst case per page...
Following Car Algorithm With Multi Agent Randomized System
Mounir Gouiouez
2013-08-01
Full Text Available We present a new Following Car Algorithm in Microscopic Urban Traffic Models which integrates some real-life factors that need to be considered, such as the effect of random distributions in the car speed, acceleration, entry of lane… Our architecture is based on Multi-Agent Randomized Systems (MARSdeveloped in earlier publications
Barral, Julien
2010-01-01
We establish large deviations properties valid for almost every sample path of a class of stationary mixing processes $(X_1,\\dots, X_n,\\dots)$. These large deviations properties are inherited from those of $S_n=\\sum_{i=1}^nX_i$ and they describe how the local fluctuations of almost every realization of $S_n$ deviate from the almost sure behavior provided by the strong law of large numbers. These results have interesting applications to the fluctuations of Brownian motion increments, the local fluctuations of Birkhoff averages on symbolic spaces and their geometric realizations, as well as the local fluctuations of branching random walks. Also, they lead to new insights into the "randomness" of the digits of their expansions in integer bases for fundamental constants such as Pi and the Euler constant. We formulate a new conjecture, supported by numerical experiments, implying the normality of these numbers.
Dafna eMerom
2016-02-01
Full Text Available Background: A physically active lifestyle has the potential to prevent cognitive decline and dementia, yet the optimal type of physical activity/exercise remains unclear. Dance is of special interest as it complex sensorimotor rhythmic activity with additional cognitive, social and affective dimensions. Objectives: to determine whether dance benefits executive function more than walking, an activity that is simple and functional. Methods: Two-arm randomised controlled trial among community-dwelling older adults. The intervention group received 1 hour of ballroom dancing twice weekly over 8 months (~69sessions in local community dance studios. The control group received a combination of a home walking program with a pedometer and optional biweekly group-based walking in local community park to facilitate socialisation. Main outcomes: Main outcomes: executive function tests: processing speed and task shift by the Trail Making Tests (TMT, response inhibition by the Stroop Colour-Word Test (SCWT, working memory by the Digit Span Backwards (DSB test, immediate and delayed verbal recall by the Rey Auditory Verbal Learning Test (RAVLT and visuospatial recall by the Brief Visuospatial Memory Test (BVST. Results: One hundred and fifteen adults (69.5 years, SD6.4 completed baseline and delayed baseline (3 weeks apart before being randomised to either dance (n=60 or walking (n=55. Of those randomized, 79 (68% completed the follow-up measurements (32 weeks from baseline. In the dance group only, ‘non-completers’ had significant lower baseline scores on all executive function tests than those completed the full program. Intention-to-treat analyses showed no group effect. In a random effects model including participants who completed all measurements, adjusted for baseline score and covariates (age, education, estimated verbal intelligence, community, a between group effect in favour of dance was noted only for BVST total learning (Cohen’s D Effect size
Randomized algorithms in automatic control and data mining
Granichin, Oleg; Toledano-Kitai, Dvora
2015-01-01
In the fields of data mining and control, the huge amount of unstructured data and the presence of uncertainty in system descriptions have always been critical issues. The book Randomized Algorithms in Automatic Control and Data Mining introduces the readers to the fundamentals of randomized algorithm applications in data mining (especially clustering) and in automatic control synthesis. The methods proposed in this book guarantee that the computational complexity of classical algorithms and the conservativeness of standard robust control techniques will be reduced. It is shown that when a problem requires "brute force" in selecting among options, algorithms based on random selection of alternatives offer good results with certain probability for a restricted time and significantly reduce the volume of operations.
Wollschlaeger, A.
1996-12-31
The presented particle tracking model is for the numerical calculation of heavy metal transport in natural waters. The Navier-Stokes-Equations are solved with the Finite-Element-Method. The advective movement of the particles is interpolated from the velocities on the discrete mesh. The influence of turbulence is simulated with a Random-Walk-Model where particles are distributed due to a given probability function. Both parts are added and lead to the new particle position. The characteristics of the heavy metals are assigned to the particules as their attributes. Dissolved heavy metals are transported only by the flow. Heavy metals which are bound to particulate matter have an additional settling velocity. The sorption and the remobilization processes are approximated through a probability law which maintains the proportionality ratio between dissolved heavy metals and those which are bound to particulate matter. At the bed heavy metals bound to particulate matter are subjected to deposition and erosion processes. The model treats these processes by considering the absorption intensity of the heavy metals to the bottom sediments. Calculations of the Weser estuary show that the particle tracking model allows the simulation of the heavy metal behaviour even under complex flow conditions. (orig.) [Deutsch] Das vorgestellte Partikelmodell dient zur numerischen Berechnung des Schwermetalltransports in natuerlichen Gewaessern. Die Navier-Stokes-Gleichungen werden mit der Methode der Finiten Elemente geloest. Die advektive Bewegung der Teilchen ergibt sich aus der Interpolation der Geschwindigkeiten auf dem diskreten Netz. Der Einfluss der Turbulenz wird mit einem Random-Walk-Modell simuliert, bei dem sich die Partikel anhand einer vorgegebenen Wahrscheinlichkeitsfunktion verteilen. Beide Bewegungsanteile werden zusammengefasst und ergeben die neue Partikelposition. Die Eigenschaften der Schwermetalle werden den Partikeln als Attribute zugeordnet. Geloeste Schwermetalle
A Random-Walk Based Privacy-Preserving Access Control for Online Social Networks
You-sheng Zhou
2016-02-01
Full Text Available Online social networks are popularized with people to connect friends, share resources etc. Meanwhile, the online social networks always suffer the problem of privacy exposure. The existing methods to prevent exposure are to enforce access control provided by the social network providers or social network users. However, those enforcements are impractical since one of essential goal of social network application is to share updates freely and instantly. To better the security and availability in social network applications, a novel random walking based access control of social network is proposed in this paper. Unlike using explicit attribute based match in the existing schemes, the results from random walking are employed to securely compute L1 distance between two social network users in the presented scheme, which not only avoids the leakage of private attributes, but also enables each social network user to define access control policy independently. The experimental results show that the proposed scheme can facilitate the access control for online social network.
BRWLDA: bi-random walks for predicting lncRNA-disease associations.
Yu, Guoxian; Fu, Guangyuan; Lu, Chang; Ren, Yazhou; Wang, Jun
2017-09-01
Increasing efforts have been done to figure out the association between lncRNAs and complex diseases. Many computational models construct various lncRNA similarity networks, disease similarity networks, along with known lncRNA-disease associations to infer novel associations. However, most of them neglect the structural difference between lncRNAs network and diseases network, hierarchical relationships between diseases and pattern of newly discovered associations. In this study, we developed a model that performs Bi-Random Walks to predict novel LncRNA-Disease Associations (BRWLDA in short). This model utilizes multiple heterogeneous data to construct the lncRNA functional similarity network, and Disease Ontology to construct a disease network. It then constructs a directed bi-relational network based on these two networks and available lncRNAs-disease associations. Next, it applies bi-random walks on the network to predict potential associations. BRWLDA achieves reliable and better performance than other comparing methods not only on experiment verified associations, but also on the simulated experiments with masked associations. Case studies further demonstrate the feasibility of BRWLDA in identifying new lncRNA-disease associations.
Super-extreme event's influence on a Weierstrass-Mandelbrot Continuous-Time Random Walk
Gubiec, Tomasz; Kutner, Ryszard; Sornette, Didier
2010-01-01
Two utmost cases of super-extreme event's influence on the velocity autocorrelation function (VAF) were considered. The VAF itself was derived within the hierarchical Weierstrass-Mandelbrot Continuous-Time Random Walk (WM-CTRW) formalism, which is able to cover a broad spectrum of continuous-time random walks. Firstly, we studied a super-extreme event in a form of a sustained drift, whose duration time is much longer than that of any other event. Secondly, we considered a super-extreme event in the form of a shock with the size and velocity much larger than those corresponding to any other event. We found that the appearance of these super-extreme events substantially changes the results determined by extreme events (the so called "black swans") that are endogenous to the WM-CTRW process. For example, changes of the VAF in the latter case are in the form of some instability and distinctly differ from those caused in the former case. In each case these changes are quite different compared to the situation with...
A random walk simulation of scalar mixing in flows through submerged vegeta-tions
梁东方
2014-01-01
The scalar transport phenomena in vertical two-dimensional flows are studied using the random walk method. The establi-shed Lagrangian model is first applied to study the idealized longitudinal dispersion in open channels, before being used to investi-gate the scalar mixing characteristics of the flows through submerged vegetations. The longitudinal dispersion coefficients of the fully-developed boundary layer flows, with and without vegetations, are calculated based on the positions of the particles. A conve-nient way of incorporating the effects of vegetations is proposed, where all the flow parameters are regarded to be continually distri-buted over the depth. The simulation results show high accuracy of the developed random walk method, and indicate that the new method of accounting for the vegetation effects is appropriate for all the test cases considered. The predicted longitudinal dispersion coefficients agree well with the measurements. The merit of the new method is highlighted by its simplicity and efficiency in com-parison with the conventional method that assumes the discontinuous distribution of the flow parameters over the depth.
A lattice-model representation of continuous-time random walks
Campos, Daniel [School of Mathematics, Department of Applied Mathematics, University of Manchester, Manchester M60 1QD (United Kingdom); Mendez, Vicenc [Grup de Fisica Estadistica, Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona) (Spain)], E-mail: daniel.campos@uab.es, E-mail: vicenc.mendez@uab.es
2008-02-29
We report some ideas for constructing lattice models (LMs) as a discrete approach to the reaction-dispersal (RD) or reaction-random walks (RRW) models. The analysis of a rather general class of Markovian and non-Markovian processes, from the point of view of their wavefront solutions, let us show that in some regimes their macroscopic dynamics (front speed) turns out to be different from that by classical reaction-diffusion equations, which are often used as a mean-field approximation to the problem. So, the convenience of a more general framework as that given by the continuous-time random walks (CTRW) is claimed. Here we use LMs as a numerical approach in order to support that idea, while in previous works our discussion was restricted to analytical models. For the two specific cases studied here, we derive and analyze the mean-field expressions for our LMs. As a result, we are able to provide some links between the numerical and analytical approaches studied.
The survival probability of a branching random walk in presence of an absorbing wall
Derrida, B.; Simon, D.
2007-06-01
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for vvc. At v=vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (vc-v)-1/2 in the quasi-stationary regime below vc. Moreover for v>vc, our data indicate that there is no quasi-stationary regime.
Random-walk mobility analysis of Lisbon's plans for the post-1755 reconstruction
de Sampayo, Mafalda Teixeira; Sousa-Rodrigues, David
2016-11-01
The different options for the reconstruction of the city of Lisbon in the aftermath of the 1755 earthquake are studied with an agent-based model based on randomwalks. This method gives a comparative quantitative measure of mobility of the circulation spaces within the city. The plans proposed for the city of Lisbon signified a departure from the medieval mobility city model. The intricacy of the old city circulation spaces is greatly reduced in the new plans and the mobility between different areas is substantially improved. The simulation results of the random-walk model show that those plans keeping the main force lines of the old city presented less improvement in terms ofmobility. The plans that had greater design freedom were, by contrast, easier to navigate. Lisbon's reconstruction followed a plan that included a shift in the traditional notions of mobility. This affected the daily lives of its citizens by potentiating an easy access to the waterfront, simplifying orientation and navigability. Using the random-walk model it is shown how to quantitatively measure the potential that synthetic plans have in terms of the permeability and navigability of different city public spaces.
Transverse momentum spectra of the produced hadrons at SPS energy and a random walk model
Bedangadas Mohanty
2014-05-01
The transverse momentum spectra of the produced hadrons have been compared to a model, which is based on the assumption that a nucleus–nucleus collision is a superposition of isotropically decaying thermal sources at a given freeze-out temperature. The freeze-out temperature in nucleus–nucleus collisions is fixed from the inverse slope of the transverse momentum spectra of hadrons in nucleon–nucleon collision. The successive collisions in the nuclear reaction lead to gain in transverse momentum, as the nucleons propagate in the nucleus following a random walk pattern. The average transverse rapidity shift per collision is determined from the nucleon–nucleus collision data. Using this information, we obtain parameter-free result for the transverse momentum distribution of produced hadrons in nucleus–nucleus collisions. It is observed that such a model is able to explain the transverse mass spectra of the produced pions at SPS energies. However, it fails to satisfactorily explain the transverse mass spectra of kaons and protons. This indicates the presence of collective effect which cannot be accounted for, by the initial state collision broadening of transverse momentum of produced hadrons, the basis of random walk model.
Estimating a Random Walk First-Passage Time from Noisy or Delayed Observations
Burnashev, Marat V
2012-01-01
A random walk (or a Wiener process), possibly with drift, is observed in a noisy or delayed fashion. The problem considered in this paper is to estimate the first time \\tau the random walk reaches a given level. Specifically, the p-moment (p\\geq 1) optimization problem \\inf_\\eta \\ex|\\eta-\\tau|^p is investigated where the infimum is taken over the set of stopping times that are defined on the observation process. When there is no drift, optimal stopping rules are characterized for both types of observations. When there is a drift, upper and lower bounds on \\inf_\\eta \\ex|\\eta-\\tau|^p are established for both types of observations. The bounds are tight in the large-level regime for noisy observations and in the large-level-large-delay regime for delayed observations. Noteworthy, for noisy observations there exists an asymptotically optimal stopping rule that is a function of a single observation. Simulation results are provided that corroborate the validity of the results for non-asymptotic settings.
Anomalous diffusion and Levy random walk of magnetic field lines in three dimensional turbulence
Zimbardo, G.; Veltri, P.; Basile, G.; Principato, S. [Dipartimento di Fisica, Universita della Calabria, I-87030 Arcavacata di Rende (Italy)
1995-07-01
The transport of magnetic field lines is studied numerically where three dimensional (3-D) magnetic fluctuations, with a power law spectrum, and periodic over the simulation box are superimposed on an average uniform magnetic field. The weak and the strong turbulence regime, {delta}{ital B}{similar_to}{ital B}{sub 0}, are investigated. In the weak turbulence case, magnetic flux tubes are separated from each other by percolating layers in which field lines undergo a chaotic motion. In this regime the field lines may exhibit Levy, rather than Gaussian, random walk, changing from Levy flights to trapped motion. The anomalous diffusion laws {l_angle}{Delta}{ital x}{sup 2}{sub {ital i}}{r_angle}{proportional_to}{ital s}{sup {alpha}} with {alpha}{gt}1 and {alpha}{lt}1, are obtained for a number of cases, and the non-Gaussian character of the field line random walk is pointed out by computing the kurtosis. Increasing the fluctuation level, and, therefore stochasticity, normal diffusion ({alpha}{congruent}1) is recovered and the kurtoses reach their Gaussian value. However, the numerical results show that neither the quasi-linear theory nor the two dimensional percolation theory can be safely extrapolated to the considered 3-D strong turbulence regime. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.
Effective degrees of freedom of a random walk on a fractal.
Balankin, Alexander S
2015-12-01
We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.
Local and global survival for nonhomogeneous random walk systems on Z
Bertacchi, Daniela; Zucca, Fabio
2012-01-01
We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site $n \\ge 1$. Particles become active when hit by another active particle. Once activated perform an asymmetric nearest neighbour random walk which depends only on the starting location of the particle. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction. Analysis of explicit examples is provided.
A Random Walk in the Park: An Individual-Based Null Model for Behavioral Thermoregulation.
Vickers, Mathew; Schwarzkopf, Lin
2016-04-01
Behavioral thermoregulators leverage environmental temperature to control their body temperature. Habitat thermal quality therefore dictates the difficulty and necessity of precise thermoregulation, and the quality of behavioral thermoregulation in turn impacts organism fitness via the thermal dependence of performance. Comparing the body temperature of a thermoregulator with a null (non-thermoregulating) model allows us to estimate habitat thermal quality and the effect of behavioral thermoregulation on body temperature. We define a null model for behavioral thermoregulation that is a random walk in a temporally and spatially explicit thermal landscape. Predicted body temperature is also integrated through time, so recent body temperature history, environmental temperature, and movement influence current body temperature; there is no particular reliance on an organism's equilibrium temperature. We develop a metric called thermal benefit that equates body temperature to thermally dependent performance as a proxy for fitness. We measure thermal quality of two distinct tropical habitats as a temporally dynamic distribution that is an ergodic property of many random walks, and we compare it with the thermal benefit of real lizards in both habitats. Our simple model focuses on transient body temperature; as such, using it we observe such subtleties as shifts in the thermoregulatory effort and investment of lizards throughout the day, from thermoregulators to thermoconformers.
Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?
Czégel, Dániel; Palla, Gergely
2015-01-01
Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology. PMID:26657012
Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?
Czégel, Dániel; Palla, Gergely
2015-12-10
Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology.
Bottlenecks, burstiness, and fat tails regulate mixing times of non-Poissonian random walks
Delvenne, Jean-Charles; Rocha, Luis E C
2013-01-01
We focus on general continuous-time random walks on networks and find that the mixing time, i.e. the relaxation time for the random process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data. These results provide a theoretical framework for the modeling of diffusion on temporal networks representing human interactions, often characterized by non-Poissonian contact patterns.
Scaling Behavior of the First Arrival Time of a Random-Walking Magnetic Domain
Im, M.-Y.; Lee, S.-H.; Kim, D.-H.; Fischer, P.; Shin, S.-C.
2008-02-04
We report a universal scaling behavior of the first arrival time of a traveling magnetic domain wall into a finite space-time observation window of a magneto-optical microscope enabling direct visualization of a Barkhausen avalanche in real time. The first arrival time of the traveling magnetic domain wall exhibits a nontrivial fluctuation and its statistical distribution is described by universal power-law scaling with scaling exponents of 1.34 {+-} 0.07 for CoCr and CoCrPt films, despite their quite different domain evolution patterns. Numerical simulation of the first arrival time with an assumption that the magnetic domain wall traveled as a random walker well matches our experimentally observed scaling behavior, providing an experimental support for the random-walking model of traveling magnetic domain walls.
Effective-medium approximation for lattice random walks with long-range jumps.
Thiel, Felix; Sokolov, Igor M
2016-07-01
We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.
Effective-medium approximation for lattice random walks with long-range jumps
Thiel, Felix; Sokolov, Igor M.
2016-07-01
We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.
Two-step memory within Continuous Time Random Walk. Description of double-action market dynamics
Gubiec, Tomasz
2013-01-01
By means of a novel version of the Continuous-Time Random Walk (CTRW) model with memory, we describe, for instance, the stochastic process of a single share price on a double-auction market within the high frequency time scale. The memory present in the model is understood as dependence between successive share price jumps, while waiting times between price changes are considered as i.i.d. random variables. The range of this memory is defined herein by dependence between three successive jumps of the process. This dependence is motivated both empirically, by analysis of empirical two-point histograms, and theoretically, by analysis of the bid-ask bounce mechanism containing some delay. Our model turns out to be analytically solvable, which enables us a direct comparison of its predictions with empirical counterparts, for instance, with so significant and commonly used quantity as velocity autocorrelation function. This work strongly extends the capabilities of the CTRW formalism.
The Study of Randomized Visual Saliency Detection Algorithm
Yuantao Chen
2013-01-01
Full Text Available Image segmentation process for high quality visual saliency map is very dependent on the existing visual saliency metrics. It is mostly only get sketchy effect of saliency map, and roughly based visual saliency map will affect the image segmentation results. The paper had presented the randomized visual saliency detection algorithm. The randomized visual saliency detection method can quickly generate the same size as the original input image and detailed results of the saliency map. The randomized saliency detection method can be applied to real-time requirements for image content-based scaling saliency results map. The randomization method for fast randomized video saliency area detection, the algorithm only requires a small amount of memory space can be detected detailed oriented visual saliency map, the presented results are shown that the method of visual saliency map used in image after the segmentation process can be an ideal segmentation results.
Han, Seung Hoon; Kim, Taikon; Jang, Seong Ho; Kim, Mi Jung; Park, Si-bog; Yoon, Seoung Ic; Choi, Bong-Kun; Lee, Michael Y
2011-01-01
Objective: To evaluate the effect of an arm sling on gait speed and energy efficiency of patients with hemiplegia. Design: A randomized crossover design. Setting: A rehabilitation department of a university hospital. Subjects: Thirty-seven outpatients with hemiplegia were included in this study. Interventions: All patients walked on a 20-m walkway twice on the same day, randomly with and without an arm sling, at a self selected speed. Main measures: The heart rate, gait speed, oxygen cost and oxygen rate were measured on all patients. We analysed all values with and without an arm sling and also compared them after all patients being stratified according to demographic and clinical characteristics. Results: When we compared the heart rate between walking with (90.7 ± 17.2 beats/min) and without (91.2 ± 18.6 beats/min) the arm sling, it was significantly decreased while walking with the arm sling. When we compared the gait speed between walking with (32.8 m/min) and without (30.1 m/min), it was significantly increased with the arm sling walking. The O2 rate in hemiplegic patients walking with the arm sling was significantly decreased by 7%, compared to walking without arm sling (5.8 mL/kg min and 6.2 mL/kg min, respectively). The O2 cost in hemiplegic patients walking without arm sling was significantly 1.4 times greater than walking with it (0.2 mL/kg m and 0.3 mL/kg m, respectively). Conclusion: An arm sling can be used to improve the gait efficiency. PMID:21059662
Liu, Q.; Liu, F.; Turner, I.; Anh, V.
2007-03-01
In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grünwald-Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.
A random search algorithm for cyclic delivery synchronization problem
Katarzyna Gdowska
2017-09-01
Full Text Available Background: The paper is devoted to the cyclic delivery synchronization problem with vehicles serving fixed routes. Each vehicle is assigned to a fixed route: the series of supplier’s and logistic centers to be visited one after another. For each route the service frequency is fixed and known in advance. A vehicle loads at a supplier’s, then it delivers goods to a logistic center and either loads other goods there and delivers them to the next logistic center along the route or goes to another logistic center. Each logistic center can belong to several routes, so goods are delivered there with one vehicle and then they departure for the further journey with another truck. The objective of this cyclic delivery synchronization problem is to maximize the total number of synchronizations of vehicles arrivals in logistic centers and their load times, so that it is possible to organize their arrivals in repeatable blocks. Methods: Basing on the previously developed mathematical model for the cyclic delivery synchronization problem we built a random search algorithm for cyclic delivery synchronization problem. The random heuristic search utilizes objective-oriented randomizing. In the paper the newly-developed random search algorithm for cyclic delivery synchronization problem is presented. Results: A computational experiment consisted of employing the newly-developed random search algorithm for solving a series of cyclic delivery synchronization problems. Results obtained with the algorithm were compared with solutions computed with the exact method. Conclusions: The newly-developed random search algorithm for cyclic delivery synchronization problem gives results which are considerably close to the ones obtained with mixed-integer programming. The main advantage of the algorithm is reduction of computing time; it is relevant for utilization of this method in practice, especially for large-sized problems.
Ageing first passage time density in continuous time random walks and quenched energy landscapes
Krüsemann, Henning; Godec, Aljaž; Metzler, Ralf
2015-07-01
We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function \\psi (t)≃ {t}-1-α (0≤slant α ≤slant 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time ta, the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0\\lt α \\lt 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1\\lt α \\lt 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.
Thiery, Thimothée; Le Doussal, Pierre
2017-01-01
We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin (BC) (2016 Probab. Theory Relat. Fields 1-16). We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time t is performed both (i) in a diffusive vicinity, x˜ {{t}1/2} , of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, x˜ t , of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as t 1/3 and to be distributed according to the Tracy-Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as x˜ {{t}3/4} . Our results are confronted to extensive numerical simulations of the model.
Høyer, Ellen; Jahnsen, Reidun; Stanghelle, Johan Kvalvik; Strand, Liv Inger
2012-01-01
Treadmill training with body weight support (TTBWS) for relearning walking ability after brain damage is an approach under current investigation. Efficiency of this method beyond traditional training is lacking evidence, especially in patients needing walking assistance after stroke. The objective of this study was to investigate change in walking and transfer abilities, comparing TTBWS with traditional walking training. A single-blinded, randomized controlled trial was conducted. Sixty patients referred for multi-disciplinary primary rehabilitation were assigned into one of two intervention groups, one received 30 sessions of TTBWS plus traditional training, the other traditional training alone. Daily training was 1 hr. Outcome measures were Functional Ambulation Categories (FAC), Walking, Functional Independence Measure (FIM); shorter transfer and stairs, 10 m and 6-min walk tests. Substantial improvements in walking and transfer were shown within both groups after 5 and 11 weeks of intervention. Overall no statistical significant differences were found between the groups, but 12 of 17 physical measures tended to show improvements in favour of the treadmill approach. Both training strategies provided significant improvements in the tested activities, suggesting that similar outcomes can be obtained in the two modalities by systematic, intensive and goal directed training.
Collapse transition of a hydrophobic self-avoiding random walk in a coarse-grained model solvent.
Gaudreault, Mathieu; Viñals, Jorge
2009-08-01
In order to study solvation effects on protein folding, we analyze the collapse transition of a self-avoiding random walk composed of hydrophobic segments that is embedded in a lattice model of a solvent. As expected, hydrophobic interactions lead to an attractive potential of mean force among chain segments. As a consequence, the random walk in solvent undergoes a collapse transition at a higher temperature than in its absence. Chain collapse is accompanied by the formation of a region depleted of solvent around the chain. In our simulation, the depleted region at collapse is as large as our computational domain.
Reuveni, Shlomi; Granek, Rony; Klafter, Joseph
2010-10-01
We present an approach to mapping between random walks and vibrational dynamics on general networks. Random walk occupation probabilities, first passage time distributions and passage probabilities between nodes are expressed in terms of thermal vibrational correlation functions. Recurrence is demonstrated equivalent to the Landau-Peierls instability. Fractal networks are analyzed as a case study. In particular, we show that the spectral dimension governs whether or not the first passage time distribution is well represented by its mean. We discuss relevance to universal features arising in protein vibrational dynamics.
Collapse transition of a hydrophobic self-avoiding random walk in a coarse-grained model solvent
Gaudreault, Mathieu; Viñals, Jorge
2009-08-01
In order to study solvation effects on protein folding, we analyze the collapse transition of a self-avoiding random walk composed of hydrophobic segments that is embedded in a lattice model of a solvent. As expected, hydrophobic interactions lead to an attractive potential of mean force among chain segments. As a consequence, the random walk in solvent undergoes a collapse transition at a higher temperature than in its absence. Chain collapse is accompanied by the formation of a region depleted of solvent around the chain. In our simulation, the depleted region at collapse is as large as our computational domain.
Probabilistic Analysis of Random Extension-Rotation Algorithms
1981-10-01
Whitney matroid. Matroid theory (see [ Tutte , 19711, (Lawler, 19761) has applicatlons to a wide class of combinatorial optimization problems: where we...observed that Posa’s proof yields a polynomial time I algorithm for constructing Hamiltonian paths in a random instance of Gn. Angluin and Valiant [1979... Tutte , W.T., Introduction to the Theory of x•atroids, American Elsevier, New York, 1971. Walkup, D.W., "On the expected value of a random assignment
Generalized Pareto for Pattern-Oriented Random Walk Modelling of Organisms' Movements.
Sophie Bertrand
Full Text Available How organisms move and disperse is crucial to understand how population dynamics relates to the spatial heterogeneity of the environment. Random walk (RW models are typical tools to describe movement patterns. Whether Lévy or alternative RW better describes forager movements is keenly debated. We get around this issue using the Generalized Pareto Distribution (GPD. GPD includes as specific cases Normal, exponential and power law distributions, which underlie Brownian, Poisson-like and Lévy walks respectively. Whereas previous studies typically confronted a limited set of candidate models, GPD lets the most likely RW model emerge from the data. We illustrate the wide applicability of the method using GPS-tracked seabird foraging movements and fishing vessel movements tracked by Vessel Monitoring System (VMS, both collected in the Peruvian pelagic ecosystem. The two parameters from the fitted GPD, a scale and a shape parameter, provide a synoptic characterization of the observed movement in terms of characteristic scale and diffusive property. They reveal and quantify the variability, among species and individuals, of the spatial strategies selected by predators foraging on a common prey field. The GPD parameters constitute relevant metrics for (1 providing a synthetic and pattern-oriented description of movement, (2 using top predators as ecosystem indicators and (3 studying the variability of spatial behaviour among species or among individuals with different personalities.
Generalized Pareto for Pattern-Oriented Random Walk Modelling of Organisms' Movements.
Bertrand, Sophie; Joo, Rocío; Fablet, Ronan
2015-01-01
How organisms move and disperse is crucial to understand how population dynamics relates to the spatial heterogeneity of the environment. Random walk (RW) models are typical tools to describe movement patterns. Whether Lévy or alternative RW better describes forager movements is keenly debated. We get around this issue using the Generalized Pareto Distribution (GPD). GPD includes as specific cases Normal, exponential and power law distributions, which underlie Brownian, Poisson-like and Lévy walks respectively. Whereas previous studies typically confronted a limited set of candidate models, GPD lets the most likely RW model emerge from the data. We illustrate the wide applicability of the method using GPS-tracked seabird foraging movements and fishing vessel movements tracked by Vessel Monitoring System (VMS), both collected in the Peruvian pelagic ecosystem. The two parameters from the fitted GPD, a scale and a shape parameter, provide a synoptic characterization of the observed movement in terms of characteristic scale and diffusive property. They reveal and quantify the variability, among species and individuals, of the spatial strategies selected by predators foraging on a common prey field. The GPD parameters constitute relevant metrics for (1) providing a synthetic and pattern-oriented description of movement, (2) using top predators as ecosystem indicators and (3) studying the variability of spatial behaviour among species or among individuals with different personalities.
Quantum Walks on the Hypercube
Moore, Cristopher; Moore, Cristopher; Russell, Alexander
2001-01-01
Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk mixes in (\\pi/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is {\\em exactly} uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds p...
Mazari, Fayyaz Ali Khan; Mockford, Katherine; Barnett, Cleveland; Khan, Junaid A; Brown, Barbara; Smith, Lynne; Polman, Remco C; Hancock, Amanda; Vanicek, Natalie K; Chetter, Ian C
2010-12-01
To compare articulated and nonarticulated early walking aids (EWAs) for clinical and quality-of-life outcomes in transtibial amputees. Patients undergoing lower limb amputation in a tertiary-care vascular surgical unit were screened over a 4-year period. Recruited patients were randomized to receive articulated amputee mobility aid (AMA) or nonarticulated pneumatic postamputation mobility aid (PPAMA) during early rehabilitation. Primary (10-meter walking velocity) and secondary clinical (number and duration of physiotherapy treatments during EWA/prosthesis use) and quality-of-life (SF-36) outcome measures were recorded at five standardized assessment visits. Inter-group and intra-group analyses were performed. Two hundred seventy-two patients were screened and 29 transtibial amputees (median age, 56 years) were recruited (14/treatment arm). No significant difference was seen in demographics and comorbidities at baseline. Inter-group analysis: Median 10-meter walking velocity was significantly (Mann-Whitney, P = .020) faster in the PPAMA group (0.245 m/s, interquartile range [IQR] 0.218-0.402 m/s) compared with the AMA group (0.165 m/s; IQR, 0.118-0.265 m/s) at visit 1. However, there was no difference between the groups at any other visit. Similarly, the number of treatments using EWA was significantly (P = .045) lower in the PPAMA group (5.0; IQR, 3.5-8.0) compared with the AMA group (6.0; IQR, 6.0-10.5). No difference was observed between the groups in duration of physiotherapy or SF-36 domain and summary scores. Intra-group analysis: Both treatment groups showed significant improvement in 10-meter walking velocity (Friedman test; AMA P = .001; PPAMA P = .007); however, other clinical outcomes did not show any statistically significant improvement. Only physical function domain of SF-36 demonstrated significant improvement (Friedman test; AMA P = .037; PPAMA P = .029). There is no difference in clinical and QOL outcomes between articulated and nonarticulated EWAs
Renormalized Random Walk Study of Oxygen Absorption in the Human Lung
Felici, M.; Filoche, M.; Sapoval, B.
2004-02-01
The possibility to renormalize random walks is used to study numerically the oxygen diffusion and permeation in the acinus, the diffusion cell terminating the mammalian airway tree. This is done in a 3D tree structure which can be studied from its topology only. The method is applied to the human acinus real morphology as studied by Haefeli-Bleuer and Weibel in order to compute the respiratory efficiency of the human lung. It provides the first quantitative evidence of the role of diffusion screening in real 3D mammalian respiration. The net result of this study is that, at rest, the efficiency of the human acinus is only of order 33%. Application of these results to CO2 clearance provides for the first time a theoretical support to the empirical relation between the O2 and CO2 partial pressures in blood.
RecRWR: a recursive random walk method for improved identification of diseases.
Arrais, Joel Perdiz; Oliveira, José Luís
2015-01-01
High-throughput methods such as next-generation sequencing or DNA microarrays lack precision, as they return hundreds of genes for a single disease profile. Several computational methods applied to physical interaction of protein networks have been successfully used in identification of the best disease candidates for each expression profile. An open problem for these methods is the ability to combine and take advantage of the wealth of biomedical data publicly available. We propose an enhanced method to improve selection of the best disease targets for a multilayer biomedical network that integrates PPI data annotated with stable knowledge from OMIM diseases and GO biological processes. We present a comprehensive validation that demonstrates the advantage of the proposed approach, Recursive Random Walk with Restarts (RecRWR). The obtained results outline the superiority of the proposed approach, RecRWR, in identifying disease candidates, especially with high levels of biological noise and benefiting from all data available.
A random-walk model for pore pressure accumulation in marine soils
Sumer, B. Mutlu; Cheng, Niang-Sheng
1999-01-01
waves. The model will apparently enable the researcher to handle complex geometries (such as a pipeline buried in a soil) relatively easily. Early results with regard to the latter example, namely the buildup of pore pressure around a buried pipeline subject to a progressive wave, are encouraging.......A numerical random-walk model has been developed for the pore-water pressure. The model is based on the analogy between the variation of the pore pressure and the diffusion process of any passive quantity such as concentration. The pore pressure in the former process is analogous...... to the concentration in the latter. In the simulation, particles are released in the soil, and followed as they travel through the statistical field variables. The model has been validated (1) against the Terzaghi consolidation process, and (2) against the process where the pore pressure builds up under progressive...
On the temporal order of first-passage times in one-dimensional lattice random walks
Sanders, J. B.; Temme, N. M.
2005-10-01
A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing modified Bessel functions of the first kind. By using several transformations, simpler integrals are obtained from which for two and three particles asymptotic approximations are derived for large values of the parameters. Expressions of the probability for n particles are also derived.I returned and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all. George Orwell, Politics and the English Language, Selected Essays, Penguin Books, 1957. (The citation is from Ecclesiastes 9:11.)
Random walk in nonhomogeneous environments: A possible approach to human and animal mobility
Srokowski, Tomasz
2017-03-01
The random walk process in a nonhomogeneous medium, characterized by a Lévy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is affected by the variable width and the variance may be finite; then all kinds of the anomalous diffusion are predicted. In the former case, only the time characteristics are sensitive to the variable width. The corresponding Langevin equation with different interpretations of the multiplicative noise is discussed. The dependence of the distribution width on position after jump is interpreted in terms of cognitive abilities and related to such problems as migration in a human population and foraging habits of animals.
Random-walk model to study cycles emerging from the exploration-exploitation trade-off
Kazimierski, Laila D.; Abramson, Guillermo; Kuperman, Marcelo N.
2015-01-01
We present a model for a random walk with memory, phenomenologically inspired in a biological system. The walker has the capacity to remember the time of the last visit to each site and the step taken from there. This memory affects the behavior of the walker each time it reaches an already visited site modulating the probability of repeating previous moves. This probability increases with the time elapsed from the last visit. A biological analog of the walker is a frugivore, with the lattice sites representing plants. The memory effect can be associated with the time needed by plants to recover its fruit load. We propose two different strategies, conservative and explorative, as well as intermediate cases, leading to nonintuitive interesting results, such as the emergence of cycles.
Upper tails of self-intersection local times of random walks: survey of proof techniques
König, Wolfgang
2010-01-01
The asymptotics of the probability that the self-intersection local time of a random walk on $\\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on {\\it Excess self-intersection local times, and related topics} in Luminy, 6-10 Dec., 2010.
Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?
Czégel, Dániel
2015-01-01
Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications...
Simply and multiply scaled diffusion limits for continuous time random walks
Gorenflo, Rudolf [Erstes Mathematisches Institut, Freie Universitaet Berlin, Arnimallee 3, D-14195 Berlin (Germany); Mainardi, Francesco [Dipartimento di Fisica, Universita di Bologna and INFN, Via Irnerio 46, I-40126 Bologna (Italy)
2005-01-01
First a survey is presented on how space-time fractional diffusion processes can be obtained by well-scaled limiting from continuous time random walks under the sole assumption of asymptotic power laws (with appropriate exponents for the tail behaviour of waiting times and jumps). The spatial operator in the limiting pseudo-differential equation is the inverse of a general Riesz-Feller potential operator. The analysis is carried out via the transforms of Fourier and Laplace. Then mixtures of waiting time distributions, likewise of jump distributions, are considered, and it is shown that correct multiple scaling in the limit yields diffusion equations with distributed order fractional derivatives (fractional operators being replaced by integrals over such ones, with the order of differentiation as variable of integration). It is outlined how in this way super-fast and super-slow diffusion can be modelled.
Most likely paths to error when estimating the mean of a reflected random walk
Duffy, Ken R
2009-01-01
It is known that simulation of the mean position of a reflected random walk $\\{W_n\\}$ exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let $I$ denote the rate function for the one dimensional increment process. If $I$ is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, $\\psi$, of the paths $n^{-1} W_{\\lfloor tn\\rfloor}$ is to be zero apart from on an interval $[T_0,T_1]\\subset[0,1]$ and to satisfy the functional equation \
Quantitative Assessment of Image Segmentation Quality by Random Walk Relaxation Times
Andres, Björn; Köthe, Ullrich; Bonea, Andreea; Nadler, Boaz; Hamprecht, Fred A.
The purpose of image segmentation is to partition the pixel grid of an image into connected components termed segments such that (i) each segment is homogenous and (ii) for any pair of adjacent segments, their union is not homogenous. (If it were homogenous the segments should be merged). We propose a rigorous definition of segment homogeneity which is scale-free and adaptive to the geometry of segments. We motivate this definition using random walk theory and show how segment homogeneity facilitates the quantification of violations of the conditions (i) and (ii) which are referred to as under-segmentation and over-segmentation, respectively. We describe the theoretical foundations of our approach and present a proof of concept on a few natural images.
Branching and annihilating random walks: exact results at low branching rate.
Benitez, Federico; Wschebor, Nicolás
2013-05-01
We present some exact results on the behavior of branching and annihilating random walks, both in the directed percolation and parity conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the nontrivial pure annihilation (PA) model, whose correlation and response function we compute exactly. With this, the nonuniversal threshold value for having a phase transition in the simplest system belonging to the directed percolation universality class is found to coincide with previous nonperturbative renormalization group (RG) approximate results. We also show that the parity conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
A random walk evolution model of wireless sensor networks and virus spreading
Wang Ya-Qi; Yang Xiao-Yuan
2013-01-01
In this paper,considering both cluster heads and sensor nodes,we propose a novel evolving a network model based on a random walk to study the fault tolerance decrease of wireless sensor networks (WSNs) due to node failure,and discuss the spreading dynamic behavior of viruses in the evolution model.A theoretical analysis shows that the WSN generated by such an evolution model not only has a strong fault tolerance,but also can dynamically balance the energy loss of the entire network.It is also found that although the increase of the density of cluster heads in the network reduces the network efficiency,it can effectively inhibit the spread of viruses.In addition,the heterogeneity of the network improves the network efficiency and enhances the virus prevalence.We confirm all the theoretical results with sufficient numerical simulations.
Cvetkovic, V.; Molin, S.
2012-02-01
We present a methodology that combines numerical simulations of groundwater flow and advective transport in heterogeneous porous media with analytical retention models for computing the infection risk probability from pathogens in aquifers. The methodology is based on the analytical results presented in [1,2] for utilising the colloid filtration theory in a time-domain random walk framework. It is shown that in uniform flow, the results from the numerical simulations of advection yield comparable results as the analytical TDRW model for generating advection segments. It is shown that spatial variability of the attachment rate may be significant, however, it appears to affect risk in a different manner depending on if the flow is uniform or radially converging. In spite of the fact that numerous issues remain open regarding pathogen transport in aquifers on the field scale, the methodology presented here may be useful for screening purposes, and may also serve as a basis for future studies that would include greater complexity.
One-dimensional random walk of nanosized liquid Pb inclusions on dislocations in Al
Johnson, E.; Levinsen, M.T.; Steenstrup, S.;
2004-01-01
to and perpendicular to the dislocations respectively. Movements parallel to the dislocation lines display properties of partially confined one-dimensional random walks where smaller inclusions can be seen to move over distances that are many times their own sizes. In contrast, the trajectories perpendicular......Migration of nanosized liquid Pb inclusions attached to dislocations in Al has been observed during in-situ transmission electron microscopy heating experiments and monitored by real-time video recordings. The movements of the inclusions can be separated into two independent components parallel...... to the dislocation lines are within narrowly confined spaces. Frame-by-frame analysis of digitized video sequences recorded at different temperatures for the same inclusion attached to a nearly horizontal dislocation illustrates the two types of movement. The step lengths parallel to the dislocation increase rapidly...
Ni Xiaohui [School of Business, East China University of Science and Technology, Shanghai 200237 (China)] [School of Science, East China University of Science and Technology, Shanghai 200237 (China)] [Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237 (China); Jiang Zhiqiang [School of Business, East China University of Science and Technology, Shanghai 200237 (China)] [School of Science, East China University of Science and Technology, Shanghai 200237 (China)] [Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237 (China)] [Chair of Entrepreneurial Risks, D-MTEC, ETH Zurich, Kreuplatz 5, CH-8032 Zurich (Switzerland); Zhou Weixing, E-mail: wxzhou@ecust.edu.c [School of Business, East China University of Science and Technology, Shanghai 200237 (China)] [School of Science, East China University of Science and Technology, Shanghai 200237 (China)] [Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237 (China)] [Engineering Research Center of Process Systems Engineering (Ministry of Education), East China University of Science and Technology, Shanghai 200237 (China)] [Research Center on Fictitious Economics and Data Science, Chinese Academy of Sciences, Beijing 100080 (China)
2009-10-12
The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent alpha is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the alphaapproxH linear relationship.
Flexible sampling large-scale social networks by self-adjustable random walk
Xu, Xiao-Ke; Zhu, Jonathan J. H.
2016-12-01
Online social networks (OSNs) have become an increasingly attractive gold mine for academic and commercial researchers. However, research on OSNs faces a number of difficult challenges. One bottleneck lies in the massive quantity and often unavailability of OSN population data. Sampling perhaps becomes the only feasible solution to the problems. How to draw samples that can represent the underlying OSNs has remained a formidable task because of a number of conceptual and methodological reasons. Especially, most of the empirically-driven studies on network sampling are confined to simulated data or sub-graph data, which are fundamentally different from real and complete-graph OSNs. In the current study, we propose a flexible sampling method, called Self-Adjustable Random Walk (SARW), and test it against with the population data of a real large-scale OSN. We evaluate the strengths of the sampling method in comparison with four prevailing methods, including uniform, breadth-first search (BFS), random walk (RW), and revised RW (i.e., MHRW) sampling. We try to mix both induced-edge and external-edge information of sampled nodes together in the same sampling process. Our results show that the SARW sampling method has been able to generate unbiased samples of OSNs with maximal precision and minimal cost. The study is helpful for the practice of OSN research by providing a highly needed sampling tools, for the methodological development of large-scale network sampling by comparative evaluations of existing sampling methods, and for the theoretical understanding of human networks by highlighting discrepancies and contradictions between existing knowledge/assumptions of large-scale real OSN data.
Qing Guo
2015-04-01
Full Text Available A gait identification method for a lower extremity exoskeleton is presented in order to identify the gait sub-phases in human-machine coordinated motion. First, a sensor layout for the exoskeleton is introduced. Taking the difference between human lower limb motion and human-machine coordinated motion into account, the walking gait is divided into five sub-phases, which are ‘double standing’, ‘right leg swing and left leg stance’, ‘double stance with right leg front and left leg back’, ‘right leg stance and left leg swing’, and ‘double stance with left leg front and right leg back’. The sensors include shoe pressure sensors, knee encoders, and thigh and calf gyroscopes, and are used to measure the contact force of the foot, and the knee joint angle and its angular velocity. Then, five sub-phases of walking gait are identified by a C4.5 decision tree algorithm according to the data fusion of the sensors’ information. Based on the simulation results for the gait division, identification accuracy can be guaranteed by the proposed algorithm. Through the exoskeleton control experiment, a division of five sub-phases for the human-machine coordinated walk is proposed. The experimental results verify this gait division and identification method. They can make hydraulic cylinders retract ahead of time and improve the maximal walking velocity when the exoskeleton follows the person’s motion.
Qing Guo
2015-04-01
Full Text Available A gait identification method for a lower extremity exoskeleton is presented in order to identify the gait sub-phases in human-machine coordinated motion. First, a sensor layout for the exoskeleton is introduced. Taking the difference between human lower limb motion and human-machine coordinated motion into account, the walking gait is divided into five sub-phases, which are ‘double standing’, ‘right leg swing and left leg stance’, ‘double stance with right leg front and left leg back’, ‘right leg stance and left leg swing’, and ‘double stance with left leg front and right leg back’. The sensors include shoe pressure sensors, knee encoders, and thigh and calf gyroscopes, and are used to measure the contact force of the foot, and the knee joint angle and its angular velocity. Then, five sub-phases of walking gait are identified by a C4.5 decision tree algorithm according to the data fusion of the sensors' information. Based on the simulation results for the gait division, identification accuracy can be guaranteed by the proposed algorithm. Through the exoskeleton control experiment, a division of five sub-phases for the human-machine coordinated walk is proposed. The experimental results verify this gait division and identification method. They can make hydraulic cylinders retract ahead of time and improve the maximal walking velocity when the exoskeleton follows the person's motion.
Hurley, Jane C; Hollingshead, Kevin E; Todd, Michael; Jarrett, Catherine L; Tucker, Wesley J; Angadi, Siddhartha S; Adams, Marc A
2015-09-11
Walking is a widely accepted and frequently targeted health promotion approach to increase physical activity (PA). Interventions to increase PA have produced only small improvements. Stronger and more potent behavioral intervention components are needed to increase time spent in PA, improve cardiometabolic risk markers, and optimize health. Our aim is to present the rationale and methods from the WalkIT Trial, a 4-month factorial randomized controlled trial (RCT) in inactive, overweight/obese adults. The main purpose of the study was to evaluate whether intensive adaptive components result in greater improvements to adults' PA compared to the static intervention components. Participants enrolled in a 2x2 factorial RCT and were assigned to one of four semi-automated, text message-based walking interventions. Experimental components included adaptive versus static steps/day goals, and immediate versus delayed reinforcement. Principles of percentile shaping and behavioral economics were used to operationalize experimental components. A Fitbit Zip measured the main outcome: participants' daily physical activity (steps and cadence) over the 4-month duration of the study. Secondary outcomes included self-reported PA, psychosocial outcomes, aerobic fitness, and cardiorespiratory risk factors assessed pre/post in a laboratory setting. Participants were recruited through email listservs and websites affiliated with the university campus, community businesses and local government, social groups, and social media advertising. This study has completed data collection as of December 2014, but data cleaning and preliminary analyses are still in progress. We expect to complete analysis of the main outcomes in late 2015 to early 2016. The Walking Interventions through Texting (WalkIT) Trial will further the understanding of theory-based intervention components to increase the PA of men and women who are healthy, insufficiently active and are overweight or obese. WalkIT is one of
Fayolle, Guy; Malyshev, Vadim
2017-01-01
This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function Theory, Boundary Value Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems. Part II borrows spec...
Walking adaptability therapy after stroke: study protocol for a randomized controlled trial
Timmermans, C.; Roerdink, M.; Ooijen, van M.W.; Meskens, C.G.; Janssen, T.W.J.; Beek, P.J.
2016-01-01
Background: Walking in everyday life requires the ability to adapt walking to the environment. This adaptability is often impaired after stroke, and this might contribute to the increased fall risk after stroke. To improve safe community ambulation, walking adaptability training might be beneficial
Kottink, A.I.R.; Kottink, Anke I.; Hermens, Hermanus J.; Nene, A.V.; Tenniglo, Martinus Johannes Bernardus; van der Aa, Hans E.; Buschman, H.P.J.; IJzerman, Maarten Joost
Objective To determine the effect of a new implantable 2-channel peroneal nerve stimulator on walking speed and daily activities, in comparison with the usual treatment in chronic stroke survivors with a drop foot. Design Randomized controlled trial. Setting All subjects were measured 5 times in the
Scholtes, Vanessa A.; Becher, Jules G.; Janssen-Potten, Yvonne J.; Dekkers, Hurnet; Smallenbroek, Linda; Dallmeijer, Annet J.
2012-01-01
The objective of the study was to evaluate the effectiveness of functional progressive resistance exercise (PRE) training on walking ability in children with cerebral palsy (CP). Fifty-one ambulant children with spastic CP (mean age 10 years 5 months, 29 boys) were randomized to an intervention (n=26) or control group (n=25, receiving usual care).…
Schippers, P.; Verboom, J.; Knaapen, J.P.; Apeldoorn, van R.
1996-01-01
A grid-based random walk model has been developed to simulate animal dispersal, taking landscape heterogeneity and linear barriers such as roads and rivers into account. The model can be used to estimate connectivity and has been parameterized for thebadger in the central part of the Netherlands. Th
Schippers, P.; Verboom, J.; Knaapen, J.P.; Apeldoorn, van R.
1996-01-01
A grid-based random walk model has been developed to simulate animal dispersal, taking landscape heterogeneity and linear barriers such as roads and rivers into account. The model can be used to estimate connectivity and has been parameterized for thebadger in the central part of the Netherlands.
de Wit, D C M; Buurke, J H; Nijlant, J M M; Ijzerman, M J; Hermens, H J
2004-08-01
Regaining walking ability is a major goal during the rehabilitation of stroke patients. To support this process an ankle-foot orthosis (AFO) is often prescribed. The aim of this study is to investigate the effect of an AFO on walking ability in chronic stroke patients. Cross-over design with randomization for the interventions. Twenty chronic stroke patients, wearing an AFO for at least six months, were included. Walking ability was operationalized as comfortable walking speed, scores on the timed up and go (TUG) test and stairs test. Patients were measured with and without their AFO, the sequence of which was randomized. Additionally, subjective impressions of self-confidence and difficulty of the tasks were scored. Clinically relevant differences based on literature were defined for walking speed (20 cm/s), the TUG test (10 s). Gathered data were statistically analysed using a paired t-test. The mean difference in favour of the AFO in walking speed was 4.8 cm/s (95% CI 0.85-8.7), in the TUG test 3.6 s (95% CI 2.4-4.8) and in the stairs test 8.6 s (95% CI 3.1-14.1). Sixty-five per cent of the patients experienced less difficulty and 70% of the patients felt more self-confident while wearing the AFO. The effect of an AFO on walking ability is statistically significant, but compared with the a priori defined differences it is too small to be clinically relevant. The effect on self-confidence suggests that other factors might play an important role in the motivation to use an AFO.
Kalron, Alon; Rosenblum, Uri; Frid, Lior; Achiron, Anat
2017-03-01
Evaluate the effects of a Pilates exercise programme on walking and balance in people with multiple sclerosis and compare this exercise approach to conventional physical therapy sessions. Randomized controlled trial. Multiple Sclerosis Center, Sheba Medical Center, Tel-Hashomer, Israel. Forty-five people with multiple sclerosis, 29 females, mean age (SD) was 43.2 (11.6) years; mean Expanded Disability Status Scale (S.D) was 4.3 (1.3). Participants received 12 weekly training sessions of either Pilates ( n=22) or standardized physical therapy ( n=23) in an outpatient basis. Spatio-temporal parameters of walking and posturography parameters during static stance. Functional tests included the Time Up and Go Test, 2 and 6-minute walk test, Functional Reach Test, Berg Balance Scale and the Four Square Step Test. In addition, the following self-report forms included the Multiple Sclerosis Walking Scale and Modified Fatigue Impact Scale. At the termination, both groups had significantly increased their walking speed ( P=0.021) and mean step length ( P=0.023). According to the 2-minute and 6-minute walking tests, both groups at the end of the intervention program had increased their walking speed. Mean (SD) increase in the Pilates and physical therapy groups were 39.1 (78.3) and 25.3 (67.2) meters, respectively. There was no effect of group X time in all instrumented and clinical balance and gait measures. Pilates is a possible treatment option for people with multiple sclerosis in order to improve their walking and balance capabilities. However, this approach does not have any significant advantage over standardized physical therapy.
Comolli, Alessandro; Hakoun, Vivien; Dentz, Marco
2017-04-01
Achieving the understanding of the process of solute transport in heterogeneous porous media is of crucial importance for several environmental and social purposes, ranging from aquifers contamination and remediation, to risk assessment in nuclear waste repositories. The complexity of this aim is mainly ascribable to the heterogeneity of natural media, which can be observed at all the scales of interest, from pore scale to catchment scale. In fact, the intrinsic heterogeneity of porous media is responsible for the arising of the well-known non-Fickian footprints of transport, including heavy-tailed breakthrough curves, non-Gaussian spatial density profiles and the non-linear growth of the mean squared displacement. Several studies investigated the processes through which heterogeneity impacts the transport properties, which include local modifications to the advective-dispersive motion of solutes, mass exchanges between some mobile and immobile phases (e.g. sorption/desorption reactions or diffusion into solid matrix) and spatial correlation of the flow field. In the last decades, the continuous time random walk (CTRW) model has often been used to describe solute transport in heterogenous conditions and to quantify the impact of point heterogeneity, spatial correlation and mass transfer on the average transport properties [1]. Open issues regarding this approach are the possibility to relate measurable properties of the medium to the parameters of the model, as well as its capability to provide predictive information. In a recent work [2] the authors have shed new light on understanding the relationship between Lagrangian and Eulerian dynamics as well as on their evolution from arbitrary initial conditions. On the basis of these results, we derive a CTRW model for the description of Darcy-scale transport in d-dimensional media characterized by spatially random permeability fields. The CTRW approach models particle velocities as a spatial Markov process, which is
Kelbert, M. Ya.; Suhov, Yu. M.
1995-02-01
A general model of a branching random walk in R 1 is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable 220_2005_Article_BF02101538_TeX2GIFE1.gif X^0 = mathop {sup max}limits_{ n ≥q 0 1 ≤q k ≤q N_n } X_{n,k} to be finite. Here, X n, k is the position of the k th particle in the n th generation, N n is the number of particles in the n th generation (regardless of their type). It turns out that the distribution of X 0 gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.
On efficient randomized algorithms for finding the PageRank vector
Gasnikov, A. V.; Dmitriev, D. Yu.
2015-03-01
Two randomized methods are considered for finding the PageRank vector; in other words, the solution of the system p T = p T P with a stochastic n × n matrix P, where n ˜ 107-109, is sought (in the class of probability distributions) with accuracy ɛ: ɛ ≫ n -1. Thus, the possibility of brute-force multiplication of P by the column is ruled out in the case of dense objects. The first method is based on the idea of Markov chain Monte Carlo algorithms. This approach is efficient when the iterative process p {/t+1 T} = p {/t T} P quickly reaches a steady state. Additionally, it takes into account another specific feature of P, namely, the nonzero off-diagonal elements of P are equal in rows (this property is used to organize a random walk over the graph with the matrix P). Based on modern concentration-of-measure inequalities, new bounds for the running time of this method are presented that take into account the specific features of P. In the second method, the search for a ranking vector is reduced to finding the equilibrium in the antagonistic matrix game where S n (1) is a unit simplex in ℝ n and I is the identity matrix. The arising problem is solved by applying a slightly modified Grigoriadis-Khachiyan algorithm (1995). This technique, like the Nazin-Polyak method (2009), is a randomized version of Nemirovski's mirror descent method. The difference is that randomization in the Grigoriadis-Khachiyan algorithm is used when the gradient is projected onto the simplex rather than when the stochastic gradient is computed. For sparse matrices P, the method proposed yields noticeably better results.
Stochastic geometry, spatial statistics and random fields models and algorithms
2015-01-01
Providing a graduate level introduction to various aspects of stochastic geometry, spatial statistics and random fields, this volume places a special emphasis on fundamental classes of models and algorithms as well as on their applications, for example in materials science, biology and genetics. This book has a strong focus on simulations and includes extensive codes in Matlab and R, which are widely used in the mathematical community. It can be regarded as a continuation of the recent volume 2068 of Lecture Notes in Mathematics, where other issues of stochastic geometry, spatial statistics and random fields were considered, with a focus on asymptotic methods.
Revisiting random walks in fractal media: on the occurrence of time discrete scale invariance.
Bab, M A; Fabricius, G; Albano, Ezequiel V
2008-01-28
This paper addresses the kinetic behavior of random walks in fractal media. We perform extensive numerical simulations of both single and annihilating random walkers on several Sierpinski carpets, in order to study the time behavior of three observables: the average number of distinct sites visited by a single walker, the mean-square displacement from the origin, and the density of annihilating random walkers. We found that the time behavior of those observables is given by a power law modulated by soft logarithmic-periodic oscillations. We conjecture that logarithmic-periodic oscillations are a manifestation of a time domain discrete scale iNvariance (DSI) that occurs as a consequence of the spatial DSI of the substrate. Our conjecture implies that the logarithmic periods of oscillations in space and time domains are linked by a dynamic exponent z, through z=log(tau)/log(b(1)), where tau and b(1) are the fundamental scaling ratios of the DSI symmetry in the time and space domains, respectively. We use this relationship in order to compute z for different observables and fractals. Furthermore, we check the values obtained with independent measurements provided by the power-law behavior of the mean-square displacement with time [R(2)(t) proportional variant t(2/z)]. The very good agreement obtained between both computations of the z exponent gives strong support to the idea of an intimate interplay between spatial and time symmetry properties that we expect will have a quite general scope. We expect that the application of the outlined concepts in the field of dynamic processes in fractal media will stimulate further research.
Finding tree symmetries using continuous-time quantum walk
Wu Jun-Jie; Zhang Bai-Da; Tang Yu-Hua; Qiang Xiao-Gang; Wang Hui-Quan
2013-01-01
Quantum walk,the quantum counterpart of random walk,is an important model and widely studied to develop new quantum algorithms.This paper studies the relationship between the continuous-time quantum walk and the symmetry of a graph,especially that of a tree.Firstly,we prove in mathematics that the symmetry of a graph is highly related to quantum walk.Secondly,we propose an algorithm based on the continuous-time quantum walk to compute the symmetry of a tree.Our algorithm has better time complexity O(N3) than the current best algorithm.Finally,through testing three types of 10024 trees,we find that the symmetry of a tree can be found with an extremely high efficiency with the help of the continuous-time quantum walk.
Financial Data Analysis by means of Coupled Continuous-Time Random Walk in Rachev-Rűschendorf Model
Jurlewicz, A.; Wyłomańska, A.; Żebrowski, P.
2008-09-01
We adapt the continuous-time random walk formalism to describe asset price evolution. We expand the idea proposed by Rachev and Rűschendorf who analyzed the binomial pricing model in the discrete time with randomization of the number of price changes. As a result, in the framework of the proposed model we obtain a mixture of the Gaussian and a generalized arcsine laws as the limiting distribution of log-returns. Moreover, we derive an European-call-option price that is an extension of the Black-Scholes formula. We apply the obtained theoretical results to model actual financial data and try to show that the continuous-time random walk offers alternative tools to deal with several complex issues of financial markets.
A random forest algorithm for nowcasting of intense precipitation events
Das, Saurabh; Chakraborty, Rohit; Maitra, Animesh
2017-09-01
Automatic nowcasting of convective initiation and thunderstorms has potential applications in several sectors including aviation planning and disaster management. In this paper, random forest based machine learning algorithm is tested for nowcasting of convective rain with a ground based radiometer. Brightness temperatures measured at 14 frequencies (7 frequencies in 22-31 GHz band and 7 frequencies in 51-58 GHz bands) are utilized as the inputs of the model. The lower frequency band is associated to the water vapor absorption whereas the upper frequency band relates to the oxygen absorption and hence, provide information on the temperature and humidity of the atmosphere. Synthetic minority over-sampling technique is used to balance the data set and 10-fold cross validation is used to assess the performance of the model. Results indicate that random forest algorithm with fixed alarm generation time of 30 min and 60 min performs quite well (probability of detection of all types of weather condition ∼90%) with low false alarms. It is, however, also observed that reducing the alarm generation time improves the threat score significantly and also decreases false alarms. The proposed model is found to be very sensitive to the boundary layer instability as indicated by the variable importance measure. The study shows the suitability of a random forest algorithm for nowcasting application utilizing a large number of input parameters from diverse sources and can be utilized in other forecasting problems.
Rosvall, M
2010-01-01
To comprehend the hierarchical organization of large integrated systems, we introduce the hierarchical map equation that reveals multilevel structures in networks. In this information-theoretic approach, we exploit the duality between compression and pattern detection; by compressing a description of a random walker as a proxy for real flow on a network, we find regularities in the network that induce this system-wide flow. Finding the shortest multilevel description of the random walker therefore gives us the best hierarchical clustering of the network, the optimal number of levels and modular partition at each level, with respect to the dynamics on the network. With a novel search algorithm, we extract and illustrate the rich multilevel organization of several large social and biological networks. For example, from the global air traffic network we uncover countries and continents, and from the pattern of scientific communication we reveal more than 100 scientific fields organized in four major disciplines:...
Jung, Kyeong-Man; Bang, Dae-Hyouk
2017-02-01
[Purpose] To investigate the effects of inspiratory muscle training on respiratory capacity and walking ability in subacute stroke patients. [Subjects and Methods] The subjects were randomly assigned to an experimental group (n=6) or a control group (n=6). Patients in the experimental group received inspiratory muscle training for 30 minutes (six sets of five-minutes) and traditional physical therapy once a day, five days a week, for four weeks. The control group received aerobic exercise for 30 minutes and traditional physical therapy for 30 minutes a day, five days a week, for four weeks. [Results] After the intervention, both groups showed significant improvements in the forced vital capacity, forced expiratory volume in one second, 10-meter walking test, and six-minute walking test over the baseline results. There were significant between-group differences for the forced vital capacity, forced expiratory volume in one second, and six-minute walking test. No statistically significant differences were observed for measures of saturation pulse oximetry oxygen and 10-meter walking test between the groups. [Conclusion] These findings gave some indications that inspiratory muscle training may benefit in patients with subacute stroke, and it is feasible to be included in rehabilitation program with this population.
Bang, Dae-Hyouk; Cho, Hyuk-Shin
2016-01-01
[Purpose] To investigate the effects of body awareness training on balance and walking ability in chronic stroke patients. [Subjects] The subjects were randomly assigned to a body awareness training group (n=6) and a control group (n=6). [Methods] Patients in the body awareness training group received body awareness training for 20 minutes, followed by walking training for 30 minutes a day, 5 days a week for 4 weeks. The control group received walking training for 30 minutes a day, 5 days a week for 4 weeks. [Results] After the intervention, both groups showed significant improvements in the Berg Balance Scale, Timed Up and Go Test, and 10 m walk test compared with baseline results. The body awareness training group showed more significant improvements in the Berg Balance Scale and Timed Up and Go Test than the control group. There was no significant difference in the 10 m walk test between the groups. [Conclusion] The results of this study suggest that body awareness training has a positive effect on balance in patients with chronic stroke.
Network Location-Aware Service Recommendation with Random Walk in Cyber-Physical Systems.
Yin, Yuyu; Yu, Fangzheng; Xu, Yueshen; Yu, Lifeng; Mu, Jinglong
2017-09-08
Cyber-physical systems (CPS) have received much attention from both academia and industry. An increasing number of functions in CPS are provided in the way of services, which gives rise to an urgent task, that is, how to recommend the suitable services in a huge number of available services in CPS. In traditional service recommendation, collaborative filtering (CF) has been studied in academia, and used in industry. However, there exist several defects that limit the application of CF-based methods in CPS. One is that under the case of high data sparsity, CF-based methods are likely to generate inaccurate prediction results. In this paper, we discover that mining the potential similarity relations among users or services in CPS is really helpful to improve the prediction accuracy. Besides, most of traditional CF-based methods are only capable of using the service invocation records, but ignore the context information, such as network location, which is a typical context in CPS. In this paper, we propose a novel service recommendation method for CPS, which utilizes network location as context information and contains three prediction models using random walking. We conduct sufficient experiments on two real-world datasets, and the results demonstrate the effectiveness of our proposed methods and verify that the network location is indeed useful in QoS prediction.
Biased random walk in spatially embedded networks with total cost constraint
Niu, Rui-Wu; Pan, Gui-Jun
2016-11-01
We investigate random walk with a bias toward a target node in spatially embedded networks with total cost restriction introduced by Li et al. (2010). Precisely, The network is built from a two-dimension regular lattice to be improved by adding long-range shortcuts with probability P(rij) ∼rij-α, where rij is the Manhattan distance between sites i and j, and α is a variable exponent, the total length of the long-range connections is restricted. Bias is represented as a probability p of the packet or particle to travel at every hop toward the node which has the smallest Manhattan distance to the target node. By studying the mean first passage time (MFPT) for different exponent log , we find that the best transportation condition is obtained with an exponent α = d + 1(d = 2) for all p. The special phenomena can be possibly explained by the theory of information entropy, we find that when α = d + 1(d = 2), the spatial network with total cost restriction becomes an optimal network which has a maximum information entropy. In addition, the scaling of the MFPT with the size of the network is also investigated, and finds that the scaling of the MFPT with L follows a linear distribution for all p > 0.
Random walk with nonuniform angular distribution biased by an external periodic pulse
Acharyya, Aranyak
2016-11-01
We studied the motion of a random walker in two dimensions with nonuniform angular distribution biased by an external periodic pulse. Here, we analytically calculated the mean square displacement (end-to-end distance of a walk after n time steps), without bias and with bias. We determined the average x-component of the final displacement of the walker. Interestingly, we noted that for a particular periodicity of the bias, this average x-component of the final displacement becomes approximately zero. The average y-component of the final displacement is found to be zero for any perodicity of the bias, and its reason can be attributed to the nature of the probability density function of the angle (subtended by the displacement vector with the x-axis). These analytical results are also supported by computer simulations. The present study may be thought of as a model for arresting the bacterial motion (along a preferred direction) by an external periodic bias. This article will be useful for undergraduate students of physics, statistics and biology as an example of an interdisciplinary approach to understand a way to control bacterial motion.
Transport of dense pollutants: nonlinear random walk modeling and experimental validation
Zoia, A.; Latrille, C.; Cartalade, A.
2009-04-01
Fickian transport with uncorrelated particles paths is recovered. We have tested the proposed random walk model on experimental measurements of dense contaminant transport obtained with the BEETI experimental device, a dichromatic X-ray source coupled with a NaI detector [5] This setup allows quantitatively assessing the contaminant concentration câ(t) inside a vertical 80 cm column (as a function of time), at various sections â. The injected contaminant is KI and the column is filled with homogeneously mixed Fontainebleau sand. As a salient feature, contaminant profiles are sensibly skewed (depending on the flow direction) and therefore non-Gaussian. Monte Carlo estimates of concentration profiles and temporal moments have been computed and a good agreement is found between simulation results and experimental data, for both downwards and upwards injection, at various flow regimes and molar concentrations. The proposed random walk model is admittedly simple, since the full spectrum of interactions that actually take place between the velocity and density fields [2-4] has been condensed in a single nonlinear coupling at the scale of particles trajectories. Yet, despite its simplicity, it compares well to the set of dense contaminant transport measurements. Finally, the random walk approach has been rephrased in terms of a more general nonlinear master equation [6], thus providing a link with the Continuous Time Random Walk (CTRW) formalism [1,7]. The CTRW framework can be used to deal with heterogenous and/or unsaturated porous media and this allows extending our model, so to make predictions about pollutants behavior in such complex materials. References [1] B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Rev. Geophys. 44, RG2003 (2006). [2] S. M. Hassanizadeh and A. Leijnse, Adv. Water Resour. 18, 203 (1995). [3] C. T. Simmons, T. R. Fenstemaker, and J. M. Sharp Jr., J. Contam. Hydrology 52, 245 (2001). [4] H.-J. G. Diersch and O. Kolditz, Adv. Water Resour
Geometric random walk of finite number of agents under constant variance
Yano, Ryosuke
2017-05-01
The characteristics of the 1D geometric random walk of a finite number of agents are investigated by assuming constant variance. Firstly, the characteristics of the steady state solution of the distribution function, which is obtained using the extended geometric Brownian motion (EGBM), are investigated in the framework of the 1D Fokker-Planck type equation. The uniqueness and existence of the steady state solution of the distribution function requires the number of particles to be finite. To avoid the divergence of the steady state solution of the distribution function at the mean value in the 1D Fokker-Planck type equation, the hybrid model, which is a combination of EGBM and normal BM, is proposed. Next, the steady state solution of the distribution function, which is obtained using the geometric Lévy flight, is investigated under constant variance in the framework of the space fractional 1D Fokker-Planck type equation. Additionally, we confirm that the solution of the distribution function obtained using the super-elastic and inelastic (SI-) Boltzmann equation under constant variance approaches the Cauchy distribution, when the power law number of the relative velocity increases. Finally, dissipation processes of the pressure deviator and heat flux are numerically investigated using the 2D space fractional Fokker-Planck type equations for Lévy flight and SI-Boltzmann equation by assuming their linear response relations.
Random Walks, Electric Networks and The Transience Class problem of Sandpiles
Choure, Ayush
2011-01-01
The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \\cite{DD90}, Dhar et al. \\cite{DD95}) which serves as the standard model of \\textit{self-organized criticality}. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent (\\cite{BT05}). We develop the theory of discrete diffusions in contrast to continuous harmonic functions on graphs and establish deep connections between standard results in the study of random walks on graphs and sandpiles on graphs. Using this connection and building other necessary machinery we improve the main result of Babai and Gorodezky (SODA 2007,\\cite{LB07}) of the bound on the transience class of an $n \\times n$ grid, from $O(n^{30})$ to $O(n^{7})$. Proving that the transience class is small validates the general notion that for most natural phenomenon, the time during which the system is transient is small. In addition, we use the machinery developed to prove a number of au...
Continuous-time random walk for open systems: fluctuation theorems and counting statistics.
Esposito, Massimiliano; Lindenberg, Katja
2008-05-01
We consider continuous-time random walks (CTRW) for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution (WTD) for times between steps is characterized by a positive parameter alpha , which is set to alpha=1 if it decays at least as fast as t{-2} at long times and therefore has a finite first moment. A WTD with alpha<1 decays as t{-alpha-1} . A fluctuation theorem for the trajectory quantity R , defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have alpha=1 and satisfy separability (also called "direction time independence"). For nonseparable WTDs with alpha=1 , R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0
Random Walks in Anderson's Garden: A Journey from Cuprates to Cooper Pair Insulators and Beyond
Baskaran, G
2016-01-01
Anderson's Garden is a drawing presented to Philip W. Anderson on the eve of his 60th birthday celebration, in 1983. This cartoon (Fig. 1), whose author is unknown, succinctly depicts some of Anderson's pre-1983 works, as a blooming garden. As an avid reader of Anderson's papers, random walk in Anderson's garden had become a part of my routine since graduate school days. This was of immense help and prepared me for a wonderful collaboration with the gardener himself, on the resonating valence bond (RVB) theory of High Tc cuprates and quantum spin liquids, at Princeton. The result was bountiful - the first (RVB mean field) theory for i) quantum spin liquids, ii) emergent fermi surfaces in Mott insulators and iii) superconductivity in doped Mott insulators. Beyond mean field theory - i) emergent gauge fields, ii) Ginzbuerg Landau theory with RVB gauge fields, iii) prediction of superconducting dome, iv) an early identification and study of a non-fermi liquid normal state of cuprates and so on. Here I narrate th...
Limitations on the recovery of the true AGN variability parameters using Damped Random Walk modeling
Kozłowski, Szymon
2016-01-01
Context: The damped random walk (DRW) stochastic process is nowadays frequently used to model aperiodic light curves of AGNs. A number of correlations between the DRW model parameters, the signal decorrelation timescale and amplitude, and the physical AGN parameters such as the black hole mass or luminosity have been reported. Aims: We are interested in whether it is plausible to correctly measure the DRW parameters from a typical ground-based survey, in particular how accurate the recovered DRW parameters are compared to the input ones. Methods: By means of Monte Carlo simulations of AGN light curves, we study the impact of the light curve length, the source magnitude, cadence, and additional light on the DRW model parameters. Results: The most significant finding is that currently existing surveys are going to return unconstrained DRW decorrelation timescales, because typical rest-frame data do not probe long enough timescales or the white noise part of PSD for DRW. The experiment length must be at least te...
Finite current stationary states of random walks on one-dimensional lattices with aperiodic disorder
Miki, Hiroshi
2016-11-01
Stationary states of random walks with finite induced drift velocity on one-dimensional lattices with aperiodic disorder are investigated by scaling analysis. Three aperiodic sequences, the Thue-Morse (TM), the paperfolding (PF), and the Rudin-Shapiro (RS) sequences, are used to construct the aperiodic disorder. These are binary sequences, composed of two symbols A and B, and the ratio of the number of As to that of Bs converges to unity in the infinite sequence length limit, but their effects on diffusional behavior are different. For the TM model, the stationary distribution is extended, as in the case without current, and the drift velocity is independent of the system size. For the PF model and the RS model, as the system size increases, the hierarchical and fractal structure and the localized structure, respectively, are broken by a finite current and changed to an extended distribution if the system size becomes larger than a certain threshold value. Correspondingly, the drift velocity is saturated in a large system while in a small system it decreases as the system size increases.