On the number of spanning trees in random regular graphs
Greenhill, Catherine; Kwan, Matthew; Wind, David Kofoed
2014-01-01
Let d >= 3 be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random d-regular graph with n vertices. (The asymptotics are as n -> infinity, restricted to even n if d is odd.) We also obtain the asymptotic distribution of the number of spanni...
Modularity of tree-like and random regular graphs
McDiarmid, Colin
2016-01-01
Clustering algorithms for large networks typically use the modularity score to compare which partitions better represent modular structure in the data. Given a network, the modularity of a partition of the vertex set is a number in [0, 1) which measures the extent to which edge density is higher within parts than between parts; and the modularity of the network is the maximum modularity of any partition. We show that random cubic graphs usually have modularity in the interval (0.666, 0.804); and random r-regular graphs for large r usually have modularity ${\\Theta}(1/\\sqrt{r})$. Our results can give thresholds for the statistical significance of clustering found in large regular networks. The modularity of cycles and low degree trees is known to be asymptotically 1. We extend these results to all graphs whose product of treewidth and maximum degree is much less than the number of edges. This shows for example that random planar graphs typically have modularity close to 1.
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.
Cycles and eigenvalues of sequentially growing random regular graphs
Johnson, Tobias
2012-01-01
Consider the sum of d many iid random permutation matrices on n labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree 2d on n vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article we explore joint asymptotic (in n) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These proc...
Competing first passage percolation on random regular graphs
Antunović, Tonći; Mossel, Elchanan; Peres, Yuval
2011-01-01
We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy Theta(1) N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the ...
Decentralized formation of random regular graphs for robust multi-agent networks
Yazicioglu, A. Yasin
2014-12-15
Multi-agent networks are often modeled via interaction graphs, where the nodes represent the agents and the edges denote direct interactions between the corresponding agents. Interaction graphs have significant impact on the robustness of networked systems. One family of robust graphs is the random regular graphs. In this paper, we present a locally applicable reconfiguration scheme to build random regular graphs through self-organization. For any connected initial graph, the proposed scheme maintains connectivity and the average degree while minimizing the degree differences and randomizing the links. As such, if the average degree of the initial graph is an integer, then connected regular graphs are realized uniformly at random as time goes to infinity.
Anderson localization and ergodicity on random regular graphs
Tikhonov, K. Â. S.; Mirlin, A. Â. D.; Skvortsov, M. Â. A.
2016-12-01
A numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. In a certain sense, the RRG ensemble can be seen as an infinite-dimensional (d →∞ ) cousin of the Anderson model in d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from a small (N ≪Nc ) to a large (N ≫Nc ) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. Our results support an analytical prediction that states in the delocalized phase (and at N ≫Nc ) are ergodic in the sense that their inverse participation ratio scales as 1 /N .
Formation of Robust Multi-Agent Networks through Self-Organizing Random Regular Graphs
Yasin Yazicioǧlu, A.
2015-11-25
Multi-Agent networks are often modeled as interaction graphs, where the nodes represent the agents and the edges denote some direct interactions. The robustness of a multi-Agent network to perturbations such as failures, noise, or malicious attacks largely depends on the corresponding graph. In many applications, networks are desired to have well-connected interaction graphs with relatively small number of links. One family of such graphs is the random regular graphs. In this paper, we present a decentralized scheme for transforming any connected interaction graph with a possibly non-integer average degree of k into a connected random m-regular graph for some m ϵ [k+k ] 2. Accordingly, the agents improve the robustness of the network while maintaining a similar number of links as the initial configuration by locally adding or removing some edges. © 2015 IEEE.
Multiple graph regularized protein domain ranking
Wang, Jim Jing-Yan
2012-11-19
Background: Protein domain ranking is a fundamental task in structural biology. Most protein domain ranking methods rely on the pairwise comparison of protein domains while neglecting the global manifold structure of the protein domain database. Recently, graph regularized ranking that exploits the global structure of the graph defined by the pairwise similarities has been proposed. However, the existing graph regularized ranking methods are very sensitive to the choice of the graph model and parameters, and this remains a difficult problem for most of the protein domain ranking methods.Results: To tackle this problem, we have developed the Multiple Graph regularized Ranking algorithm, MultiG-Rank. Instead of using a single graph to regularize the ranking scores, MultiG-Rank approximates the intrinsic manifold of protein domain distribution by combining multiple initial graphs for the regularization. Graph weights are learned with ranking scores jointly and automatically, by alternately minimizing an objective function in an iterative algorithm. Experimental results on a subset of the ASTRAL SCOP protein domain database demonstrate that MultiG-Rank achieves a better ranking performance than single graph regularized ranking methods and pairwise similarity based ranking methods.Conclusion: The problem of graph model and parameter selection in graph regularized protein domain ranking can be solved effectively by combining multiple graphs. This aspect of generalization introduces a new frontier in applying multiple graphs to solving protein domain ranking applications. 2012 Wang et al; licensee BioMed Central Ltd.
Minimal regular 2-graphs and applications
FAN; Hongbing; LIU; Guizhen; LIU; Jiping
2006-01-01
A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n+3-i/3where 1 ≤i≤6, i=n (mod 6) andn≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, f2(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.
No finite $5$-regular matchstick graph exists
2014-01-01
A graph $G=(V,E)$ is called a unit-distance graph in the plane if there is an injective embedding of $V$ in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing and all vertices have the same degree $r$ we talk of a regular matchstick graph. Due to Euler's polyhedron formula we have $r\\le 5$. The smallest known $4$-regular matchstick graph is the so called Harborth graph consisting of $52$ vertices. In this ar...
Spectral partitioning of random regular blockmodels
Barucca, Paolo
2016-01-01
Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of random graphs with regular block structure is introduced, for which analytical results can be obtained. In particular, the spectral density of such random regular blockmodels is computed exactly for a modular, bipartite and core-periphery structure. McKay's law for random regular graphs is found analytically to apply also for regular modular and bipartite structures when blocks are homogeneous. In core-periphery structures, where blocks are intrinsically heterogeneous, a new law is found to apply for the spectral density. Exact solution to the inference problem is provided for the models discussed. All analytical results show perfect agreement with numerical experiments. Final discussion summarizes results and outlines the relevance of the results for the solution of graph partitioning problems in other graph en...
On the total domatic number of regular graphs
H. Aram
2012-03-01
Full Text Available A set S of vertices of a graph G = (V;E without isolated vertex is a total dominating set if every vertex of V (G is adjacent to some vertex in S. The total domatic number of a graph G is the maximum number of total dominating sets into which the vertex set of G can be partitioned. We show that the total domatic number of a random r-regular graph is almost surely at most r
Counting colorings of a regular graph
Galvin, David
2012-01-01
At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copies of K_{d,d}. An auxiliary result is an upper bound on the number of colorings of a regular graph in terms of its independence number. For example, we show that for all even q and fixed \\epsilon > 0 there is \\delta=\\delta(\\epsilon,q) such that the number of proper q-colorings admitted by an n-vertex, d-regular graph with no independent set of size n(1-\\epsilon)/2 is at most (a-\\delta)^n.
Local Interaction on Random Graphs
Hans Haller
2010-08-01
Full Text Available We analyze dynamic local interaction in population games where the local interaction structure (modeled as a graph can change over time: A stochastic process generates a random sequence of graphs. This contrasts with models where the initial interaction structure (represented by a deterministic graph or the realization of a random graph cannot change over time.
A distance regular graph of type E1 Ed
无
2000-01-01
In this note, the distance regular graph of type E1 Ed is considered and some characterization of the type graph is given. The results generalize the characterization of tight distance regular graphs.
Adjacent strong edge colorings and total colorings of regular graphs
WOODALL; Douglas; R
2009-01-01
It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.
*-Regular Leavitt Path Algebras of Arbitrary Graphs
Gonzalo ARANDA PINO; Kulumani RANGASWAMY; Lia VA(S)
2012-01-01
If K is a field with involution and E an arbitrary graph,the involution from K naturally induces an involution of the Leavitt path algebra LK(E).We show that the involution on LK(E) is proper if the involution on K is positive-definite,even in the case when the graph E is not necessarily finite or row-finite.It has been shown that the Leavitt path algebra LK(E) is regular if and only if E is acyclic.We give necessary and sufficient conditions for LK(E) to be *-regular (i.e.,regular with proper involution).This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K,not just a graph-theoretic property of E.This differs from the.known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone.As a corollary,we show that Handelman's conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras.Moreover,its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.
Exponential random graph models
Fronczak, Agata
2012-01-01
Nowadays, exponential random graphs (ERGs) are among the most widely-studied network models. Different analytical and numerical techniques for ERG have been developed that resulted in the well-established theory with true predictive power. An excellent basic discussion of exponential random graphs addressed to social science students and researchers is given in [Anderson et al., 1999][Robins et al., 2007]. This essay is intentionally designed to be more theoretical in comparison with the well-known primers just mentioned. Given the interdisciplinary character of the new emerging science of complex networks, the essay aims to give a contribution upon which network scientists and practitioners, who represent different research areas, could build a common area of understanding.
Multiple graph regularized nonnegative matrix factorization
Wang, Jim Jing-Yan
2013-10-01
Non-negative matrix factorization (NMF) has been widely used as a data representation method based on components. To overcome the disadvantage of NMF in failing to consider the manifold structure of a data set, graph regularized NMF (GrNMF) has been proposed by Cai et al. by constructing an affinity graph and searching for a matrix factorization that respects graph structure. Selecting a graph model and its corresponding parameters is critical for this strategy. This process is usually carried out by cross-validation or discrete grid search, which are time consuming and prone to overfitting. In this paper, we propose a GrNMF, called MultiGrNMF, in which the intrinsic manifold is approximated by a linear combination of several graphs with different models and parameters inspired by ensemble manifold regularization. Factorization metrics and linear combination coefficients of graphs are determined simultaneously within a unified object function. They are alternately optimized in an iterative algorithm, thus resulting in a novel data representation algorithm. Extensive experiments on a protein subcellular localization task and an Alzheimer\\'s disease diagnosis task demonstrate the effectiveness of the proposed algorithm. © 2013 Elsevier Ltd. All rights reserved.
Avetisov, V; Nechaev, S; Valba, O
2016-01-01
We consider from the localization perspective the new critical behavior discovered recently for the regular random graphs (RRG) and constrained Erd\\H{o}s-Renyi networks (CERN). The diagonal disorder for standard models, we replace by the fugacity $\\mu$ of triads in the RRG and CERN. At some critical value of $\\mu$ the network decays into the maximally possible number of almost full graphs, and the adjacency matrix acquires the two-gapped structure. We find that the eigenvalue statistics corresponds to delocalized states in the central zone, and to the localized states in the side one. The mobility edge lies between zones. We apply these findings to the many-body localization assuming the approximation of the hierarchical structure of the Fock space (for some interacting many-body system) by the RGG and by CERN with some vertex degree. We allow the 3-cycles in the Fock space and identify particles in the many-body system above the phase transition with clusters in the RRG. We discuss the controversial issue of...
Estrada, Ernesto
2015-01-01
A generalization of the random geometric graph (RGG) model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square \\left[0,1\\right]^{2}. The topological properties, such as connectivity, average degree, average path length and clustering, of the random rectangular graphs (RRGs) generated by this model are then studied as a function of the rectangle sides lengths a and b=1/a, and the radius r used to connect the nodes. When a=1 we recover the RGG, and when a\\rightarrow\\infty the very elongated rectangle generated resembles a one-dimensional RGG. We provided computational and analytical evidence that the topological properties of the RRG differ significantly from those of the RGG. The connectivity of the RRG depends not only on the number of nodes as in the case of the RGG, but also on the side length of the rectangle. As the rectangle is more elongated the critical radius for connectivity increases following first a power-law an...
Estrada, Ernesto; de la Pena, Jose A.
2013-01-01
Let G be a graph with set of vertices 1,...,n and adjacency matrix A of size nxn. Let d(i,j)=d, we say that f_d:N->N is a d-function on G if for every pair of vertices i,j and k>=d, we have a_ij^(k)=f_d(k). If this function f_d exists on G we say that G is d-walk regular. We prove that G is d-walk regular if and only if for every pair of vertices i,j at distance
Vertex-antimagic Labelings of Regular Graphs
Ali AHMAD; Kashif ALI; Martin BA(C)A; Petr KOV(A)(R); Andrea SEMANI(C)OV(A)-FE(N)OV(C)(I)KOV(A)
2012-01-01
Let G =(V,E) be a finite,simple and undirected graph with p vertices and q edges.An (a,d)-vertex-antimagic total labeling of G is a bijection f from V(G).∪E(G) onto the set of consecutive integers 1,2,...,p + q,such that the vertex-weights form an arithmetic progression with the initial term a and difference d,where the vertex-weight of x is the sum of the value f(x) assigned to the vertex x together with all values f(xy) assigned to edges xy incident to x.Such labeling is called super if the smallest possible labels appear on the vertices.In this paper,we study the properties of such labelings and examine their existence for 2r-regular graphs when the difference d is 0,1,...,r + 1.
Spectral Characterizations of Some Distance-Regular Graphs
van Dam, E.R.; Haemers, W.H.
2000-01-01
When can one see from the spectrum of a graph whether it is distance-regular or not?We give some new results for when this is the case.As a consequence we find (among others) that the following distance-regular graphs are uniquely determined by their spectrum: The collinearity graphs of the generali
Genus Ranges of 4-Regular Rigid Vertex Graphs.
Buck, Dorothy; Dolzhenko, Egor; Jonoska, Nataša; Saito, Masahico; Valencia, Karin
2015-01-01
A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with 2n vertices (n > 1), we prove that all intervals [a, b] for all a genus ranges. For graphs with 2n - 1 vertices (n ≥ 1), we prove that all intervals [a, b] for all a genus ranges. We also provide constructions of graphs that realize these ranges.
Generating random networks and graphs
Coolen, Ton; Roberts, Ekaterina
2017-01-01
This book supports researchers who need to generate random networks, or who are interested in the theoretical study of random graphs. The coverage includes exponential random graphs (where the targeted probability of each network appearing in the ensemble is specified), growth algorithms (i.e. preferential attachment and the stub-joining configuration model), special constructions (e.g. geometric graphs and Watts Strogatz models) and graphs on structured spaces (e.g. multiplex networks). The presentation aims to be a complete starting point, including details of both theory and implementation, as well as discussions of the main strengths and weaknesses of each approach. It includes extensive references for readers wishing to go further. The material is carefully structured to be accessible to researchers from all disciplines while also containing rigorous mathematical analysis (largely based on the techniques of statistical mechanics) to support those wishing to further develop or implement the theory of rand...
Hamiltonian Cycles in Regular 2-Connected Claw-Free Graphs
李明楚
2003-01-01
A known result by Jackson Bill is that every 2-connected k-regular graph on at most 3k vertices is Hamiltonian. In this paper,it is proved that every 2-connected k-regular claw-free graph on at most 5k(k≥10)vertices is Hamiltonian. Moreover, the bound 5k is best possible. A counterexample of a 2-connected k-regular claw-free non-Hamiltonian graph on 5k+1 vertices is given, and it is conjectured that every 3-connected k-regular claw-free graph on at most 12k-7 vertices is Hamiltonian.
Regularity in Vague Intersection Graphs and Vague Line Graphs
Muhammad Akram
2014-01-01
Full Text Available Fuzzy graph theory is commonly used in computer science applications, particularly in database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capturing. A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduce the notion of vague line graphs, and certain types of vague line graphs and present some of their properties. We also discuss an example application of vague digraphs.
Chromatic polynomials of random graphs
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION
Muhammad IMRAN; Syed Ahtsham ul Haq BOKHARY; Ali AHMAD; Andrea SEMANI(C)OV(A)-FE(N)OV(C)(I)KOV(A)
2013-01-01
In this paper,we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn,a class of cubic convex polytopes considering the open problem raised in [M.Imran et al.,families of plane graphs with constant metric dimension,Utilitas Math.,in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in [I.Javaid et al.,Families of regular graphs with constant metric dimension,Utilitas Math.,2012,88:43-57].We prove that these classes of regular graphs have constant metric dimension.
On perturbations of almost distance-regular graphs
Dalfó, Cristina; Fiol, Miquel Angel; 10.1016/j.laa.2011.05.004
2012-01-01
In this paper we show that certain almost distance-regular graphs, the so-called $h$-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is called $h$-punctually walk-regular, for a given $h\\le D$, if the number of paths of length $\\ell$ between a pair of vertices $u,v$ at distance $h$ depends only on $\\ell$. The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regular graphs some of these operations are equivalent, in the sense that one perturbation produces cospectral graphs if and only if the others do. Our study is based on the theory of graph perturbations developed by Cvetkovi\\'c, Godsil, McKay, Rowlinson, Schwenk, and others. As a consequence, some new characterizations of distance-regular graphs are obtained.
Random broadcast on random geometric graphs
Bradonjic, Milan [Los Alamos National Laboratory; Elsasser, Robert [UNIV OF PADERBORN; Friedrich, Tobias [ICSI/BERKELEY; Sauerwald, Tomas [ICSI/BERKELEY
2009-01-01
In this work, we consider the random broadcast time on random geometric graphs (RGGs). The classic random broadcast model, also known as push algorithm, is defined as: starting with one informed node, in each succeeding round every informed node chooses one of its neighbors uniformly at random and informs it. We consider the random broadcast time on RGGs, when with high probability: (i) RGG is connected, (ii) when there exists the giant component in RGG. We show that the random broadcast time is bounded by {Omicron}({radical} n + diam(component)), where diam(component) is a diameter of the entire graph, or the giant component, for the regimes (i), or (ii), respectively. In other words, for both regimes, we derive the broadcast time to be {Theta}(diam(G)), which is asymptotically optimal.
Regular Maps of Graphs of Order 4p
Jin Xin ZHOU; Yan Quan FENG
2012-01-01
A 2-cell embedding f:X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair M =(X; p) called a map,where p is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S.It is well known that the automorphism group of M acts semi-regularly on the arc set of X and if the action is regular,then the map Mand the embedding f are called regular.Let p and q be primes.Duet al.[J.Algebraic Combin.,19,123-141 (2004)] classified the regular maps of graphs of order pq.In this paper all pairwise non-isomorphic regular maps of graphs of order 4p are constructed explicitly and the genera of such regular maps are computed.As a result,there are twelve sporadic and six infinite families of regular maps of graphs of order 4p; two of the infinite families are regular maps with the complete bipartite graphs K2p,2p as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups Z4p,Z22 × Zp and D4p.
Bounds on the clique-transversal number of regular graphs
CHENG; T.C.E
2008-01-01
A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.
Bounds on the clique-transversal number of regular graphs
SHAN ErFang; CHENG T.C.E.; KANG LiYing
2008-01-01
A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted Tc(G), is the minimum cardinality of a clique-transversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.
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...
Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph
P. Anusha Devi
2015-10-01
Full Text Available An ecard of a graph $G$ is a subgraph formed by deleting an edge. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph $G,~dern(G,$ is the minimum number of da-ecards that uniquely determines $G.$ The adversary degree associated edge reconstruction number of a graph $G, adern(G,$ is the minimum number $k$ such that every collection of $k$ da-ecards of $G$ uniquely determines $G.$ The maximal subgraph without end vertices of a graph $G$ which is not a tree is the pruned graph of $G.$ It is shown that $dern$ of complete multipartite graphs and some connected graphs with regular pruned graph is $1$ or $2.$ We also determine $dern$ and $adern$ of corona product of standard graphs.
The total chromatic number of regular graphs of high degree
2009-01-01
The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) 32 |V (G)| + 263 , where d(G) denotes the degree of a vertex in G, then χT (G) d(G) + 2.
Tetravalent one-regular graphs of order 4p^{2}
Feng, Yan-Quan; Kutnar, Klavdija; Marusic, Dragan;
2014-01-01
A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper tetravalent one-regular graphs of order 4p2, where p is a prime, are classified.......A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper tetravalent one-regular graphs of order 4p2, where p is a prime, are classified....
Perfect Matchings in O(n \\log n) Time in Regular Bipartite Graphs
Goel, Ashish; Khanna, Sanjeev
2009-01-01
In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m\\sqrt{n}). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to \\tilde O(min{m, n^{2.5}/d}) and then to \\tilde O(min{m, n^2/d}). It was also shown that the latter algorithm is optimal up to polylogarithmic factors among all algorithms that use non-adaptive uniform sampling to reduce the size of the graph as a first step. In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n\\log n) time (both in expectation and with high probability). The algorithm performs an appropriately truncated random walk on a modified graph to successively find augmenting paths. Our...
Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs
Goel, Ashish; Khanna, Sanjeev
2008-01-01
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\\"{o}nig's work in 1916 (here $m=nd$ is the number of edges in the graph, $2n$ is the number of vertices, and $d$ is the degree of each node). The currently most efficient algorithm takes time $O(m)$, and is due to Cole, Ost, and Schirra. We improve this running time to $O(\\min\\{m, \\frac{n^{2.5}\\ln n}{d}\\})$; this minimum can never be larger than $O(n^{1.75}\\sqrt{\\ln n})$. We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a $d$-regular bipartite graph independently with a probability $p = O(\\frac{n\\ln n}{d^2})$ then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential ...
Construction of directed strongly regular graphs using finite incidence structures
Olmez, O
2010-01-01
We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters \\[(l(q-1)q^l,\\ l(q-1)q^{l-1},\\ (lq-l+1)q^{l-2},\\ (l-1)(q-1)q^{l-2},\\ (lq-l+1)q^{l-2})\\] for integers $q$ and $l$ ($q, l\\ge 2$), and \\[(lq^2(q-1),\\ lq(q-1),\\ lq-l+1,\\ (l-1)(q-1),\\ lq-l+1)\\] for all prime powers $q$ and $l\\in \\{1, 2, \\dots, q\\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4, 7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8, 14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on "Parameters of directed strongly regular graphs," at ${http://homepages.cwi.nl/^\\sim aeb/math/dsrg/dsrg.html}$ by S. Hobart and A. E. Brouwer. We review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs.
Universality for the Distance in Finite Variance Random Graphs
Van den Esker, H.; Van der Hofstad, R.; Hooghiemstra, G.
2008-01-01
We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the
Aspects of randomness in neural graph structures
Rudolph-Lilith, Michelle
2013-01-01
In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the presence of serious uncertainties, such as major undersampling of the experimental data. In the specific case of neural systems, however, a few moderately robust experimental reconstructions do exist, and these have long served as fundamental prototypes for studying connectivity patterns in the nervous system. In this paper, we provide a comparative analysis of these "historical" graphs, both in (unmodified) directed and (often symmetrized) undirected forms, and focus on simple structural characterizations of their connectivity. We find that in most measures the networks studied are captured by simple random graph models; in a few key measures, however, we observe a marked departure from the random graph prediction. Our results suggest that the mechanism of graph formation in th...
On the game chromatic number of sparse random graphs
Frieze, Alan; Lavrov, Mikhail
2012-01-01
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are colored. The game chromatic number \\chi_g(G) is the minimum k for which the first player has a winning strategy. The paper \\cite{BFS} began the analysis of the asymptotic behavior of this parameter for a random graph G_{n,p}. This paper provides some further analysis for graphs with constant average degree i.e. np=O(1) and for random regular graphs.
Generating Random Graphs with Large Girth
Bayati, Mohsen; Saberi, Amin
2008-01-01
We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance Low-Density Parity -Check (LDPC) codes. For any constant k, alpha<1/2k(k+3) and m=O(n^{1+alpha}), our algorithm generate s an asymptotically uniform random graph with n vertices, m edges, and girth larger tha n k in polynomial time. To the best of our knowledge this is the first polynomial-algorith m for the problem. Our algorithm generates a graph by sequentially adding m edges to an empty graph with n vertices. Recently, these types of sequential methods for counting and random generation have been very successful.
Spectral statistics of random geometric graphs
Dettmann, Carl P; Knight, Georgie
2016-01-01
We study the spectrum of random geometric graphs using random matrix theory. We look at short range correlations in the level spacings via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity $\\Delta_3$ statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find that the spectral statistics of random geometric graphs fits the universality of random matrix theory. In particular, the short range correlations are very close to those found in the Gaussian orthogonal ensemble of random matrix theory. For long range correlations we find deviations from Gaussian orthogonal ensemble statistics towards Poisson. We compare with previous results for Erd\\H{o}s-R\\'{e}nyi, Barab{\\'a}si-Albert and Watts-Strogatz random graphs where similar random matrix theory universality has been found.
Graph Regularized Auto-Encoders for Image Representation.
Yiyi Liao; Yue Wang; Yong Liu
2017-06-01
Image representation has been intensively explored in the domain of computer vision for its significant influence on the relative tasks such as image clustering and classification. It is valuable to learn a low-dimensional representation of an image which preserves its inherent information from the original image space. At the perspective of manifold learning, this is implemented with the local invariant idea to capture the intrinsic low-dimensional manifold embedded in the high-dimensional input space. Inspired by the recent successes of deep architectures, we propose a local invariant deep nonlinear mapping algorithm, called graph regularized auto-encoder (GAE). With the graph regularization, the proposed method preserves the local connectivity from the original image space to the representation space, while the stacked auto-encoders provide explicit encoding model for fast inference and powerful expressive capacity for complex modeling. Theoretical analysis shows that the graph regularizer penalizes the weighted Frobenius norm of the Jacobian matrix of the encoder mapping, where the weight matrix captures the local property in the input space. Furthermore, the underlying effects on the hidden representation space are revealed, providing insightful explanation to the advantage of the proposed method. Finally, the experimental results on both clustering and classification tasks demonstrate the effectiveness of our GAE as well as the correctness of the proposed theoretical analysis, and it also suggests that GAE is a superior solution to the current deep representation learning techniques comparing with variant auto-encoders and existing local invariant methods.
Generating Realistic Labelled, Weighted Random Graphs
Davis, Michael Charles; Liu, Weiru; Miller, Paul; Hunter, Ruth; Kee, Frank
2015-01-01
Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs) with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI) approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs). Our results allow us to draw conclusions about the contribution of vertex labels a...
Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups
XU ShangJin; WU ZhengFei; DENG YunPing
2009-01-01
A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.
k-Connectivity of Random Key Graphs
Zhao, Jun; Gligor, Virgil
2015-01-01
Random key graphs represent topologies of secure wireless sensor networks that apply the seminal Eschenauer-Gligor random key predistribution scheme to secure communication between sensors. These graphs have received much attention and also been used in diverse application areas beyond secure sensor networks; e.g., cryptanalysis, social networks, and recommender systems. Formally, a random key graph with $n$ nodes is constructed by assigning each node $X_n$ keys selected uniformly at random from a pool of $Y_n$ keys and then putting an undirected edge between any two nodes sharing at least one key. Considerable progress has been made in the literature to analyze connectivity and $k$-connectivity of random key graphs, where $k$-connectivity of a graph ensures connectivity even after the removal of $k$ nodes or $k$ edges. Yet, it still remains an open question for $k$-connectivity in random key graphs under $X_n \\geq 2$ and $X_n = o(\\sqrt{\\ln n})$ (the case of $X_n=1$ is trivial). In this paper, we answer the a...
The evolution of random reversal graph
Jin, Emma Y
2010-01-01
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $\\lambda_n=1/\\binom{n+1}{2}$. For $\\lambda_n=(1-\\epsilon)/\\binom{n+1}{2}$, the random graph consists of components of size at most $O(n\\ln(n))$ a.s. and for $(1+\\epsilon)/\\binom{n+1}{2}$, there emerges a unique largest component of size $\\sim \\wp(\\epsilon) \\cdot 2^n\\cdot n$!$ a.s.. This "giant" component is furthermore dense in the reversal graph.
Multiple Kernel Learning for adaptive graph regularized nonnegative matrix factorization
Wang, Jim Jing-Yan
2012-01-01
Nonnegative Matrix Factorization (NMF) has been continuously evolving in several areas like pattern recognition and information retrieval methods. It factorizes a matrix into a product of 2 low-rank non-negative matrices that will define parts-based, and linear representation of non-negative data. Recently, Graph regularized NMF (GrNMF) is proposed to find a compact representation, which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In GNMF, an affinity graph is constructed from the original data space to encode the geometrical information. In this paper, we propose a novel idea which engages a Multiple Kernel Learning approach into refining the graph structure that reflects the factorization of the matrix and the new data space. The GrNMF is improved by utilizing the graph refined by the kernel learning, and then a novel kernel learning method is introduced under the GrNMF framework. Our approach shows encouraging results of the proposed algorithm in comparison to the state-of-the-art clustering algorithms like NMF, GrNMF, SVD etc.
Quantum graphs and random-matrix theory
Pluhař, Z.; Weidenmüller, H. A.
2015-07-01
For simple connected graphs with incommensurate bond lengths and with unitary symmetry we prove the Bohigas-Giannoni-Schmit (BGS) conjecture in its most general form. Using supersymmetry and taking the limit of infinite graph size, we show that the generating function for every (P,Q) correlation function for both closed and open graphs coincides with the corresponding expression of random-matrix theory. We show that the classical Perron-Frobenius operator is bistochastic and possesses a single eigenvalue +1. In the quantum case that implies the existence of a zero (or massless) mode of the effective action. That mode causes universal fluctuation properties. Avoiding the saddle-point approximation we show that for graphs that are classically mixing (i.e. for which the spectrum of the classical Perron-Frobenius operator possesses a finite gap) and that do not carry a special class of bound states, the zero mode dominates in the limit of infinite graph size.
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)
Oren, Idan
2010-01-01
Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from the known Poisson distribution of cycle counts in regular graphs, in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely. The result is analogous to the so called "diagonal approximation" in Quantum Chaos. We also show that by assuming that the spectral correlations are given by RMT to all orders, we can compute the leading deviations from the Poisson distribution for cycle counts. We provide numerical evidence which supports this conjecture.
Giant component in random multipartite graphs with given degree sequences
David Gamarnik
2015-12-01
Full Text Available We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with p parts generated according to a given degree sequence ndi(n, n≥1 which denotes the number of vertices in part i of the multipartite graph with degree given by the vector d in an n-node graph. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Reed Molloy and Reed (1995 to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the random processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimensional Galton-Watson processes along with Lyapunov function technique to overcome the challenges.
Graph Regularized Nonnegative Matrix Factorization for Hyperspectral Data Unmixing
Rajabi, Roozbeh; Ghassemian, Hassan
2011-01-01
Spectral unmixing is an important tool in hyperspectral data analysis for estimating endmembers and abundance fractions in a mixed pixel. This paper examines the applicability of a recently developed algorithm called graph regularized nonnegative matrix factorization (GNMF) for this aim. The proposed approach exploits the intrinsic geometrical structure of the data besides considering positivity and full additivity constraints. Simulated data based on the measured spectral signatures, is used for evaluating the proposed algorithm. Results in terms of abundance angle distance (AAD) and spectral angle distance (SAD) show that this method can effectively unmix hyperspectral data.
Infinite Random Graphs as Statistical Mechanical Models
Durhuus, Bergfinnur Jøgvan; Napolitano, George Maria
2011-01-01
We discuss two examples of infinite random graphs obtained as limits of finite statistical mechanical systems: a model of two-dimensional dis-cretized quantum gravity defined in terms of causal triangulated surfaces, and the Ising model on generic random trees. For the former model we describe...
Generating Realistic Labelled, Weighted Random Graphs
Michael Charles Davis
2015-12-01
Full Text Available Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs. Our results allow us to draw conclusions about the contribution of vertex labels and edge weights to graph structure.
Approximately Counting Embeddings into Random Graphs
Furer, Martin
2008-01-01
Let H be a graph, and let C(H,G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C(H,G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices generating a sequence of bipartite graphs. The decomposition permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipa...
Clique percolation in random graphs
Li, Ming; Deng, Youjin; Wang, Bing-Hong
2015-10-01
As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two k cliques means that they share at least l clique percolation in Erdős-Rényi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction ψ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction ϕ of vertices in the giant clique cluster for l >1 makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for l =1 . More interesting, our analysis shows that at the critical point, the order parameter ϕc for l >1 is neither 0 nor 1, but a constant depending on k and l . All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.
Color normalization of histology slides using graph regularized sparse NMF
Sha, Lingdao; Schonfeld, Dan; Sethi, Amit
2017-03-01
Computer based automatic medical image processing and quantification are becoming popular in digital pathology. However, preparation of histology slides can vary widely due to differences in staining equipment, procedures and reagents, which can reduce the accuracy of algorithms that analyze their color and texture information. To re- duce the unwanted color variations, various supervised and unsupervised color normalization methods have been proposed. Compared with supervised color normalization methods, unsupervised color normalization methods have advantages of time and cost efficient and universal applicability. Most of the unsupervised color normaliza- tion methods for histology are based on stain separation. Based on the fact that stain concentration cannot be negative and different parts of the tissue absorb different stains, nonnegative matrix factorization (NMF), and particular its sparse version (SNMF), are good candidates for stain separation. However, most of the existing unsupervised color normalization method like PCA, ICA, NMF and SNMF fail to consider important information about sparse manifolds that its pixels occupy, which could potentially result in loss of texture information during color normalization. Manifold learning methods like Graph Laplacian have proven to be very effective in interpreting high-dimensional data. In this paper, we propose a novel unsupervised stain separation method called graph regularized sparse nonnegative matrix factorization (GSNMF). By considering the sparse prior of stain concentration together with manifold information from high-dimensional image data, our method shows better performance in stain color deconvolution than existing unsupervised color deconvolution methods, especially in keeping connected texture information. To utilized the texture information, we construct a nearest neighbor graph between pixels within a spatial area of an image based on their distances using heat kernal in lαβ space. The
Warmth and mobility of random graphs
Fadnavis, Sukhada
2010-01-01
Brightwell and Winkler introduced the graph parameters warmth and mobility in the context of combinatorial statistical physics. They related both parameters to lower bounds on chromatic number. Although warmth is not a monotone graph property we show it is nevertheless "statistically monotone" in the sense that it tends to increase with added random edges, and that for sparse graphs ($p=O(n^{-\\alpha})$, $\\alpha > 0$) the warmth is concentrated on at most two values, and for most $p$ it is concentrated on one value. We also put bounds on warmth and mobility in the dense regime, and as a corollary obtain that a conjecture of Lov\\'asz holds for almost all graphs.
Asynchronous Rumor Spreading on Random Graphs
Panagiotou, Konstantinos
2016-01-01
We perform a thorough study of various characteristics of the asynchronous push-pull protocol for spreading a rumor on Erd\\H{o}s-R\\'enyi random graphs $G_{n,p}$, for any $p>c\\ln(n)/n$ with $c>1$. In particular, we provide a simple strategy for analyzing the asynchronous push-pull protocol on arbitrary graph topologies and apply this strategy to $G_{n,p}$. We prove tight bounds of logarithmic order for the total time that is needed until the information has spread to all nodes. Surprisingly, the time required by the asynchronous push-pull protocol is asymptotically almost unaffected by the average degree of the graph. Similarly tight bounds for Erd\\H{o}s-R\\'enyi random graphs have previously only been obtained for the synchronous push protocol, where it has been observed that the total running time increases significantly for sparse random graphs. Finally, we quantify the robustness of the protocol with respect to transmission and node failures. Our analysis suggests that the asynchronous protocols are particu...
Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups
无
2009-01-01
A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.
Scale-invariant geometric random graphs
Xie, Zheng
2015-01-01
We introduce and analyze a class of growing geometric random graphs that are invariant under rescaling of space and time. Directed connections between nodes are drawn according to an influence zone that depends on node position in space and time, capturing the heterogeneity and increased specialization found in growing networks. Through calculations and numerical simulations we explore the consequences of scale-invariance for geometric graphs generated this way. Our analysis reveals a dichotomy between scale-free and Poisson distributions of in- and out-degree, the existence of a random number of hub nodes, high clustering, and unusual percolation behaviour. Moreover, we show how these properties provide a good fit to those of empirically observed web graphs.
Largest sparse subgraphs of random graphs
Fountoulakis, N.; Kang, R.J.; McDiarmid, C.J.H.; Nešetřil, J.; Győri, E.; Sali, A.
2011-01-01
For the Erd\\H{o}s-R\\'enyi random graph $G_{n,p}$, we consider the order of a largest vertex subset that induces a subgraph with average degree at most $t$. For the case when both $p$ and $t$ are fixed, this value is asymptotically almost surely concentrated on at most two explicitly given points. Th
Clique colouring of binomial random graphs
Mcdiarmid, Colin; Mitsche, Dieter; Pralat, Pawel
2016-01-01
A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.
Bootstrap Percolation on Random Geometric Graphs
Bradonjić, Milan
2012-01-01
Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, bootstrap percolation starts by random and independent "activation" of nodes with a fixed probability $p$, followed by a deterministic process for additional activations based on the density of active nodes in each neighborhood ($\\th$ activated nodes). Here, we study bootstrap percolation on random geometric graphs in the regime when the latter are (almost surely) connected. Random geometric graphs provide an appropriate model in settings where the neighborhood structure of each node is determined by geographical distance, as in wireless {\\it ad hoc} ...
s-REGULAR DIHEDRAL COVERINGS OF THE COMPLETE GRAPH OF ORDER 4
FENG YANQUAN(冯衍全); J.H. KWAK
2004-01-01
A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s ≥ 1, whose fibre preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s ≥ 1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.
A note on a conjecture concerning tree-partitioning 3-regular graphs
Bohme, T.; Broersma, Hajo; Tuinstra, Hilde
1998-01-01
If G is a 4-connected maximal planar graph, then G is hamiltonian (by a theorem of Whitney), implying that its dual graph G� is a cyclically 4-edge connected 3- regular planar graph admitting a partition of the vertex set into two parts, each inducing a tree in G�, a so-called tree-partition. It is
A note on a conjecture concerning tree-partitioning 3-regular graphs
Bohme, T.; Bohme, T.; Broersma, Haitze J.; Tuinstra, Hilde
1998-01-01
If $G$ is a 4-connected maximal planar graph, then $G$ is Hamiltonian (by a theorem of Whitney), implying that its dual graph $G^*$ is a cyclically 4-edge connected 3-regular planar graph admitting a partition of the vertex set into two parts, each inducing a tree in $G^*$, a so-called
The rigidity transition in random graphs
Kasiviswanathan, Shiva Prasad; Theran, Louis
2010-01-01
As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c c_2, w.h.p. there is a giant rigid component. The constant c_2 \\approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.
Cubic one-regular graphs of order twice a square-free integer
2008-01-01
A graph is one-regular if its automorphism group acts regularly on the set of its arcs.Let n be a square-free integer.In this paper,we show that a cubic one-regular graph of order 2n exists if and only if n=3tp1p2…ps≥13,where t≤1,s≥1 and pi’s are distinct primes such that 3|（Pi—1）. For such an integer n,there are 2s-1 non-isomorphic cubic one-regular graphs of order 2n,which are all Cayley graphs on the dihedral group of order 2n.As a result,no cubic one-regular graphs of order 4 times an odd square-free integer exist.
The computational hardness of counting in two-spin models on d-regular graphs
Sly, Allan
2012-01-01
The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has non-uniqueness on the d-regular tree. Together with results of Jerrum--Sinclair, Weitz, and Sinclair--Srivastava--Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs. Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting "free energy density" which coincides with the (non-rigorous) Bethe prediction of statistical physics. We use this result to characterize the lo...
Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain.
Pang, Jiahao; Cheung, Gene
2017-04-01
Inverse imaging problems are inherently underdetermined, and hence, it is important to employ appropriate image priors for regularization. One recent popular prior-the graph Laplacian regularizer-assumes that the target pixel patch is smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. To address this problem, in this paper, we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization for image denoising. Specifically, we first show the convergence of the graph Laplacian regularizer to a continuous-domain functional, integrating a norm measured in a locally adaptive metric space. Focusing on image denoising, we derive an optimal metric space assuming non-local self-similarity of pixel patches, leading to an optimal graph Laplacian regularizer for denoising in the discrete domain. We then interpret graph Laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify our analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods, such as BM3D for natural images, and outperforms them significantly for piecewise smooth images.
MINIMUM CONGESTION SPANNING TREES IN BIPARTITE AND RANDOM GRAPHS
M.L Ostrovskii
2011-01-01
The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.
On the limit points of the smallest eigenvalues of regular graphs
Yu, Hyonju
2011-01-01
In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[\\alpha_1, -1-\\sqrt2)$ where $\\alpha_1$ is the smallest root$(\\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than -2. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than -2 and we give the unique graph with this supremum value as its smallest eigenvalue.
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 ...
Efficient broadcast on random geometric graphs
Bradonjic, Milan [Los Alamos National Laboratory; Elsasser, Robert [UNIV OF PADERBORN; Friedrich, Tobias [INTERNATIONAL COMPUTER SCI.; Sauerwald, Thomas [INTERNATIONAL COMPUTER SCI.
2009-01-01
A Randon Geometric Graph (RGG) is constructed by distributing n nodes uniformly at random in the unit square and connecting two nodes if their Euclidean distance is at most r, for some prescribed r. They analyze the following randomized broadcast algorithm on RGGs. At the beginning, there is only one informed node. Then in each round, each informed node chooses a neighbor uniformly at random and informs it. They prove that this algorithm informs every node in the largest component of a RGG in {Omicron}({radical}n/r) rounds with high probability. This holds for any value of r larger than the critical value for the emergence of a giant component. In particular, the result implies that the diameter of the giant component is {Theta}({radical}n/r).
Component evolution in general random intersection graphs
Bradonjic, Milan [Los Alamos National Laboratory; Hagberg, Aric [Los Alamos National Laboratory; Hengartner, Nick [Los Alamos National Laboratory; Percus, Allon G [CLAREMONT GRADUATE UNIV.
2010-01-01
We analyze component evolution in general random intersection graphs (RIGs) and give conditions on existence and uniqueness of the giant component. Our techniques generalize the existing methods for analysis on component evolution in RIGs. That is, we analyze survival and extinction properties of a dependent, inhomogeneous Galton-Watson branching process on general RIGs. Our analysis relies on bounding the branching processes and inherits the fundamental concepts from the study on component evolution in Erdos-Renyi graphs. The main challenge becomes from the underlying structure of RIGs, when the number of offsprings follows a binomial distribution with a different number of nodes and different rate at each step during the evolution. RIGs can be interpreted as a model for large randomly formed non-metric data sets. Besides the mathematical analysis on component evolution, which we provide in this work, we perceive RIGs as an important random structure which has already found applications in social networks, epidemic networks, blog readership, or wireless sensor networks.
Edge-partitioning graphs into regular and locally irregular components
Bensmail, Julien; Stevens, Brett
2016-01-01
A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph G, there exists a minimum integer χ'irr(G) such that G admits an edge-pa...
Clique-Transversal Sets in 4-Regular Claw-Free Graphs
Er Fang SHAN; Li Ying KANG
2011-01-01
A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by T(G), is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.
Wang, Jim Jing-Yan
2014-09-20
Nonnegative matrix factorization (NMF), a popular part-based representation technique, does not capture the intrinsic local geometric structure of the data space. Graph regularized NMF (GNMF) was recently proposed to avoid this limitation by regularizing NMF with a nearest neighbor graph constructed from the input data set. However, GNMF has two main bottlenecks. First, using the original feature space directly to construct the graph is not necessarily optimal because of the noisy and irrelevant features and nonlinear distributions of data samples. Second, one possible way to handle the nonlinear distribution of data samples is by kernel embedding. However, it is often difficult to choose the most suitable kernel. To solve these bottlenecks, we propose two novel graph-regularized NMF methods, AGNMFFS and AGNMFMK, by introducing feature selection and multiple-kernel learning to the graph regularized NMF, respectively. Instead of using a fixed graph as in GNMF, the two proposed methods learn the nearest neighbor graph that is adaptive to the selected features and learned multiple kernels, respectively. For each method, we propose a unified objective function to conduct feature selection/multi-kernel learning, NMF and adaptive graph regularization simultaneously. We further develop two iterative algorithms to solve the two optimization problems. Experimental results on two challenging pattern classification tasks demonstrate that the proposed methods significantly outperform state-of-the-art data representation methods.
Betweenness centrality patterns in random planar graphs
Lion, Benjamin
2016-01-01
Random planar graphs appear in a variety of context and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display non-trivial patterns, such as loops of very central nodes. We discuss briefly empirical results for different random planar graphs and we propose a toy model which allows us to discuss the condition for the emergence of non-trivial patterns such as central loops. This toy model is made of a star network with $N_b$ branches of size $n$ and links of weight $1$, superimposed to a loop at distance $\\ell$ from the center and with links of weight $w$. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if $w
Infinite Random Graphs as Statistical Mechanical Models
Durhuus, Bergfinnur Jøgvan; Napolitano, George Maria
2011-01-01
We discuss two examples of infinite random graphs obtained as limits of finite statistical mechanical systems: a model of two-dimensional dis-cretized quantum gravity defined in terms of causal triangulated surfaces, and the Ising model on generic random trees. For the former model we describe...... a relation to the so-called uniform infinite tree and results on the Hausdorff and spectral dimension of two-dimensional space-time obtained in B. Durhuus, T. Jonsson, J.F. Wheater, J. Stat. Phys. 139, 859 (2010) are briefly outlined. For the latter we discuss results on the absence of spontaneous...... magnetization and argue that, in the generic case, the values of the Hausdorff and spectral dimension of the underlying infinite trees are not influenced by the coupling to an Ising model in a constant magnetic field (B. Durhuus, G.M. Napolitano, in preparation)...
Spatially-Coupled Random Access on Graphs
Liva, Gianluigi; Lentmaier, Michael; Chiani, Marco
2012-01-01
In this paper we investigate the effect of spatial coupling applied to the recently-proposed coded slotted ALOHA (CSA) random access protocol. Thanks to the bridge between the graphical model describing the iterative interference cancelation process of CSA over the random access frame and the erasure recovery process of low-density parity-check (LDPC) codes over the binary erasure channel (BEC), we propose an access protocol which is inspired by the convolutional LDPC code construction. The proposed protocol exploits the terminations of its graphical model to achieve the spatial coupling effect, attaining performance close to the theoretical limits of CSA. As for the convolutional LDPC code case, large iterative decoding thresholds are obtained by simply increasing the density of the graph. We show that the threshold saturation effect takes place by defining a suitable counterpart of the maximum-a-posteriori decoding threshold of spatially-coupled LDPC code ensembles. In the asymptotic setting, the proposed s...
Zero-one laws for connectivity in random key graphs
Yagan, Osman
2009-01-01
The random key graph is a random graph naturally associated with the random key predistribution scheme of Eschenauer and Gligor for wireless sensor networks. For this class of random graphs we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to wireless sensor networks.
Cross over of recurrence networks to random graphs and random geometric graphs
RINKU JACOB; K P HARIKRISHNAN; R MISRA; G AMBIKA
2017-02-01
Recurrence networks are complex networks constructed from the time series of chaotic dynamical systems where the connection between two nodes is limited by the recurrence threshold. This condition makes the topology of every recurrence network unique with the degree distribution determined by the probability densityvariations of the representative attractor from which it is constructed. Here we numerically investigate the properties of recurrence networks from standard low-dimensional chaotic attractors using some basic network measuresand show how the recurrence networks are different from random and scale-free networks. In particular, we show that all recurrence networks can cross over to random geometric graphs by adding sufficient amount of noise tothe time series and into the classical random graphs by increasing the range of interaction to the system size. We also highlight the effectiveness of a combined plot of characteristic path length and clustering coefficient in capturing the small changes in the network characteristics.
Cross over of recurrence networks to random graphs and random geometric graphs
Jacob, Rinku; Harikrishnan, K. P.; Misra, R.; Ambika, G.
2017-02-01
Recurrence networks are complex networks constructed from the time series of chaotic dynamical systems where the connection between two nodes is limited by the recurrence threshold. This condition makes the topology of every recurrence network unique with the degree distribution determined by the probability density variations of the representative attractor from which it is constructed. Here we numerically investigate the properties of recurrence networks from standard low-dimensional chaotic attractors using some basic network measures and show how the recurrence networks are different from random and scale-free networks. In particular, we show that all recurrence networks can cross over to random geometric graphs by adding sufficient amount of noise to the time series and into the classical random graphs by increasing the range of interaction to the system size. We also highlight the effectiveness of a combined plot of characteristic path length and clustering coefficient in capturing the small changes in the network characteristics.
Configuring Random Graph Models with Fixed Degree Sequences
Fosdick, Bailey K; Nishimura, Joel; Ugander, Johan
2016-01-01
Random graph null models have found widespread application in diverse research communities analyzing network datasets. The most popular family of random graph null models, called configuration models, are defined as uniform distributions over a space of graphs with a fixed degree sequence. Commonly, properties of an empirical network are compared to properties of an ensemble of graphs from a configuration model in order to quantify whether empirical network properties are meaningful or whether they are instead a common consequence of the particular degree sequence. In this work we study the subtle but important decisions underlying the specification of a configuration model, and investigate the role these choices play in graph sampling procedures and a suite of applications. We place particular emphasis on the importance of specifying the appropriate graph labeling---stub-labeled or vertex-labeled---under which to consider a null model, a choice that closely connects the study of random graphs to the study of...
Agreement dynamics on directed random graphs
Lipowski, Adam; Ferreira, Antonio L
2016-01-01
When agreement-dynamics models are placed on a directed random graph, a fraction of sites $\\exp(-z)$, where $z$ is the average degree, becomes permanently fixed or flickering. In the Voter model, which has no surface tension, such zealots or flickers freely spread their opinions and that makes the system disordered. For models with a surface tension, like the Ising model or the Naming Game model, their role is limited and such systems are ordered at large~$z$. However, when $z$ decreases, the density of zealots or flickers increases, and below a certain threshold ($z\\sim 1.9-2.0$) the system becomes disordered. Our results show that the agreement dynamics on directed networks is much different from their undirected analogues.
Random graph models for dynamic networks
Zhang, Xiao; Newman, M E J
2016-01-01
We propose generalizations of a number of standard network models, including the classic random graph, the configuration model, and the stochastic block model, to the case of time-varying networks. We assume that the presence and absence of edges are governed by continuous-time Markov processes with rate parameters that can depend on properties of the nodes. In addition to computing equilibrium properties of these models, we demonstrate their use in data analysis and statistical inference, giving efficient algorithms for fitting them to observed network data. This allows us, for instance, to estimate the time constants of network evolution or infer community structure from temporal network data using cues embedded both in the probabilities over time that node pairs are connected by edges and in the characteristic dynamics of edge appearance and disappearance. We illustrate our methods with a selection of applications, both to computer-generated test networks and real-world examples.
Regularity via topology of the lcm-lattice for $C_4$-free graphs
Nevo, Eran
2009-01-01
We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the primal graph is claw free with no induced 4-cycle in its complement. For the later family we show that the second power of the edge ideal has a linear resolution.
The bondage number of $(n-3)$-regular graphs of order $n$
Hu, Fu-Tao
2011-01-01
Let $G=(V,E)$ be a graph. A subset $D\\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$ is the smallest cardinality of a dominating set of $G$. The bondage number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number of $G$. In this paper, we determine that the exact value of the bondage number of $(n-3)$-regular graph $G$ of order $n$ is $n-3$.
Perfect Matchings in \\~O(n^{1.5}) Time in Regular Bipartite Graphs
Goel, Ashish
2009-01-01
We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes O(m\\sqrt{n}) time, in regular bipartite graphs, a perfect matching is known to be computable in O(m) time. Very recently, the O(m) bound was improved to O(min{m, n^{2.5} log n/d}) expected time, an expression that is bounded by \\~O(n^{1.75}). In this paper, we present an \\~O(n^{1.5}) expected time algorithm for finding a perfect matching in regular bipartite graphs. To obtain this result, we prove a correspondence theorem between cuts and Hall's theorem "witnesses" for a perfect matching in a bipartite graph. We then design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in regular bipartite graphs to the same problem on arbitrary bipartite graphs with O(n log n) edges.
Randomized Consensus Processing over Random Graphs: Independence and Convergence
Shi, Guodong
2011-01-01
Various consensus algorithms over random networks have been investigated in the literature. In this paper, we focus on the role that randomized individual decision-making plays to consensus seeking under stochastic communications. At each time step, each node will independently choose to follow the consensus algorithm, or to stick to current state by a simple Bernoulli trial with time-dependent success probabilities. This node decision strategy characterizes the random node-failures on a communication networks, or a biased opinion selection in the belief evolution over social networks. Connectivity-independent and arc-independent graphs are defined, respectively, to capture the fundamental nature of random network processes with regard to the convergence of the consensus algorithms. A series of sufficient and/or necessary conditions are given on the success probability sequence for the network to reach a global consensus with probability one under different stochastic connectivity assumptions, by which a comp...
Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices
Bandyopadhyay, Antar; Sajadi, Farkhondeh
2012-01-01
For connectivity of \\emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \\emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2\\phi$ where $0 ...
Waldorp, Lourens J
2016-01-01
It was recently shown how graphs can be used to provide descriptions of psychopathologies, where symptoms of, say, depression, affect each other and certain configurations determine whether someone could fall into a sudden depression. To analyse changes over time and characterise possible future behaviour is rather difficult for large graphs. We describe the dynamics of networks using one-dimensional discrete time dynamical systems theory obtained from a mean field approach to (elementary) probabilistic cellular automata (PCA). Often the mean field approach is used on a regular graph (a grid or torus) where each node has the same number of edges and the same probability of becoming active. We show that we can use variations of the mean field of the grid to describe the dynamics of the PCA on a random and small-world graph. Bifurcation diagrams for the mean field of the grid, random, and small-world graphs indicate possible phase transitions for certain parameter settings. Extensive simulations indicate for di...
SIS epidemics on Triadic Random Graphs
Rausch, Ilja
2016-01-01
It has been shown in the past that many real-world networks exhibit community structures and non-trivial clustering which comes with the occurrence of a notable number of triangular connections. Yet the influence of such connection patterns on the dynamics of disease transmission is not fully understood. In order to study their role in the context of Susceptible-Infected-Susceptible (SIS) epidemics we use the Triadic Random Graph (TRG) model to construct small networks (N=49) from distinct, closed, directed triadic subpatterns. We compare various global properties of TRGs and use the N-intertwined mean-field approximation (NIMFA) model to perform numerical simulations of SIS-dynamics on TRGs. The results show that the infection spread on undirected TRGs displays very similar behavior to TRGs with an abundance of (directed) feed-back-loops, while using (directed) feed-forward-loops as network-entities significantly slows down the epidemic and lowers the number of infected individuals in the endemic state. More...
On the One Dimensional Poisson Random Geometric Graph
L. Decreusefond
2011-01-01
Full Text Available Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process, and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.
Two-Level Bregman Method for MRI Reconstruction with Graph Regularized Sparse Coding
刘且根; 卢红阳; 张明辉
2016-01-01
In this paper, a two-level Bregman method is presented with graph regularized sparse coding for highly undersampled magnetic resonance image reconstruction. The graph regularized sparse coding is incorporated with the two-level Bregman iterative procedure which enforces the sampled data constraints in the outer level and up-dates dictionary and sparse representation in the inner level. Graph regularized sparse coding and simple dictionary updating applied in the inner minimization make the proposed algorithm converge with a relatively small number of iterations. Experimental results demonstrate that the proposed algorithm can consistently reconstruct both simulated MR images and real MR data efficiently, and outperforms the current state-of-the-art approaches in terms of visual comparisons and quantitative measures.
Random graph states, maximal flow and Fuss-Catalan distributions
Collins, BenoIt; Nechita, Ion [Department of Mathematics and Statistics, University of Ottawa, Ontario K1N8M2 (Canada); Zyczkowski, Karol [Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krakow (Poland)
2010-07-09
For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems. Each edge of the graph represents a bipartite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.
Random graph states, maximal flow and Fuss-Catalan distributions
Collins, Benoit; Zyczkowski, Karol
2010-01-01
For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by thes...
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...
A relationship between the diameter and the intersection number c2 for a distance-regular graph
Koolen, Jack H
2011-01-01
In this paper we will look at the relationship between the intersection number c2 and its diameter for a distance-regular graph. And also, we give some tools to show that a distance-regular graph with large c2 is bipartite, and a tool to show that if kD is too small then the distance-regular graph has to be antipodal.
Outlier Edge Detection Using Random Graph Generation Models and Applications
Zhang, Honglei; Gabbouj, Moncef
2016-01-01
Outliers are samples that are generated by different mechanisms from other normal data samples. Graphs, in particular social network graphs, may contain nodes and edges that are made by scammers, malicious programs or mistakenly by normal users. Detecting outlier nodes and edges is important for data mining and graph analytics. However, previous research in the field has merely focused on detecting outlier nodes. In this article, we study the properties of edges and propose outlier edge detection algorithms using two random graph generation models. We found that the edge-ego-network, which can be defined as the induced graph that contains two end nodes of an edge, their neighboring nodes and the edges that link these nodes, contains critical information to detect outlier edges. We evaluated the proposed algorithms by injecting outlier edges into some real-world graph data. Experiment results show that the proposed algorithms can effectively detect outlier edges. In particular, the algorithm based on the Prefe...
Semisymmetric Cubic Graphs as Regular Covers of K3,3
Chang Qun WANG; Tie Sheng CHEN
2008-01-01
A regular graph X is called semisymmetric if it is edge-transitive but not vertex-transitive.For G≤Aut X wecalla G-cover X semisymmetric if X is semisymmetric,and call a G-cover X one-regular if Aut X acts regularly on its arc-set.In this paper,we give the sufficient and necessary conditions for the existence of one-regular or semisymmetric Zn-covers of K3,3 Also,an in finite family of semisymmetric Zn ×Zn-covers of K3,3 are constructed.
Threshold Functions in Random s-Intersection Graphs
Zhao, Jun; Gligor, Virgil
2015-01-01
Random $s$-intersection graphs have recently received considerable attention in a wide range of application areas. In such a graph, each vertex is equipped with a set of items in some random manner, and any two vertices establish an undirected edge in between if and only if they have at least $s$ common items. In particular, in a uniform random $s$-intersection graph, each vertex independently selects a fixed number of items uniformly at random from a common item pool, while in a binomial random $s$-intersection graph, each item in some item pool is independently attached to each vertex with the same probability. For binomial/uniform random $s$-intersection graphs, we establish threshold functions for perfect matching containment, Hamilton cycle containment, and $k$-robustness, where $k$-robustness is in the sense of Zhang and Sundaram [IEEE Conf. on Decision & Control '12]. We show that these threshold functions resemble those of classical Erd\\H{o}s-R\\'{e}nyi graphs, where each pair of vertices has an un...
Phase diagrams for three-strategy evolutionary prisoner's dilemma games on regular graphs.
Szolnoki, Attila; Perc, Matjaz; Szabó, György
2009-11-01
Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.
REGULAR PATTERN MINING (WITH JITTER) ON WEIGHTED-DIRECTED DYNAMIC GRAPHS
Gupta, A.; H. K. THAKUR; Gupta, T.; Yadav, S.
2017-01-01
Real world graphs are mostly dynamic in nature, exhibiting time-varying behaviour in structure of the graph, weight on the edges and direction of the edges. Mining regular patterns in the occurrence of edge parameters gives an insight into the consumer trends over time in ecommerce co-purchasing networks. But such patterns need not necessarily be precise as in the case when some product goes out of stock or a group of customers becomes unavailable for a short period of time. Ignoring them ...
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Bayati, Mohsen; Tetali, Prasad
2009-01-01
We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erd{\\"o}s-R\\'{e}nyi graph $\\G(N,c/N)$ and random $r$-regular graph $\\G(N,r)$. For a variety of models, including independent sets, MAX-CUT, Coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p., thus resolving an open problem, (see Conjecture 2.20 in \\cite{WormaldModelsRandomGraphs}, as well as \\cite{Aldous:FavoriteProblems}, \\cite{BollobasRiordanMetrics}, \\cite{JansonThomason}, and \\cite{AldousSteele:survey}). Our approach is based on extending and simplifying the interpolation method developed by Guerra and Toninelli \\cite{GuerraTon} and Franz and Leone \\cite{FranzLeone},...
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.
Ensemble nonequivalence in random graphs with modular structure
Garlaschelli, Diego; den Hollander, Frank; Roccaverde, Andrea
2017-01-01
Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In Squartini, de Mol, den Hollander and Garlaschelli (2015 New J. Phys. 17 023052) it was shown that ensembles of random graphs are nonequivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, time-varying, block-model or community structure. We give a full classification of ensemble equivalence in the sparse regime, proving that breakdown occurs as soon as the number of local constraints (i.e. the number of constrained degrees) is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive an explicit formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.
Ensemble nonequivalence in random graphs with modular structure
Garlaschelli, Diego; Roccaverde, Andrea
2016-01-01
Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In Squartini et al. (2015) it was shown that ensembles of random graphs are non-equivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, block-model or community structure. We give a full classification of ensemble equivalence, proving that breakdown occurs if and only if the number of local constraints...
Multi-Graph Matching via Affinity Optimization with Graduated Consistency Regularization.
Yan, Junchi; Cho, Minsu; Zha, Hongyuan; Yang, Xiaokang; Chu, Stephen M
2016-06-01
This paper addresses the problem of matching common node correspondences among multiple graphs referring to an identical or related structure. This multi-graph matching problem involves two correlated components: i) the local pairwise matching affinity across pairs of graphs; ii) the global matching consistency that measures the uniqueness of the pairwise matchings by different composition orders. Previous studies typically either enforce the matching consistency constraints in the beginning of an iterative optimization, which may propagate matching error both over iterations and across graph pairs; or separate affinity optimization and consistency enforcement into two steps. This paper is motivated by the observation that matching consistency can serve as a regularizer in the affinity objective function especially when the function is biased due to noises or inappropriate modeling. We propose composition-based multi-graph matching methods to incorporate the two aspects by optimizing the affinity score, meanwhile gradually infusing the consistency. We also propose two mechanisms to elicit the common inliers against outliers. Compelling results on synthetic and real images show the competency of our algorithms.
REGULAR PATTERN MINING (WITH JITTER ON WEIGHTED-DIRECTED DYNAMIC GRAPHS
A. GUPTA
2017-02-01
Full Text Available Real world graphs are mostly dynamic in nature, exhibiting time-varying behaviour in structure of the graph, weight on the edges and direction of the edges. Mining regular patterns in the occurrence of edge parameters gives an insight into the consumer trends over time in ecommerce co-purchasing networks. But such patterns need not necessarily be precise as in the case when some product goes out of stock or a group of customers becomes unavailable for a short period of time. Ignoring them may lead to loss of useful information and thus taking jitter into account becomes vital. To the best of our knowledge, no work has been yet reported to extract regular patterns considering a jitter of length greater than unity. In this article, we propose a novel method to find quasi regular patterns on weight and direction sequences of such graphs. The method involves analysing the dynamic network considering the inconsistencies in the occurrence of edges. It utilizes the relation between the occurrence sequence and the corresponding weight and direction sequences to speed up this process. Further, these patterns are used to determine the most central nodes (such as the most profit yielding products. To accomplish this we introduce the concept of dynamic closeness centrality and dynamic betweenness centrality. Experiments on Enron e-mail dataset and a synthetic dynamic network show that the presented approach is efficient, so it can be used to find patterns in large scale networks consisting of many timestamps.
Dynamics of excitable nodes on random graphs
K Manchanda; T Umeshkanta Singh; R Ramaswamy
2011-11-01
We study the interplay of topology and dynamics of excitable nodes on random networks. Comparison is made between systems grown by purely random (Erd˝os–Rényi) rules and those grown by the Achlioptas process. For a given size, the growth mechanism affects both the thresholds for the emergence of different structural features as well as the level of dynamical activity supported on the network.
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...
Greedy Local Search and Vertex Cover in Sparse Random Graphs
Witt, Carsten
2009-01-01
. This work starts with a rigorous explanation for this claim based on the refined analysis of the Karp-Sipser algorithm by Aronson et al. Subsequently, theoretical supplements are given to experimental studies of search heuristics on random graphs. For c
Zhao, Jun; Gligor, Virgil
2015-01-01
Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of social networks including common-interest networks, collaboration networks, and actor networks. Simply put, a random intersection graph is constructed by assigning each node a set of items in some random manner and then putting an edge between any two nodes that share a certain number of items. Broadly speaking, our work is about analyzing random intersection graphs, and models generated by composing it with other random graph models including random geometric graphs and Erd\\H{o}s-R\\'enyi graphs. These compositional models are introduced to capture the characteristics of various complex natural or man-made networks more accurately than the existing models in the literature. For random intersection graphs and their compositions with other random graphs, we study properties su...
Randomly Orthogonal (g,f)-factorizations in Graphs
Gui-zhen Liu; He-ping Long
2002-01-01
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valued functions defined on V(G) such that 2k - 1 ≤ g(x) ≤ f(x) for all x ∈ V(G). Let H be a subgraph of G with mk edges . In this paper it is proved that every (mg + m - 1, mf - m + 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H and shown that the result is best possible.
Continuity of the integrated density of states on random length metric graphs
Lenz, Daniel; Post, Olaf; Veselic', Ivan
2008-01-01
We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge length are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.
A zero-one law for the existence of triangles in random key graphs
Yagan, Osman
2009-01-01
Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph.
Paths of specified length in random k-partite graphs
C. R. Subramanian
2001-12-01
Full Text Available Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V i, (i=1, …,k each having size Ω(n and choosing each possible edge with probability p. Consider any vertex x in any V i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V i (as x does or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.
Investigating Facebook Groups through a Random Graph Model
Dinithi Pallegedara; Lei Pan
2014-01-01
Facebook disseminates messages for billions of users everyday. Though there are log files stored on central servers, law enforcement agencies outside of the U.S. cannot easily acquire server log files from Facebook. This work models Facebook user groups by using a random graph model. Our aim is to facilitate detectives quickly estimating the size of a Facebook group with which a suspect is involved. We estimate this group size according to the number of immediate friends and the number of ext...
Random Matrices and Lyapunov Coefficients Regularity
Gallavotti, Giovanni
2017-02-01
Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a "cone property" are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.
Visibility graphs of random scalar fields and spatial data
Lacasa, Lucas; Iacovacci, Jacopo
2017-07-01
We extend the family of visibility algorithms to map scalar fields of arbitrary dimension into graphs, enabling the analysis of spatially extended data structures as networks. We introduce several possible extensions and provide analytical results on the topological properties of the graphs associated to different types of real-valued matrices, which can be understood as the high and low disorder limits of real-valued scalar fields. In particular, we find a closed expression for the degree distribution of these graphs associated to uncorrelated random fields of generic dimension. This result holds independently of the field's marginal distribution and it directly yields a statistical randomness test, applicable in any dimension. We showcase its usefulness by discriminating spatial snapshots of two-dimensional white noise from snapshots of a two-dimensional lattice of diffusively coupled chaotic maps, a system that generates high dimensional spatiotemporal chaos. The range of potential applications of this combinatorial framework includes image processing in engineering, the description of surface growth in material science, soft matter or medicine, and the characterization of potential energy surfaces in chemistry, disordered systems, and high energy physics. An illustration on the applicability of this method for the classification of the different stages involved in carcinogenesis is briefly discussed.
Ensembles of physical states and random quantum circuits on graphs
Hamma, Alioscia; Zanardi, Paolo
2012-01-01
In this paper we continue and extend the investigations of the ensembles of random physical states introduced in A. Hamma et al arXiv:1109.4391. These ensembles are constructed by finite-length random quantum circuits (RQC) acting on (hyper)edges of an underlying (hyper)graph structure. The latter encodes for the locality structure associated with finite-time quantum evolutions generated by physical i.e., local, Hamiltonians. Our goal is to analyze physical properties of typical states in these ensembles, in particular here we focus on proxies of quantum entanglement as purity and $\\alpha$-Renyi entropies. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. In the $\\alpha=2$ case these superoperators act on a restricted multi-qubit space generated by permutation operators associated to the subsets of vertices of the graph. For permutationally invariant interactions the dynamics c...
Evolution of tag-based cooperation on Erdős-Rényi random graphs
Lima, F. W. S.; Hadzibeganovic, Tarik; Stauffer, Dietrich
2014-12-01
Here, we study an agent-based model of the evolution of tag-mediated cooperation on Erdős-Rényi random graphs. In our model, agents with heritable phenotypic traits play pairwise Prisoner's Dilemma-like games and follow one of the four possible strategies: Ethnocentric, altruistic, egoistic and cosmopolitan. Ethnocentric and cosmopolitan strategies are conditional, i.e. their selection depends upon the shared phenotypic similarity among interacting agents. The remaining two strategies are always unconditional, meaning that egoists always defect while altruists always cooperate. Our simulations revealed that ethnocentrism can win in both early and later evolutionary stages on directed random graphs when reproduction of artificial agents was asexual; however, under the sexual mode of reproduction on a directed random graph, we found that altruists dominate initially for a rather short period of time, whereas ethnocentrics and egoists suppress other strategists and compete for dominance in the intermediate and later evolutionary stages. Among our results, we also find surprisingly regular oscillations which are not damped in the course of time even after half a million Monte Carlo steps. Unlike most previous studies, our findings highlight conditions under which ethnocentrism is less stable or suppressed by other competing strategies.
Coloring random graphs online without creating monochromatic subgraphs
Mütze, Torsten; Spöhel, Reto
2011-01-01
Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \\emph{online vertex-Ramsey density} of $F$ and $r$. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$ is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial-time...
Topics in networks: Community detection, random graphs, and network epidemiology
Karrer, Brian C.
In this dissertation, we present research on several topics in networks including community detection, random graphs, and network epidemiology. Traditional stochastic blockmodels may produce inaccurate fits to complex networks with heterogeneous degree distributions and we devise a degree-corrected block-model that alleviates this problematic behavior. The resulting objective function for community detection using the degree-corrected version outperforms the traditional model at finding communities on a variety of real-world and synthetic tests. Then we study a different generative model that associates communities to the edges of the network and naturally includes overlapping vertex communities. We create a fast and accurate algorithm to fit this model to empirical networks and show that it can be used to quickly find non-overlapping communities as well. We also develop random graph models for directed acyclic graphs, a class of networks including family trees and citation networks. We argue that the lack of cycles comes from an ordering constraint and then generalize the configuration model to incorporate this constraint. We calculate many properties of these models and demonstrate that sonic of the model predictions agree quite well with real-world networks, emphasizing the importance of vertex ordering to generating directed acyclic networks with realistic properties. Finally, we examine the spread of disease over networks, starting with a simple model of two diseases spreading with cross-immunity, where infection by one disease makes an individual immune to the other disease and vice versa. Utilizing a timescale separation argument, we map the system to consecutive bond percolation, one disease spreading after the other. The resulting phase diagram includes discontinuous and continuous phase transitions and a coexistence region where both diseases can spread to a substantial fraction of the population. Then we analyze a flexible susceptible
Graph Regularized Within-Class Sparsity Preserving Projection for Face Recognition
Songjiang Lou
2015-04-01
Full Text Available As a dominant method for face recognition, the subspace learning algorithm shows desirable performance. Manifold learning can deal with the nonlinearity hidden in the data, and can project high dimensional data onto low dimensional data while preserving manifold structure. Sparse representation shows its robustness for noises and is very practical for face recognition. In order to extract the facial features from face images effectively and robustly, in this paper, a method called graph regularized within-class sparsity preserving analysis (GRWSPA is proposed, which can preserve the within-class sparse reconstructive relationship and enhances separatability for different classes. Specifically, for each sample, we use the samples in the same class (except itself to represent it, and keep the reconstructive weight unchanged during projection. To preserve the manifold geometry structure of the original space, one adjacency graph is constructed to characterize the interclass separability and is incorporated into its criteria equation as a constraint in a supervised manner. As a result, the features extracted are sparse and discriminative and helpful for classification. Experiments are conducted on the two open face databases, the ORL and YALE face databases, and the results show that the proposed method can effectively and correctly find the key facial features from face images and can achieve better recognition rate compared with other existing ones.
Plasmodial vein networks of the slime mold Physarum polycephalum form regular graphs
Baumgarten, Werner; Ueda, Tetsuo; Hauser, Marcus J. B.
2010-10-01
The morphology of a typical developing biological transportation network, the vein network of the plasmodium of the myxomycete Physarum polycephalum is analyzed during its free extension. The network forms a classical, regular graph, and has exclusively nodes of degree 3. This contrasts to most real-world transportation networks which show small-world or scale-free properties. The complexity of the vein network arises from the weighting of the lengths, widths, and areas of the vein segments. The lengths and areas follow exponential distributions, while the widths are distributed log-normally. These functional dependencies are robust during the entire evolution of the network, even though the exponents change with time due to the coarsening of the vein network.
Structure of Thin Irreducible Modules of a Q-polynomial Distance-Regular Graph
Cerzo, Diana R
2010-01-01
Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the subalgebra of Mat_X(C) generated by A and A*. Let W denote a thin irreducible T-module. It is known that the action of A and A* on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, these results will be applied to obtain a detailed description of W. In particular, we give a description of W in terms of its intersection numbers, dual intersection numbers and parameter array. Finally, we apply our results to the case in which Gamma has q-Racah type or classical parameters.
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....
Antiferromagnetic Potts model on the Erdos-Renyi random graph
Contucci, Pierluig; Giardina', Cristian; Starr, Shannon
2011-01-01
We study the antiferromagnetic Potts model on the Erdos-Renyi random graph. By identifying a suitable interpolation structure and proving an extended variational principle we show that the replica symmetric solution is an upper bound for the limiting pressure which can be recovered in the framework of Derrida-Ruelle probability cascades. A comparison theorem with a mixed model made of a mean field Potts-antiferromagnet plus a Potts-Sherrington-Kirkpatrick model allows to show that the replica symmetric solution is exact at high temperatures.
强正则自补图的一个注记%A Note on Strongly Regular Self-complementary Graphs
田方
2006-01-01
K(o)tzig put forward a question on strongly-regular self-complementary graphs,that is, for any natural number k, whether there exists a strongly-regular self- complementary graph whose order is 4k + 1, where 4k + 1 = x2 + y2, x and y are positive integers; what is the minimum number that made there exist at least two non-isomorphic strongly-regular self-complementary graphs. In this paper, we use two famous lemmas to generalize the existential conditions for strongly-regular self-complementary circular graphs with 4k + 1 orders.
Motifs in triadic random graphs based on Steiner triple systems
Winkler, Marco; Reichardt, Jörg
2013-08-01
Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade, the overabundance of certain subnetwork patterns, i.e., the so-called motifs, has attracted much attention. It has been hypothesized that these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information, is the central theme of network science, there is still a lack of sound generative models needed for testing the functional role of subgraph motifs. Our work aims to overcome this limitation. We employ the framework of exponential random graph models (ERGMs) to define models based on triadic substructures. The fact that only a small portion of triads can actually be set independently poses a challenge for the formulation of such models. To overcome this obstacle, we use Steiner triple systems (STSs). These are partitions of sets of nodes into pair-disjoint triads, which thus can be specified independently. Combining the concepts of ERGMs and STSs, we suggest generative models capable of generating ensembles of networks with nontrivial triadic Z-score profiles. Further, we discover inevitable correlations between the abundance of triad patterns, which occur solely for statistical reasons and need to be taken into account when discussing the functional implications of motif statistics. Moreover, we calculate the degree distributions of our triadic random graphs analytically.
Critical Behaviour of Spanning Forests on Random Planar Graphs
Bondesan, Roberto; Sportiello, Andrea
2016-01-01
As a follow-up of previous work of the authors, we analyse the statistical mechanics model of random spanning forests on random planar graphs. Special emphasis is given to the analysis of the critical behaviour. Exploiting an exact relation with a model of O(-2)-loops and dimers, previously solved by Kostov and Staudacher, we identify critical and multicritical loci, and find them consistent with recent results of Bousquet-M\\'elou and Courtiel. This is also consistent with the KPZ relation, and the Berker-Kadanoff phase in the anti-ferromagnetic regime of the Potts Model on periodic lattices, predicted by Saleur. To our knowledge, this is the first known example of KPZ appearing explicitly to work within a Berker-Kadanoff phase. We set up equations for the generating function, at the value t=-1 of the fugacity, which is of combinatorial interest, and we investigate the resulting numerical series, a Tony Guttmann's favourite problem.
Phase Transitions on Fixed Connected Graphs and Random Graphs in the Presence of Noise
Liu, Jialing; Sehgal, Hullas; Olson, Joshua M; Liu, Haifeng; Elia, Nicola
2008-01-01
In this paper, we study the phase transition behavior emerging from the interactions among multiple agents in the presence of noise. We propose a simple discrete-time model in which a group of non-mobile agents form either a fixed connected graph or a random graph process, and each agent, taking bipolar value either +1 or -1, updates its value according to its previous value and the noisy measurements of the values of the agents connected to it. We present proofs for the occurrence of the following phase transition behavior: At a noise level higher than some threshold, the system generates symmetric behavior (vapor or melt of magnetization) or disagreement; whereas at a noise level lower than the threshold, the system exhibits spontaneous symmetry breaking (solid or magnetization) or consensus. The threshold is found analytically. The phase transition occurs for any dimension. Finally, we demonstrate the phase transition behavior and all analytic results using simulations. This result may be found useful in t...
Auxiliary Parameter MCMC for Exponential Random Graph Models
Byshkin, Maksym; Stivala, Alex; Mira, Antonietta; Krause, Rolf; Robins, Garry; Lomi, Alessandro
2016-11-01
Exponential random graph models (ERGMs) are a well-established family of statistical models for analyzing social networks. Computational complexity has so far limited the appeal of ERGMs for the analysis of large social networks. Efficient computational methods are highly desirable in order to extend the empirical scope of ERGMs. In this paper we report results of a research project on the development of snowball sampling methods for ERGMs. We propose an auxiliary parameter Markov chain Monte Carlo (MCMC) algorithm for sampling from the relevant probability distributions. The method is designed to decrease the number of allowed network states without worsening the mixing of the Markov chains, and suggests a new approach for the developments of MCMC samplers for ERGMs. We demonstrate the method on both simulated and actual (empirical) network data and show that it reduces CPU time for parameter estimation by an order of magnitude compared to current MCMC methods.
Motifs in Triadic Random Graphs based on Steiner Triple Systems
Winkler, Marco
2013-01-01
Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade the overabundance of certain sub-network patterns, so called motifs, has attracted high attention. It has been hypothesized, these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information is the central theme of network science, there is still a lack of sound generative models needed for testing the functional role of subgraph motifs. Our work aims to overcome this limitation. We employ the framework of exponential random graphs (ERGMs) to define novel models based on triadic substructures. The fact that only a small portion of triads can actually be set independently poses a challenge for the formulation of such models. To overcome this obst...
Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs
Chen, Yu-Ting; McAvoy, Alex; Nowak, Martin A.
2016-12-01
Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.
On $k$-connectivity and minimum vertex degree in random $s$-intersection graphs
Zhao, Jun; Gligor, Virgil
2014-01-01
Random s-intersection graphs have recently received much interest [1-9]. In a random s-intersection graph, each vertex is assigned to a set of items in some manner, and two vertices have an edge in between if and only if they share at least s items. In particular, in a uniform random s-intersection graph, each vertex independently selects the same number of items uniformly at random from a common item pool, while in a binomial random s-intersection graph, each item in some item pool is independently attached to each vertex with the same probability. These two graph models have numerous applications; e.g., using uniform random s-intersection graph for cryptanalysis [14,15], and to model secure wireless sensor networks [8-10] and online social networks [11,12], and using a binomial random s-intersection graph for clustering analysis [17], classification [18] and the design of integrated circuits [34]. For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum ver...
Recent developments in exponential random graph (p*) models for social networks
Robins, Garry; Snijders, Tom; Wang, Peng; Handcock, Mark; Pattison, Philippa
2007-01-01
This article reviews new specifications for exponential random graph models proposed by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology] and demonstrates their improvement over homogen
Random noise attenuation using an improved anisotropic total variation regularization
Gemechu, Diriba; Yuan, Huan; Ma, Jianwei
2017-09-01
In seismic data processing, attenuation of random noise from the observed data is the basic step which improves the signal-to-noise ratio (SNR) of seismic data. In this paper, we proposed an anisotropic total bounded variation regularization approach to attenuate noise. An improved constraint convex optimization model is formulated for this approach and then the split Bregman algorithm is used to solve the optimization model. Generalized cross validation (GCV) technique is used to estimate the regularization parameter. Synthetic and real seismic data are considered to show the out performance of the proposed method in terms of event-preserving denoising, in comparison with FX deconvolution, shearlet hard thresholding, and anisotropic total variation methods. The numerical results indicate that the proposed method effectively attenuates random noise by preserving the structure and important features of seismic data.
Ising Critical Behavior of Inhomogeneous Curie-Weiss Models and Annealed Random Graphs
Dommers, Sander; Giardinà, Cristian; Giberti, Claudio; van der Hofstad, Remco; Prioriello, Maria Luisa
2016-11-01
We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant {J_{ij}(β)} for the edge {ij} on the complete graph is given by {J_{ij}(β)=β w_iw_j/( {sum_{kin[N]}w_k})}. We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises [with inverse temperature {β} replaced by {sinh(β)} ] from the annealed Ising model on the generalized random graph. We assume that the vertex weights {(w_i)_{iin[N]}} are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent {τ} with {τin(3,5)}, then the critical exponents depend sensitively on {τ}. In addition, at criticality, the total spin {S_N} satisfies that {S_N/N^{(τ-2)/(τ-1)}} converges in law to some limiting random variable whose distribution we explicitly characterize.
On the strengths of connectivity and robustness in general random intersection graphs
Zhao, Jun; Gligor, Virgil
2014-01-01
Random intersection graphs have received much attention for nearly two decades, and currently have a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. In this paper, we investigate the strengths of connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for $k$-connectivity and $k$-robustness, as well as the asymptotically exact probability of $k$-connectivity, for any positive integer $k$. The $k$-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures. On the other hand, $k$-robustness measures the effectiveness of local diffusion strategies (that do not use global graph topology information) in spreading information over the graph in the presence of misbehaving nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomi...
Critical behaviour of spanning forests on random planar graphs
Bondesan, Roberto; Caracciolo, Sergio; Sportiello, Andrea
2017-02-01
As a follow-up of previous work of the authors, we analyse the statistical mechanics model of random spanning forests on random planar graphs. Special emphasis is given to the analysis of the critical behaviour. Exploiting an exact relation with a model of \\text{O}(-2) -loops and dimers, previously solved by Kostov and Staudacher, we identify critical and multicritical loci, and find them consistent with recent results of Bousquet-Mélou and Courtiel. This is also consistent with the KPZ relation, and the Berker-Kadanoff phase in the anti-ferromagnetic regime of the Potts Model on periodic lattices, predicted by Saleur. To our knowledge, this is the first known example of KPZ appearing explicitly to work within a Berker-Kadanoff phase. We set up equations for the generating function, at the value t = -1 of the fugacity, which is of combinatorial interest, and we investigate the resulting numerical series, a favourite problem of Tony Guttmann’s. Dedicated to Tony Guttmann on the occasion of his 70th birthday.
Simplicity and Complexity, Regularity and Randomness : exceptional CERN colloquium
CERN. Geneva; Alvarez-Gaumé, Luís; Landua, Rolf
2005-01-01
The concept of effective complexity, which involves discussing bit strings, utilizing algorithmic information content (AIC), and making a distinction between regularity and randomness, will be explored. Like entropy, the quantities involved all depend crucially on coarse graining and may have other context dependence as well. Besides AIC, which involves the length of programs for a universal computer, it is important to consider also quantities that depend on the execution times for such programs. In that way one can get at pseudo-ramdomness and pseudo-complexity. The presumably simple fundamental laws of physics contribute very little to the AIC of the history of the universe. Instead, almost all of that AIC comes from the results of chance events. Thus it is deeply misleading to refer to the future unified theory of the elementary particles and their interactions as a "theory of everything." Nevertheless, the search for that unified theory, the ultimate regularity in nature, remains a magnificent challenge....
Applications of Random Graphs to 2D Quantum Gravity
Atkin, Max R
2011-01-01
The central topic of this thesis is two dimensional Quantum Gravity and its properties. The term Quantum Gravity itself is ambiguous as there are many proposals for its correct formulation and none of them have been verified experimentally. In this thesis we consider a number of closely related approaches to two dimensional quantum gravity that share the property that they may be formulated in terms of random graphs. In one such approach known as Causal Dynamical Triangulations, numerical computations suggest an interesting phenomenon in which the effective spacetime dimension is reduced in the UV. In this thesis we first address whether such a dynamical reduction in the number of dimensions may be understood in a simplified model. We introduce a continuum limit where this simplified model exhibits a reduction in the effective dimension of spacetime in the UV, in addition to having rich cross-over behaviour. In the second part of this thesis we consider an approach closely related to causal dynamical triangul...
Scale-free random graphs and Potts model
D-S Lee; K-I Goh; B Kahng; D Kim
2005-06-01
We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertex has a prescribed weight − (0 < < 1) and an edge can connect vertices and with rate . Corresponding equilibrium ensemble is identified and the problem is solved by the → 1 limit of the -state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of > 3 and 2 < < 3, where = 1 + -1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite shows double peaks.
Navigability of Random Geometric Graphs in the Universe and Other Spacetimes.
Cunningham, William; Zuev, Konstantin; Krioukov, Dmitri
2017-08-18
Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.
Localization in random bipartite graphs: Numerical and empirical study
Slanina, František
2017-05-01
We investigate adjacency matrices of bipartite graphs with a power-law degree distribution. Motivation for this study is twofold: first, vibrational states in granular matter and jammed sphere packings; second, graphs encoding social interaction, especially electronic commerce. We establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph. At the jamming threshold, where the two parts have the same size, localization vanishes. We found that the multifractal spectrum is nontrivial in the delocalized phase, but still near the mobility edge. We also study an empirical bipartite graph, namely, the Amazon reviewer-item network. We found that in this specific graph the mobility edge disappears, and we draw a conclusion from this fact regarding earlier empirical studies of the Amazon network.
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
Local dependence in random graph models: characterization, properties and statistical inference.
Schweinberger, Michael; Handcock, Mark S
2015-06-01
Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with 'ground truth'.
Degree-degree correlations in random graphs with heavy-tailed degrees
Litvak, Nelli; van der Hofstad, Remco
2012-01-01
We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity
Degree-degree correlations in random graphs with heavy-tailed degrees
Litvak, Nelli; van der Hofstad, Remco
2012-01-01
We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coe
Fast solution of NP-hard coloring problems on large random graphs
Bedini, Andrea
2010-01-01
Combining tree decomposition and transfer matrix techniques provides a highly efficient and very general algorithm for computing exact partition functions of statistical models defined on large graphs. We illustrate this by considering the hard problem of computing the exact number of vertex colorings for randomly generated planar graphs with up to N = 100 vertices.
Hitting times, commute distances and the spectral gap for large random geometric graphs
von Luxburg, Ulrike; Hein, Matthias
2010-01-01
The commute distance between two vertices in a graph is the expected time it takes a random walk to travel from the first to the second vertex and back. According to folklore opinion, it has the property that vertices in the same cluster of the graph are close to each other while vertices in different clusters are far from each other. We study the behavior of the commute distance and hitting times on random geometric graphs ($\\epsilon$-graphs, $k$-nearest neighbor graphs and Gaussian similarity graphs). It turns out that as the size of the graph increases, the suitably rescaled hitting times and commute distances can be approximated by extremely simple expressions. However, these expressions no longer take into account the cluster structure of the graph, which leads to a completely meaningless distance function. Consequently, the use of the commute distance for machine learning purposes is discouraged for large graphs and in high dimensions. Our paper also makes several important technical contributions such ...
Growth of Preferential Attachment Random Graphs Via Continuous-Time Branching Processes
Krishna B Athreya; Arka P Ghosh; Sunder Sethuraman
2008-08-01
Some growth asymptotics of a version of `preferential attachment’ random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.
Computing the Expected Values of some Properties of Randomly Weighted Graphs
Emek, Yuval; Shavitt, Yuval
2009-01-01
Consider the setting of \\emph{randomly weighted graphs}, namely, graphs whose edge weights are independent discrete random variables with finite support over the non-negative reals. Given a randomly weighted graph $G$, we are interested in computing the expected values of various graph properties of $G$. In particular, we focus on the problem of computing the expected diameter of $G$. It is easy to show that this problem is \\SharpP-hard even in the restricted case in which all edge weights are identically distributed. In this paper we prove that this problem admits a \\emph{fully polynomial time randomized approximation scheme (FPRAS)}. Our technique can also be used to derive an FPRAS for the problem of computing the expected weight of an MST of $G$.
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.
2012-02-28
IEEE Transactions on Automatic Control (to appear). • A. Chapman and M. Mesbahi, Influence models for consensus-type networks, IEEE Transactions on Automatic Control (to...analysis and synthesis of relative sensing networks, IEEE Transactions on Automatic control , 56 (5): 971-982, 2011. • D. Zelazo and M. Mesbahi, Edge...agreement: graph-theoretic performance bounds and passivity anal- ysis, IEEE
Zhao, Jun; Gligor, Virgil
2014-01-01
Random key graphs have been used in secure wireless sensor networks (WSNs) and various other applications. Random key graphs, denoted $\\mathbb{G}(n;K,P)$, form a class of random intersection graphs and can be described as follows: With $\\mathcal{V}_n=\\{v_1, \\ldots, v_n\\}$ denoting the set of vertices, each vertex $v_i$ is assigned a set $S_i$ of $K$ distinct keys that are selected uniformly at random from a key pool of size $P$. An undirected edge is then drawn between any pair of distinct vertices $v_i$ and $v_j$ if $S_i \\cap S_j \
Schmidt, Deena R; Thomas, Peter J
2014-04-17
Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion-channel models, such as the Hodgkin-Huxley or other conductance-based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion-channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán's approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.
Tumor stratification by a novel graph-regularized bi-clique finding algorithm.
Ahmadi Adl, Amin; Qian, Xiaoning
2015-08-01
Due to involved disease mechanisms, many complex diseases such as cancer, demonstrate significant heterogeneity with varying behaviors, including different survival time, treatment responses, and recurrence rates. The aim of tumor stratification is to identify disease subtypes, which is an important first step towards precision medicine. Recent advances in profiling a large number of molecular variables such as in The Cancer Genome Atlas (TCGA), have enabled researchers to implement computational methods, including traditional clustering and bi-clustering algorithms, to systematically analyze high-throughput molecular measurements to identify tumor subtypes as well as their corresponding associated biomarkers. In this study we discuss critical issues and challenges in existing computational approaches for tumor stratification. We show that the problem can be formulated as finding densely connected sub-graphs (bi-cliques) in a bipartite graph representation of genomic data. We propose a novel algorithm that takes advantage of prior biology knowledge through a gene-gene interaction network to find such sub-graphs, which helps simultaneously identify both tumor subtypes and their corresponding genetic markers. Our experimental results show that our proposed method outperforms current state-of-the-art methods for tumor stratification.
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.
A probabilistic paradigm for handling uncertain objects in GIS by randomized graph algebra
SHI Wenzhong; WU Huayi
2003-01-01
Probability theory faces difficulties when it is applied to describing uncertain objects in geographic information system (GIS). This is mainly due to the fact that an object in GIS is normally described by a series of discrete vertexes. Modeling uncertainty objects should be therefore based on error of the composed vertexes. This type of model is normally complex and relatively difficult to implement because of many unknown factors, such as the number of vertexes of a polygon, error nature of each individual vertex and error correlation among the vertexes. In this paper, a probabilistic paradigm for handling uncertain objects in GIS by randomized graph algebra is presented. The theoretical basis for this paradigm is the randomized graph algebra-a probability theory for graph-which is newly proposed in this study. Classical probability theory is based on numerical algebra and is also an extension of numerical algebra by further defining probability density within a numerical domain. In the same token, this study begins with defining graph algebra as the basis for probability theory for graph. First, we adopt the theory of graph algebra and further refine the theory by defining the modulo operation for graph. As a result, a graph can thereafter be treated as a "number" and operated by "addition", "subtraction" and others. Second, we construct a measure space by generating sigma-algebra and defining measurable function upon it. The measure space becomes a probability space when the measurable function is a probability density function. Third, we propose the probabilistic paradigm for describing and inferring the uncertainty of geometric objects in GIS by applying the developed randomized graph algebra.
COMPARING RANDOM AND REGULAR NETWORK RESILIANCE AGAINST RANDOM ATTACK ON THEIR NODES
Ilya B. Gertsbakh
2013-01-01
Full Text Available We consider a family of connected networks whose nodes are subject to randomfailures ("attacks". Node failure means elimination of all links incident to the attacked node.Each node, independently of others, fails with probability q . Network failure (DOWN state isdefined as the situation when the largest connected component has "critical" size L . Wecompare the probabilistic resilience of a simulated network (obtained by a preferentialassignment-type algorithm versus a regular network having the same number of nodes and links.This comparison is carried out for three types of regular networks: the dodecahedron (20 nodes,30 links, square torus-type grid (25 nodes, 50 links and five-dimensional cubic network (32nodes, 80 links. For all three types of networks the critical value of L was approximately equalone third of the nodes. It turns out that the network with regular structure and node degree 5 d has higher resilience than a network with random structure, i.e. a regular network has smallerDOWN probability than a random network for the same q value and for the same number offailed nodes x It turns out, however, that the advantage of a regular network over a randomnetwork vanishes with the decrease of the average node degree. So, for 3, d random networkand its regular counterpart (so-called dodecahedron have approximately the same resilience. Ourinvestigation is based on comparing the so-called cumulative D-spectra and the network DOWNprobabilities as a function of node failure probability . q
Motif based hierarchical random graphs: structural properties and critical points of an Ising model
Kotorowicz, M; 10.5488/CMP.14.13801
2011-01-01
A class of random graphs is introduced and studied. The graphs are constructed in an algorithmic way from five motifs which were found in [Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science, 2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski M., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the short-range bonds are non-random, whereas the long-range bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, small-world property. For one of the motifs, the critical point of the Ising model defined on the corresponding graph has been studied.
A study of serial ranks via random graphs
Haeusler, Erich; Mason, David M.; Turova, Tatyana S.
2000-01-01
Serial ranks have long been used as the basis for nonparametric tests of independence in time series analysis. We shall study the underlying graph structure of serial ranks. This will lead us to a basic martingale which will allow us to construct a weighted approximation to a serial rank process. To show the applicability of this approximation, we will use it to prove two very general central limit theorems for Wald-Wolfowitz-type serial rank statistics.
Halperin, S.; Zwick, U. [Tel Aviv Univ. (Israel)
1996-12-31
We present the first randomized O(log n) time and O(m + n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G = (V, E) with n vertices and m edges. Our algorithm is optimal with respect to time, work and space. As a consequence we get optimal randomized EREW PRAM algorithms for other basic connectivity problems such as finding a bipartite partition, finding bridges and biconnected components, and finding Euler tours in Eulerean graphs. For other problems such as finding an ear decomposition, finding an open ear decomposition, finding a strong orientation, and finding an st-numbering we get optimal randomized CREW PRAM algorithms.
A novel configuration model for random graphs with given degree sequence
Xu Xin-Ping; Liu Feng
2007-01-01
Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. This paper presents a specific realization of a class of random network models in which the connection probability between two vertices (i, j) is a specific function of degrees ki and kj. In the framework of the configuration model of random graphs, we find the analytical expressions for the degree correlation and clustering as a function of the variance of the desired degree distribution. The obtained expressions are checked by means of numerical simulations. Possible applications of our model are discussed.
Ahmed, Faraz; Liu, Alex X
2013-01-01
Online social networks are being increasingly used for analyzing various societal phenomena such as epidemiology, information dissemination, marketing and sentiment flow. Popular analysis techniques such as clustering and influential node analysis, require the computation of eigenvectors of the real graph's adjacency matrix. Recent de-anonymization attacks on Netflix and AOL datasets show that an open access to such graphs pose privacy threats. Among the various privacy preserving models, Differential privacy provides the strongest privacy guarantees. In this paper we propose a privacy preserving mechanism for publishing social network graph data, which satisfies differential privacy guarantees by utilizing a combination of theory of random matrix and that of differential privacy. The key idea is to project each row of an adjacency matrix to a low dimensional space using the random projection approach and then perturb the projected matrix with random noise. We show that as compared to existing approaches for ...
Liu, Cheng-Wei; Polkovnikov, Anatoli; Sandvik, Anders W
2015-04-10
We discuss an Ising spin glass where each S=1/2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
Suliman, Mohamed
2016-01-01
In this supplementary appendix we provide proofs and additional simulation results that complement the paper (constrained perturbation regularization approach for signal estimation using random matrix theory).
Analysis of an iterated local search algorithm for vertex cover in sparse random graphs
Witt, Carsten
2012-01-01
algorithm by Aronson et al. (1998) [1]. Subsequently, theoretical supplements are given to experimental studies of search heuristics on random graphs. For c...Recently, various randomized search heuristics have been studied for the solution of the minimum vertex cover problem, in particular for sparse random instances according to the G(n,c/n) model, where c>0 is a constant. Methods from statistical physics suggest that the problem is easy if c...
Xue, Zhaohui; Du, Peijun; Li, Jun; Su, Hongjun
2017-02-01
The generally limited availability of training data relative to the usually high data dimension pose a great challenge to accurate classification of hyperspectral imagery, especially for identifying crops characterized with highly correlated spectra. However, traditional parametric classification models are problematic due to the need of non-singular class-specific covariance matrices. In this research, a novel sparse graph regularization (SGR) method is presented, aiming at robust crop mapping using hyperspectral imagery with very few in situ data. The core of SGR lies in propagating labels from known data to unknown, which is triggered by: (1) the fraction matrix generated for the large unknown data by using an effective sparse representation algorithm with respect to the few training data serving as the dictionary; (2) the prediction function estimated for the few training data by formulating a regularization model based on sparse graph. Then, the labels of large unknown data can be obtained by maximizing the posterior probability distribution based on the two ingredients. SGR is more discriminative, data-adaptive, robust to noise, and efficient, which is unique with regard to previously proposed approaches and has high potentials in discriminating crops, especially when facing insufficient training data and high-dimensional spectral space. The study area is located at Zhangye basin in the middle reaches of Heihe watershed, Gansu, China, where eight crop types were mapped with Compact Airborne Spectrographic Imager (CASI) and Shortwave Infrared Airborne Spectrogrpahic Imager (SASI) hyperspectral data. Experimental results demonstrate that the proposed method significantly outperforms other traditional and state-of-the-art methods.
Performance Analysis for Mobile Ad Hoc Network in Random Graph Models with Spatial Reuse
Han-xing Wang; Xi Hu; Qin Zhang
2007-01-01
In this paper,we present a random graph model with spatial reuse for a mobile ad hoc network (MANET) based on the dynamic source routing protocol.Many important performance parameters of the MANET are obtained,such as the average flooding distance (AFD),the probability generating function of the flooding distance,and the probability of a flooding route to be symmetric.Compared with the random graph model without spatial reuse,this model is much more effective because it has a smaller value of AFD and a larger probability for finding a symmetric valid route.
Random Graphs for Performance Evaluation of Recommender Systems
Chojnacki, Szymon
2010-01-01
The purpose of this article is to introduce a new analytical framework dedicated to measuring performance of recommender systems. The standard approach is to assess the quality of a system by means of accuracy related statistics. However, the specificity of the environments in which recommender systems are deployed requires to pay much attention to speed and memory requirements of the algorithms. Unfortunately, it is implausible to assess accurately the complexity of various algorithms with formal tools. This can be attributed to the fact that such analyses are usually based on an assumption of dense representation of underlying data structures. Whereas, in real life the algorithms operate on sparse data and are implemented with collections dedicated for them. Therefore, we propose to measure the complexity of recommender systems with artificial datasets that posses real-life properties. We utilize recently developed bipartite graph generator to evaluate how state-of-the-art recommender systems' behavior is d...
Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities
Todeschini, Adrien
2016-01-01
We propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process, and naturally generalizes existing probabilistic models with overlapping block-structure to the sparse regime. Our construction builds on vectors of completely random measures, and has interpretable parameters, each node being assigned a vector representing its level of affiliation to some latent communities. We develop methods for simulating this class of random graphs, as well as to perform posterior inference. We show that the proposed approach can recover interpretable structure from two real-world networks and can handle graphs with thousands of nodes and tens of thousands of edges.
An efficient algorithm for the vertex-disjoint paths problem in random graphs
Broder, A.Z. [Digital Systems Research Center, Palo Alto, CA (United States); Frieze, A.M.; Suen, S. [Carnegie-Mellon Univ., Pittsburgh, PA (United States); Upfal, E. [IBM Almaden Research Center, San Jose, CA (United States)
1996-12-31
Given a graph G = (V, E) and a set of pairs of vertices in V, we are interested in finding for each pair (a{sub i}, b{sub i}) a path connecting a{sub i} to b{sub i}, such that the set of paths so found is vertex-disjoint. (The problem is NP-complete for general graphs as well as for planar graphs. It is in P if the number of pairs is fixed.) Our model is that the graph is chosen first, then an adversary chooses the pairs of endpoints, subject only to obvious feasibility constraints, namely, all pairs must be disjoint, no more than a constant fraction of the vertices could be required for the paths, and not {open_quotes}too many{close_quotes} neighbors of a vertex can be endpoints. We present a randomized polynomial time algorithm that works for almost all graphs; more precisely in the G{sub n,m} or G{sub n,p} models, the algorithm succeeds with high probability for all edge densities above the connectivity threshold. The set of pairs that can be accommodated is optimal up to constant factors. Although the analysis is intricate, the algorithm itself is quite simple and suggests a practical heuristic. We include two applications of the main result, one in the context of circuit switching communication, the other in the context of topological embeddings of graphs.
Norrenbrock, Christoph; Hartmann, Alexander K
2015-01-01
The pivotal quality of proximity graphs is connectivity, i.e. all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as it may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like number and sizes of connected components, and more complex ones like the hop diameter and the backup capacity, which is needed to make a network N-1 resilient. The actual fragmentation turns out to be described by a second order phase transition. Using finite-size scaling analyses we numerically assess the threshold frac...
Phase Transitions for the Cavity Approach to the Clique Problem on Random Graphs
Gaudillière, Alexandre; Scoppola, Benedetto; Scoppola, Elisabetta; Viale, Massimiliano
2011-12-01
We give a rigorous proof of two phase transitions for a disordered statistical mechanics system used to define an algorithm to find large cliques inside Erdös random graphs. Such a system is a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.
Graph-Based Transform for 2D Piecewise Smooth Signals With Random Discontinuity Locations.
Zhang, Dong; Liang, Jie
2017-04-01
The graph-based block transform recently emerged as an effective tool for compressing some special signals such as depth images in 3D videos. However, in existing methods, overheads are required to describe the graph of the block, from which the decoder has to calculate the transform via time-consuming eigendecomposition. To address these problems, in this paper, we aim to develop a single graph-based transform for a class of 2D piecewise smooth signals with similar discontinuity patterns. We first consider the deterministic case with a known discontinuity location in each row. We propose a 2D first-order autoregression (2D AR1) model and a 2D graph for this type of signals. We show that the closed-form expression of the inverse of a biased Laplacian matrix of the proposed 2D graph is exactly the covariance matrix of the proposed 2D AR1 model. Therefore, the optimal transform for the signal are the eigenvectors of the proposed graph Laplacian. Next, we show that similar results hold in the random case, where the locations of the discontinuities in different rows are randomly distributed within a confined region, and we derive the closed-form expression of the corresponding optimal 2D graph Laplacian. The theory developed in this paper can be used to design both pre-computed transforms and signal-dependent transforms with low complexities. Finally, depth image coding experiments demonstrate that our methods can achieve similar performance to the state-of-the-art method, but our complexity is much lower.
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
(g, f)-Factorizations Randomly Orthogonal to a Subgraph in Graphs
Hao ZHAO; Gui Zhen LIU; Xiao Xia YAN
2005-01-01
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer valued functions defined on V(G) such that 2k - 2 ≤ g(x) ≤ f(x) for all x ∈ V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg + m - 1, mf - m + 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions.
Szemeredi's Regularity Lemma and Its Applications to Pairwise Clustering and Segmentation
Sperotto, A.; Pelillo, M.
2007-01-01
Szemeredi’s regularity lemma is a deep result from extremal graph theory which states that every graph can be well-approximated by the union of a constant number of random-like bipartite graphs, called regular pairs. Although the original proof was non-constructive, efficient (i.e., polynomial-time)
Ge Chen; Tian-de Guo; Chang-long Yao
2009-01-01
In this paper we consider the standard Poisson Boolean model of random geometric graphs G(Hλ,s; 1) in Rd and study the properties of the order of the largest component L1(G(Hλ,s; 1)) .We prove that E[L1(G(Hλ,s; 1))]is smooth with respect to λ,and is derivable with respect to s.Also,we give the expression of these derivatives.These studies provide some new methods for the theory of the largest component of finite random geometric graphs (not asymptotic graphs as s→∞) in the high dimensional space (d≥2).Moreover,we investigate the convergence rate of E[L1(G(Hλ,s; 1))].These results have significance for theory develop-ment of random geometric graphs and its practical application.Using our theories,we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks,which has the significance of theoretical foundation and guidance for the design of network layout.
Emergence of the giant weak component in directed random graphs with arbitrary degree distributions
Kryven, Ivan
2016-07-01
The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition, we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step-polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.
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...
Maximum Matchings in Random Bipartite Graphs and the Space Utilization of Cuckoo Hashtables
Frieze, Alan
2009-01-01
We study the the following question in Random Graphs. We are given two disjoint sets $L,R$ with $|L|=n=\\alpha m$ and $|R|=m$. We construct a random graph $G$ by allowing each $x\\in L$ to choose $d$ random neighbours in $R$. The question discussed is as to the size $\\mu(G)$ of the largest matching in $G$. When considered in the context of Cuckoo Hashing, one key question is as to when is $\\mu(G)=n$ whp? We answer this question exactly when $d$ is at least four. We also establish a precise threshold for when Phase 1 of the Karp-Sipser Greedy matching algorithm suffices to compute a maximum matching whp.
Khristoforov, Mikhail; Kleptsyn, Victor; Triestino, Michele
2016-07-01
This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite graphs: these are the so-called hierarchical graphs. They possess a well-defined level structure and any level is built using a simple recursion. Stopping the construction at any finite level, we have a discrete random metric space when we set the edges to have random length (using a multiplicative cascade with fixed law {m}). We introduce a tool, the cut-off process, by means of which one finds that renormalizing the sequence of metrics by an exponential factor, they converge in law to a non-trivial metric on the limit space. Such limit law is stationary, in the sense that glueing together a certain number of copies of the random limit space, according to the combinatorics of the brick graph, the obtained random metric has the same law when rescaled by a random factor of law {m} . In other words, the stationary random metric is the solution of a distributional equation. When the measure m has continuous positive density on {mathbf{R}+}, the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation. We also investigate topological and geometric properties of the random space when m is log-normal, detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade.
Image denoising based on adaptive graph regularization%图上自适应正则化的图像去噪
刘国金; 曾孝平; 刘刈
2012-01-01
Adaptive regularization can select different parameters based on the features of local areas in an image, which can differentiate the edges and noise in an image flexibly. An adaptive graph regularization is proposed based on graph spectral theory and adaptive regularization, which uses the Non local means to generate the weighting function of graph. The adaptive graph regularization equation is used to filter the noisy image. Simulation results show that the proposed method can effectively remove the noise and is superior to other graph theory based partial differential equation methods.%自适应正则化方法在不同的局部区域能够选取不同的正则化参数和正则化约束,因而能够灵活地对边缘和噪声进行区别处理。将自适应正则化建立在图上,提出了一种定义在加权图上的,具有自适应参数的正则化模型。用nonlocal means算法构造图的权重函数,用建立在图上的自适应正则化方程实现图像的去噪处理,仿真实验结果表明：该方法能有效地去除图像中的噪声,在去噪性能上优于部分基于图论的偏微分方程方法。
Jasmine Norman
2011-10-01
Full Text Available Random Geometric Graphs have been a very influential and well-studied model of large networks, such assensor networks, where the network nodes are represented by the vertices of the RGG, and the direct connectivity between nodes is represented by the edges. This assumes homogeneous wireless nodes with uniform transmission ranges. In real life, there exist heterogeneous wireless networks in which devices have dramatically different capabilities. The connectivity of a WSN is related to the positions of nodes, and those positions are heavily affected by the method of sensor deployment. As sensors may be spread in an arbitrary manner, one of the fundamental issues in a wireless sensor network is the coverage problem. In this paper, I study connectivity and coverage in hybrid WSN based on dynamic random geometric graph.
Effect of disorder on condensation in the lattice gas model on a random graph
Handford, Thomas P.; Dear, Alexander; Pérez-Reche, Francisco J.; Taraskin, Sergei N.
2014-07-01
The lattice gas model of condensation in a heterogeneous pore system, represented by a random graph of cells, is studied using an exact analytical solution. A binary mixture of pore cells with different coordination numbers is shown to exhibit two phase transitions as a function of chemical potential in a certain temperature range. Heterogeneity in interaction strengths is demonstrated to reduce the critical temperature and, for large-enough degreeS of disorder, divides the cells into ones which are either on average occupied or unoccupied. Despite treating the pore space loops in a simplified manner, the random-graph model provides a good description of condensation in porous structures containing loops. This is illustrated by considering capillary condensation in a structural model of mesoporous silica SBA-15.
Bayesian analysis for exponential random graph models using the adaptive exchange sampler
Jin, Ick Hoon
2013-01-01
Exponential random graph models have been widely used in social network analysis. However, these models are extremely difficult to handle from a statistical viewpoint, because of the existence of intractable normalizing constants. In this paper, we consider a fully Bayesian analysis for exponential random graph models using the adaptive exchange sampler, which solves the issue of intractable normalizing constants encountered in Markov chain Monte Carlo (MCMC) simulations. The adaptive exchange sampler can be viewed as a MCMC extension of the exchange algorithm, and it generates auxiliary networks via an importance sampling procedure from an auxiliary Markov chain running in parallel. The convergence of this algorithm is established under mild conditions. The adaptive exchange sampler is illustrated using a few social networks, including the Florentine business network, molecule synthetic network, and dolphins network. The results indicate that the adaptive exchange algorithm can produce more accurate estimates than approximate exchange algorithms, while maintaining the same computational efficiency.
H 2-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise
Chun-xiao GUO; Bo-ling GUO; Hui YANG
2014-01-01
In this paper, the authors study the long time behavior of solutions to stochastic non-Newtonian fluids in a two-dimensional bounded domain, and prove the existence of H 2-regularity random attractor.
Constrained Perturbation Regularization Approach for Signal Estimation Using Random Matrix Theory
Suliman, Mohamed; Ballal, Tarig; Kammoun, Abla; Al-Naffouri, Tareq Y.
2016-12-01
In this supplementary appendix we provide proofs and additional extensive simulations that complement the analysis of the main paper (constrained perturbation regularization approach for signal estimation using random matrix theory).
Integral trees and integral graphs
Wang, Ligong
2005-01-01
This monograph deals with integral graphs, Laplacian integral regular graphs, cospectral graphs and cospectral integral graphs. The organization of this work, which consists of eight chapters, is as follows.
Influence of Inhomogeneity on Critical Behavior of Earthquake Model on Random Graph
ZHANG Duan-Ming; SUN Fan; YU Bo-Ming; PAN Gui-Jun; YIN Yan-Ping; LI Rui; SU Xiang-Ying
2006-01-01
We consider the earthquake model on a random graph. A detailed analysis of the probability distribution of the size of the avalanches will be given. The model with different inhomogeneities is studied in order to compare the critical behavior of different systems. The results indicate that with the increase of the inhomogeneities, the avalanche exponents reduce, i.e., the different numbers of defects cause different critical behaviors of the system. This is virtually ascribed to the dynamical perturbation.
Statistical mechanics on isoradial graphs
Boutillier, Cédric
2010-01-01
Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then, we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.
A graph theory practice on transformed image: a random image steganography.
Thanikaiselvan, V; Arulmozhivarman, P; Subashanthini, S; Amirtharajan, Rengarajan
2013-01-01
Modern day information age is enriched with the advanced network communication expertise but unfortunately at the same time encounters infinite security issues when dealing with secret and/or private information. The storage and transmission of the secret information become highly essential and have led to a deluge of research in this field. In this paper, an optimistic effort has been taken to combine graceful graph along with integer wavelet transform (IWT) to implement random image steganography for secure communication. The implementation part begins with the conversion of cover image into wavelet coefficients through IWT and is followed by embedding secret image in the randomly selected coefficients through graph theory. Finally stegoimage is obtained by applying inverse IWT. This method provides a maximum of 44 dB peak signal to noise ratio (PSNR) for 266646 bits. Thus, the proposed method gives high imperceptibility through high PSNR value and high embedding capacity in the cover image due to adaptive embedding scheme and high robustness against blind attack through graph theoretic random selection of coefficients.
On the evolution of random graphs on spaces of negative curvature
Fountoulakis, Nikolaos
2012-01-01
In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points, with the weight being proportional to a real parameter \\beta>0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. This class of random graphs was introduced by Krioukov et al. We provide a rigorous analysis of aspects of this model and its dependence on the parameter \\beta, verifying some of their observations. We show that a phase transition occurs around \\beta =1. More specifically, we show that when \\beta > 1 the degre...
Monotone Increasing Properties and Their Phase Transitions in Uniform Random Intersection Graphs
Zhao, Jun; Gligor, Virgil
2015-01-01
Uniform random intersection graphs have received much interest and been used in diverse applications. A uniform random intersection graph with $n$ nodes is constructed as follows: each node selects a set of $K_n$ different items uniformly at random from the same pool of $P_n$ distinct items, and two nodes establish an undirected edge in between if and only if they share at least one item. For such graph denoted by $G(n, K_n, P_n)$, we present the following results in this paper. First, we provide an exact analysis on the probabilities of $G(n, K_n, P_n)$ having a perfect matching and having a Hamilton cycle respectively, under $P_n = \\omega\\big(n (\\ln n)^5\\big)$ (all asymptotic notation are understood with $n \\to \\infty$). The analysis reveals that just like ($k$-)connectivity shown in prior work, for both properties of perfect matching containment and Hamilton cycle containment, $G(n, K_n, P_n)$ also exhibits phase transitions: for each property above, as $K_n$ increases, the limit of the probability that $G...
Betweenness Centrality in Graphs
2014-01-01
The first book devoted exclusively to quantitative graph theory, Quantitative Graph Theory: Mathematical Foundations and Applications presents and demonstrates existing and novel methods for analyzing graphs quantitatively. Incorporating interdisciplinary knowledge from graph theory, information theory, measurement theory, and statistical techniques, this book covers a wide range of quantitative-graph theoretical concepts and methods, including those pertaining to real and random graphs such ...
Estrada, Ernesto
2016-01-01
We propose a new model to account for the main structural characteristics of rock fracture networks (RFNs). The model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces. We study a series of 29 real-world RFNs and find the best fit with the random rectangular neighborhood graphs (RRNGs) proposed here. We show that this model captures most of the structural characteristics of the RFNs and allows a distinction between small and more spherical rocks and large and more elongated ones. We use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the RFNs. We find a small set of structural parameters that highly correlates with the average diffusion time in the RFNs. In particular, the second smallest eigenvalue of the Laplacian matrix is a good predictor of the average diffusion time on RFNs, showing a Pearson correlation coefficient larger than $0.99$ with the average diffusion time on RFNs. ...
Criticality in Two-Variable Earthquake Model on a Random Graph
SUN Fan; ZHANG Duan-Ming
2008-01-01
A two-variable earthquake model on a quenched random graph is established here. It can be seen as a generalization of the OFC models. We numerically study the critical behavior of the model when the system is nonconservative: the result indicates that the model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling. We compare our model with the model introduced by Stefano Lise and Maya Paczuski [Phys. Rev. Lett. 88 (2002) 228301], it is proved that they are not in the same universality class.
非本原强正则图的充分条件%The Sufficient Conditions of Imprimitive Strongly Regular Graph
扈生彪
2002-01-01
Let G be a ( n ,k ,a ,c )-strongly regular graph, (n,-k,-a,-c)be its complement graph. If its parameters satisfies one of the following condition: 1) k and n -1 coprime; 2) k and k coprime; 3) a = k - 1 ,then G is imprimitive. G is imprimitive of and only if-c = -k or-c = 0.%设G是一个(n,k,a,c)-强正则图,(n,-k,-a,-c)是它的补图.若它们的参数满足下列条件之一:1)k,n-1互素;2)k,-k互素;3)a=k-1,那么G是非本原的.G是非本原的当且仅当-c=-k或-c=0.
Contact processes on random graphs with power law degree distributions have critical value 0
Chatterjee, Shirshendu; 10.1214/09-AOP471
2009-01-01
If we consider the contact process with infection rate $\\lambda$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $\\lambda_c$ of the infection rate is positive if the power $\\alpha>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by G\\'{o}mez-Garde\\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $\\lambda_c$ is zero for any value of $\\alpha>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\\to\\infty$, maintains a positive density of infected sites for time at least $\\exp(n^{1-\\delta})$ for any $\\delta>0$. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability $\\rho(\\lambda)$. It is expected that $\\rho(\\lambda)\\sim ...
Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees
Preciado, Victor M
2015-01-01
In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\\bar{w}_n^{^{T}} = (w^{(n)}_1,\\ldots,w^{(n)}_n)$ is prescribed on the ensemble. Let $\\mathbf{a}_{i,j} =1$ if there is an edge between the nodes $\\{i,j\\}$ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: $\\mathbf{A}_n$ $=$ $ [\\mathbf{a}_{i,j}/\\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution of $\\mathbf{A}_n$ denoted by $\\mathbf{F}_n(\\mathord{\\cdot})$ is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\\mathbf{A}_n$. Under some technical conditions on the expected degrees sequence, we show that with probability one, $\\mathbf{F}_n(\\mathord{\\cdot})$ converges weakly to a deterministic distribution $F(\\mathord{\\cdot})$. Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of $...
Explosive Percolation in Erd\\"os-R\\'enyi-Like Random Graph Processes
Panagiotou, Konstantinos; Steger, Angelika; Thomas, Henning
2011-01-01
The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erd\\"os-R\\'enyi process. It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of the Erd\\"os-R\\'enyi process are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer (2009) gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke (2011) very recently showed that all Achlioptas processes have a continuous phase tran...
Weger, R. C.; Lee, J.; Zhu, Tianri; Welch, R. M.
1992-01-01
The current controversy existing in reference to the regularity vs. clustering in cloud fields is examined by means of analysis and simulation studies based upon nearest-neighbor cumulative distribution statistics. It is shown that the Poisson representation of random point processes is superior to pseudorandom-number-generated models and that pseudorandom-number-generated models bias the observed nearest-neighbor statistics towards regularity. Interpretation of this nearest-neighbor statistics is discussed for many cases of superpositions of clustering, randomness, and regularity. A detailed analysis is carried out of cumulus cloud field spatial distributions based upon Landsat, AVHRR, and Skylab data, showing that, when both large and small clouds are included in the cloud field distributions, the cloud field always has a strong clustering signal.
Zhao, Jun; Gligor, Virgil
2015-01-01
One-dimensional geometric random graphs are constructed by distributing $n$ nodes uniformly and independently on a unit interval and then assigning an undirected edge between any two nodes that have a distance at most $r_n$. These graphs have received much interest and been used in various applications including wireless networks. A threshold of $r_n$ for connectivity is known as $r_n^{*} = \\frac{\\ln n}{n}$ in the literature. In this paper, we prove that a threshold of $r_n$ for the absence of isolated node is $\\frac{\\ln n}{2 n}$ (i.e., a half of the threshold $r_n^{*}$). Our result shows there is a curious gap between thresholds of connectivity and the absence of isolated node in one-dimensional geometric random graphs; in particular, when $r_n$ equals $\\frac{c\\ln n}{ n}$ for a constant $c \\in( \\frac{1}{2}, 1)$, a one-dimensional geometric random graph has no isolated node but is not connected. This curious gap in one-dimensional geometric random graphs is in sharp contrast to the prevalent phenomenon in man...
Random packing of regular polygons and star polygons on a flat two-dimensional surface.
Cieśla, Michał; Barbasz, Jakub
2014-08-01
Random packing of unoriented regular polygons and star polygons on a two-dimensional flat continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine the saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.
Random matrix theory for mixed regular-chaotic dynamics in the super-extensive regime
El-Hady, A Abd
2011-01-01
We apply Tsallis's q-indexed nonextensive entropy to formulate a random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. We consider the super-extensive regime of q < 1. We obtain analytical expressions for the level-spacing distributions, which are strictly valid for 2 \\times 2 random-matrix ensembles, as usually done in the standard RMT. We compare the results with spacing distributions, numerically calculated for random matrix ensembles describing a harmonic oscillator perturbed by Gaussian orthogonal and unitary ensembles.
Bernoulli-based random undersampling schemes for 2D seismic data regularization
Cai Rui; Zhao Qun; She De-Ping; Yang Li; Cao Hui; Yang Qin-Yong
2014-01-01
Seismic data regularization is an important preprocessing step in seismic signal processing. Traditional seismic acquisition methods follow the Shannon-Nyquist sampling theorem, whereas compressive sensing (CS) provides a fundamentally new paradigm to overcome limitations in data acquisition. Besides the sparse representation of seismic signal in some transform domain and the 1-norm reconstruction algorithm, the seismic data regularization quality of CS-based techniques strongly depends on random undersampling schemes. For 2D seismic data, discrete uniform-based methods have been investigated, where some seismic traces are randomly sampled with an equal probability. However, in theory and practice, some seismic traces with different probability are required to be sampled for satisfying the assumptions in CS. Therefore, designing new undersampling schemes is imperative. We propose a Bernoulli-based random undersampling scheme and its jittered version to determine the regular traces that are randomly sampled with different probability, while both schemes comply with the Bernoulli process distribution. We performed experiments using the Fourier and curvelet transforms and the spectral projected gradient reconstruction algorithm for 1-norm (SPGL1), and ten different random seeds. According to the signal-to-noise ratio (SNR) between the original and reconstructed seismic data, the detailed experimental results from 2D numerical and physical simulation data show that the proposed novel schemes perform overall better than the discrete uniform schemes.
Markerless human motion capture by Markov random field and dynamic graph cuts with color constraints
LI Jia; WAN ChengKai; ZHANG DianYong; MIAO ZhenJiang; YUAN BaoZong
2009-01-01
Currently, many vision-based motion capture systems require passive markers attached to key lca-tions on the human body. However, such systems are intrusive with limited application. The algorithm that we use for human motion capture in this paper is based on Markov random field (MRF) and dynamic graph cuts. It takes full account of the impact of 3D reconstruction error and integrates human motion capture and 3D reconstruction into MRF-MAP framework. For more accurate and robust performance, we extend our algorithm by incorporating color constraints Into the pose estimation process. The ad-vantages of incorporating color constraints are demonstrated by experimental results on several video sequences.
Li, Xueliang; Gutman, Ivan
2012-01-01
This book is about graph energy. The authors have included many of the important results on graph energy, such as the complete solution to the conjecture on maximal energy of unicyclic graphs, the Wagner-Heuberger's result on the energy of trees, the energy of random graphs or the approach to energy using singular values. It contains an extensive coverage of recent results and a gradual development of topics and the inclusion of complete proofs from most of the important recent results in the area. The latter fact makes it a valuable reference for researchers looking to get into the field of g
Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models
Pachon, Angelica; Polito, Federico; Sacerdote, Laura
2016-03-01
We give a common description of Simon, Barabási-Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási-Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter α ) goes to infinity, a portion of them behave as a Yule model with parameters (λ ,β ) = (1-α ,1), and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in Newman (Contemp Phys 46:323-351, 2005). References to traditional and recent applications of the these models are also discussed.
Regularities with random interactions in energy centroids defined by group symmetries
Kota, V K B
2005-01-01
Regular structures generated by random interactions in energy centroids defined over irreducible representations (irreps) of some of the group symmetries of the interacting boson models $sd$IBM, $sdg$IBM, $sd$IBM-$T$ and $sd$IBM-$ST$ are studied by deriving trace propagations equations for the centroids. It is found that, with random interactions, the lowest and highest group irreps in general carry most of the probability for the corresponding centroids to be lowest in energy. This generalizes the result known earlier, via numerical diagonalization, for the more complicated fixed spin ($J$) centroids where simple trace propagation is not possible.
Probability of graphs with large spectral gap by multicanonical Monte Carlo
Saito, Nen; Iba, Yukito
2010-01-01
Graphs with large spectral gap are important in various fields such as biology, sociology and computer science. In designing such graphs, an important question is how the probability of graphs with large spectral gap behaves. A method based on multicanonical Monte Carlo is introduced to quantify the behavior of this probability, which enables us to calculate extreme tails of the distribution. The proposed method is successfully applied to random 3-regular graphs and large deviation probabilit...
Probability of graphs with large spectral gap by multicanonical Monte Carlo
Saito, Nen; Iba, Yukito
2011-01-01
Graphs with large spectral gap are important in various fields such as biology, sociology and computer science. In designing such graphs, an important question is how the probability of graphs with large spectral gap behaves. A method based on multicanonical Monte Carlo is introduced to quantify the behavior of this probability, which enables us to calculate extreme tails of the distribution. The proposed method is successfully applied to random 3-regular graphs and large deviation probability is estimated.
Statistical mechanics of random geometric graphs: Geometry-induced first-order phase transition.
Ostilli, Massimo; Bianconi, Ginestra
2015-04-01
Random geometric graphs (RGGs) can be formalized as hidden-variables models where the hidden variables are the coordinates of the nodes. Here we develop a general approach to extract the typical configurations of a generic hidden-variables model and apply the resulting equations to RGGs. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a nontrivial satisfaction problem: Given N nodes, a domain D, and a desired average connectivity 〈k〉, find, if any, the distribution of nodes having support in D and average connectivity 〈k〉. We find out that, in the thermodynamic limit, nodes are either uniformly distributed or highly condensed in a small region, the two regimes being separated by a first-order phase transition characterized by a O(N) jump of 〈k〉. Other intermediate values of 〈k〉 correspond to very rare graph realizations. The phase transition is observed as a function of a parameter a∈[0,1] that tunes the underlying geometry. In particular, a=1 indicates a rigid geometry where only close nodes are connected, while a=0 indicates a rigid antigeometry where only distant nodes are connected. Consistently, when a=1/2 there is no geometry and no phase transition. After discussing the numerical analysis, we provide a combinatorial argument to fully explain the mechanism inducing this phase transition and recognize it as an easy-hard-easy transition. Our result shows that, in general, ad hoc optimized networks can hardly be designed, unless to rely to specific heterogeneous constructions, not necessarily scale free.
A multi-directional rapidly exploring random graph (mRRG) for protein folding
Nath, Shuvra Kanti
2012-01-01
Modeling large-scale protein motions, such as those involved in folding and binding interactions, is crucial to better understanding not only how proteins move and interact with other molecules but also how proteins misfold, thus causing many devastating diseases. Robotic motion planning algorithms, such as Rapidly Exploring Random Trees (RRTs), have been successful in simulating protein folding pathways. Here, we propose a new multi-directional Rapidly Exploring Random Graph (mRRG) specifically tailored for proteins. Unlike traditional RRGs which only expand a parent conformation in a single direction, our strategy expands the parent conformation in multiple directions to generate new samples. Resulting samples are connected to the parent conformation and its nearest neighbors. By leveraging multiple directions, mRRG can model the protein motion landscape with reduced computational time compared to several other robotics-based methods for small to moderate-sized proteins. Our results on several proteins agree with experimental hydrogen out-exchange, pulse-labeling, and F-value analysis. We also show that mRRG covers the conformation space better as compared to the other computation methods. Copyright © 2012 ACM.
On the random access performance of Cell Broadband Engine with graph analysis application
Chen, Mingyu; Kang, Seunghua
2011-01-01
The Cell Broad Engine (BE) Processor has unique memory access architecture besides its powerful computing engines. Many computing-intensive applications have been ported to Cell/BE successfully. But memory-intensive applications are rarely investigated except for several micro benchmarks. Since Cell/BE has powerful software visible DMA engine, this paper studies on whether Cell/BE is suit for applica- tions with large amount of random memory accesses. Two benchmarks, GUPS and SSCA#2, are used. The latter is a rather complex one that in representative of real world graph analysis applications. We find both benchmarks have good performance on Cell/BE based IBM QS20/22. Com- pared with 2 conventional multi-processor systems with the same core/thread number, GUPS is about 40-80% fast and SSCA#2 about 17-30% fast. The dynamic load balanc- ing and software pipeline for optimizing SSCA#2 are intro- duced. Based on the experiment, the potential of Cell/BE for random access is analyzed in detail as well as its limita-...
Hul, Oleh; Seba, Petr; Sirko, Leszek
2009-06-01
Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c[over ](omega,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the inverse participation ratio and the amplitude distribution P(Psi(x)) .
The spectrum of the random environment and localization of noise
Cheliotis, D Dimitris; Virág, B
2008-01-01
We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.
Network motif identification and structure detection with exponential random graph models
Munni Begum
2014-12-01
Full Text Available Local regulatory motifs are identified in the transcription regulatory network of the most studied model organism Escherichia coli (E. coli through graphical models. Network motifs are small structures in a network that appear more frequently than expected by chance alone. We apply social network methodologies such as p* models, also known as Exponential Random Graph Models (ERGMs, to identify statistically significant network motifs. In particular, we generate directed graphical models that can be applied to study interaction networks in a broad range of databases. The Markov Chain Monte Carlo (MCMC computational algorithms are implemented to obtain the estimates of model parameters to the corresponding network statistics. A variety of ERGMs are fitted to identify statistically significant network motifs in transcription regulatory networks of E. coli. A total of nine ERGMs are fitted to study the transcription factor - transcription factor interactions and eleven ERGMs are fitted for the transcription factor-operon interactions. For both of these interaction networks, arc (a directed edge in a directed network and k-istar (or incoming star structures, for values of k between 2 and 10, are found to be statistically significant local structures or network motifs. The goodness of fit statistics are provided to determine the quality of these models.
Approximating the XY model on a random graph with a q -state clock model
Lupo, Cosimo; Ricci-Tersenghi, Federico
2017-02-01
Numerical simulations of spin glass models with continuous variables set the problem of a reliable but efficient discretization of such variables. In particular, the main question is how fast physical observables computed in the discretized model converge toward the ones of the continuous model when the number of states of the discretized model increases. We answer this question for the XY model and its discretization, the q -state clock model, in the mean-field setting provided by random graphs. It is found that the convergence of physical observables is exponentially fast in the number q of states of the clock model, so allowing a very reliable approximation of the XY model by using a rather small number of states. Furthermore, such an exponential convergence is found to be independent from the disorder distribution used. Only at T =0 , the convergence is slightly slower (stretched exponential). Thanks to the analytical solution to the q -state clock model, we compute accurate phase diagrams in the temperature versus disorder strength plane. We find that, at zero temperature, spontaneous replica symmetry breaking takes place for any amount of disorder, even an infinitesimal one. We also study the one step of replica symmetry breaking (1RSB) solution in the low-temperature spin glass phase.
Sharpness in the k-nearest neighbours random geometric graph model
Falgas-Ravry, Victor
2011-01-01
Let $S_{n,k}$ denote the random geometric graph obtained by placing points in a square box of area $n$ according to a Poisson process of intensity $1$ and joining each point to its $k$ nearest neighbours. Balister, Bollob\\'as, Sarkar and Walters conjectured that for every $0< \\epsilon <1$ and all $n$ sufficiently large there exists $C=C(\\epsilon)$ such that whenever the probability $S_{n,k}$ is connected is at least $\\epsilon $ then the probability $S_{n,k+C}$ is connected is at least $1-\\epsilon $. In this paper we prove this conjecture. As a corollary we prove that there is a constant $C'$ such that whenever $k=k(n)$ is a sequence of integers such that the probability $S_{n,k(n)}$ is connected tends to one as $n$ tends to infinity, then for any $s(n)$ with $s(n)=o(\\log n)$, the probability that $S_{n,k(n)+C's\\log \\log n}$ is $s$-connected tends to one This proves another conjecture of Balister, Bollob\\'as, Sarkar and Walters.
Exact two-point resistance, and the simple random walk on the complete graph minus N edges
Chair, Noureddine, E-mail: n.chair@ju.edu.jo
2012-12-15
An analytical approach is developed to obtain the exact expressions for the two-point resistance and the total effective resistance of the complete graph minus N edges of the opposite vertices. These expressions are written in terms of certain numbers that we introduce, which we call the Bejaia and the Pisa numbers; these numbers are the natural generalizations of the bisected Fibonacci and Lucas numbers. The correspondence between random walks and the resistor networks is then used to obtain the exact expressions for the first passage and mean first passage times on this graph. - Highlights: Black-Right-Pointing-Pointer We obtain exact formulas for the two-point resistance of the complete graph minus N edges. Black-Right-Pointing-Pointer We obtain also the total effective resistance of this graph. Black-Right-Pointing-Pointer We modified Schwatt's formula on trigonometrical power sum to suit our computations. Black-Right-Pointing-Pointer We introduced the generalized bisected Fibonacci and Lucas numbers: the Bejaia and the Pisa numbers. Black-Right-Pointing-Pointer The first passage and mean first passage times of the random walks have exact expressions.
Critical value for the contact process with random recovery rates and edge weights on regular tree
Xue, Xiaofeng
2016-11-01
In this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees TN. Let ρ and ξ be two nonnegative random variables such that P(ɛ ≤ ξ 0. For each vertex x on TN, ξ(x) is an independent copy of ξ while for each edge e on TN, ρ(e) is an independent copy of ρ. An infected vertex x becomes healthy at rate ξ(x) while an infected vertex y infects an healthy neighbor z at rate proportional to ρ(y , z) . For this model, we prove that the critical value under the annealed measure approximately equals (N E ρ E 1/ξ )-1 as N grows to infinity. Furthermore, we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite.
Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity
Haji Ali, Abdul Lateef
2016-08-26
We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis. © 2016 SFoCM
Greenland, Sander; Mansournia, Mohammad Ali
2015-10-01
We describe how ordinary interpretations of causal models and causal graphs fail to capture important distinctions among ignorable allocation mechanisms for subject selection or allocation. We illustrate these limitations in the case of random confounding and designs that prevent such confounding. In many experimental designs individual treatment allocations are dependent, and explicit population models are needed to show this dependency. In particular, certain designs impose unfaithful covariate-treatment distributions to prevent random confounding, yet ordinary causal graphs cannot discriminate between these unconfounded designs and confounded studies. Causal models for populations are better suited for displaying these phenomena than are individual-level models, because they allow representation of allocation dependencies as well as outcome dependencies across individuals. Nonetheless, even with this extension, ordinary graphical models still fail to capture distinctions between hypothetical superpopulations (sampling distributions) and observed populations (actual distributions), although potential-outcome models can be adapted to show these distinctions and their consequences.
Ping WANG; Jiong Sheng LI
2005-01-01
Let G be a finite simple graph with adjacency matrix A, and let P(A) be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P(A). In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.
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,...
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.
C. Dalfo
2015-10-01
Full Text Available We study a family of graphs related to the $n$-cube. The middle cube graph of parameter k is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors.
On Characterizing the Local Pooling Factor of Greedy Maximal Scheduling in Random Graphs
Wildman, Jeffrey; Weber, Steven
2014-01-01
The study of the optimality of low-complexity greedy scheduling techniques in wireless communications networks is a very complex problem. The Local Pooling (LoP) factor provides a single-parameter means of expressing the achievable capacity region (and optimality) of one such scheme, greedy maximal scheduling (GMS). The exact LoP factor for an arbitrary network graph is generally difficult to obtain, but may be evaluated or bounded based on the network graph's particular structure. In this pa...
Ahmed, Faraz; Jin, Rong; Liu, Alex X.
2013-01-01
Online social networks are being increasingly used for analyzing various societal phenomena such as epidemiology, information dissemination, marketing and sentiment flow. Popular analysis techniques such as clustering and influential node analysis, require the computation of eigenvectors of the real graph's adjacency matrix. Recent de-anonymization attacks on Netflix and AOL datasets show that an open access to such graphs pose privacy threats. Among the various privacy preserving models, Dif...
Regularities of many-body systems interacting by a two-body random ensemble
Zhao, Y.M. [Department of Physics, Shanghai Jiao-Tong University, Shanghai 200030 (China) and Cyclotron Center, Institute of Physical and Chemical Research - RIKEN, Hirosawa 2-1, Wako-shi, Saitama 351-0198 (Japan) and Department of Physics, Southeast University, Nanjing 210018 (China)]. E-mail: ymzhao@riken.jp; Arima, A. [Science Museum, Japan Science Foundation, 2-1 Kitanomaru-Koen, Chiyodaku, Tokyo 102-0091 (Japan); Yoshinaga, N. [Department of Physics, Saitama University, Saitama 338-0625 (Japan)
2004-10-01
The ground states of all even-even nuclei have angular momentum, I, equal to zero, I=0, and positive parity, {pi}=+. This feature was believed to be a consequence of the attractive short-range interaction between nucleons. However, in the presence of two-body random interactions, the predominance of I{pi}=0+ ground states (0 g.s.) was found to be robust both for bosons and for an even number of fermions. For simple systems, such as d bosons, sp bosons, sd bosons, and a few fermions in single-j shells for small j, there are a few approaches to predict and/or explain spin I ground state (I g.s.) probabilities. An empirical approach to predict I g.s. probabilities is available for general cases, such as fermions in a single-j (j>72) or many-j shells and various boson systems, but a more fundamental understanding of the robustness of 0 g.s. dominance is still out of reach. Further interesting results are also reviewed concerning other robust phenomena of many-body systems in the presence of random two-body interactions, such as the odd-even staggering of binding energies, generic collectivity, the behavior of average energies, correlations, and regularities of many-body systems interacting by a displaced two-body random ensemble.
Bapat, Ravindra B
2014-01-01
This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reo...
Särkämö, Teppo; Tervaniemi, Mari; Laitinen, Sari; Numminen, Ava; Kurki, Merja; Johnson, Julene K; Rantanen, Pekka
2014-08-01
During aging, musical activities can help maintain physical and mental health and cognitive abilities, but their rehabilitative use has not been systematically explored in persons with dementia (PWDs). Our aim was to determine the efficacy of a novel music intervention based on coaching the caregivers of PWDs to use either singing or music listening regularly as a part of everyday care. Eighty-nine PWD-caregiver dyads were randomized to a 10-week singing coaching group (n = 30), a 10-week music listening coaching group (n = 29), or a usual care control group (n = 30). The coaching sessions consisted primarily of singing/listening familiar songs coupled occasionally with vocal exercises and rhythmic movements (singing group) and reminiscence and discussions (music listening group). In addition, the intervention included regular musical exercises at home. All PWDs underwent an extensive neuropsychological assessment, which included cognitive tests, as well as mood and quality of life (QOL) scales, before and after the intervention period and 6 months later. In addition, the psychological well-being of family members was repeatedly assessed with questionnaires. Compared with usual care, both singing and music listening improved mood, orientation, and remote episodic memory and to a lesser extent, also attention and executive function and general cognition. Singing also enhanced short-term and working memory and caregiver well-being, whereas music listening had a positive effect on QOL. Regular musical leisure activities can have long-term cognitive, emotional, and social benefits in mild/moderate dementia and could therefore be utilized in dementia care and rehabilitation. © The Author 2013. Published by Oxford University Press on behalf of The Gerontological Society of America. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
Thimble regularization at work besides toy models: from Random Matrix Theory to Gauge Theories
Eruzzi, G
2015-01-01
Thimble regularization as a solution to the sign problem has been successfully put at work for a few toy models. Given the non trivial nature of the method (also from the algorithmic point of view) it is compelling to provide evidence that it works for realistic models. A Chiral Random Matrix theory has been studied in detail. The known analytical solution shows that the model is non-trivial as for the sign problem (in particular, phase quenched results can be very far away from the exact solution). This study gave us the chance to address a couple of key issues: how many thimbles contribute to the solution of a realistic problem? Can one devise algorithms which are robust as for staying on the correct manifold? The obvious step forward consists of applications to gauge theories.
Quantitative graph theory mathematical foundations and applications
Dehmer, Matthias
2014-01-01
The first book devoted exclusively to quantitative graph theory, Quantitative Graph Theory: Mathematical Foundations and Applications presents and demonstrates existing and novel methods for analyzing graphs quantitatively. Incorporating interdisciplinary knowledge from graph theory, information theory, measurement theory, and statistical techniques, this book covers a wide range of quantitative-graph theoretical concepts and methods, including those pertaining to real and random graphs such as:Comparative approaches (graph similarity or distance)Graph measures to characterize graphs quantitat
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....
Relun, Anne; Grosbois, Vladimir; Alexandrov, Tsviatko; Sánchez-Vizcaíno, Jose M; Waret-Szkuta, Agnes; Molia, Sophie; Etter, Eric Marcel Charles; Martínez-López, Beatriz
2017-01-01
In most European countries, data regarding movements of live animals are routinely collected and can greatly aid predictive epidemic modeling. However, the use of complete movements' dataset to conduct policy-relevant predictions has been so far limited by the massive amount of data that have to be processed (e.g., in intensive commercial systems) or the restricted availability of timely and updated records on animal movements (e.g., in areas where small-scale or extensive production is predominant). The aim of this study was to use exponential random graph models (ERGMs) to reproduce, understand, and predict pig trade networks in different European production systems. Three trade networks were built by aggregating movements of pig batches among premises (farms and trade operators) over 2011 in Bulgaria, Extremadura (Spain), and Côtes-d'Armor (France), where small-scale, extensive, and intensive pig production are predominant, respectively. Three ERGMs were fitted to each network with various demographic and geographic attributes of the nodes as well as six internal network configurations. Several statistical and graphical diagnostic methods were applied to assess the goodness of fit of the models. For all systems, both exogenous (attribute-based) and endogenous (network-based) processes appeared to govern the structure of pig trade network, and neither alone were capable of capturing all aspects of the network structure. Geographic mixing patterns strongly structured pig trade organization in the small-scale production system, whereas belonging to the same company or keeping pigs in the same housing system appeared to be key drivers of pig trade, in intensive and extensive production systems, respectively. Heterogeneous mixing between types of production also explained a part of network structure, whichever production system considered. Limited information is thus needed to capture most of the global structure of pig trade networks. Such findings will be useful
Nonextensive random matrix theory approach to mixed regular-chaotic dynamics.
Abul-Magd, A Y
2005-06-01
We apply Tsallis' q -indexed entropy to formulate a nonextensive random matrix theory, which may be suitable for systems with mixed regular-chaotic dynamics. The joint distribution of the matrix elements is given by folding the corresponding quantity in the conventional random matrix theory by a distribution of the inverse matrix-element variance. It keeps the basis invariance of the standard theory but violates the independence of the matrix elements. We consider the subextensive regime of q more than unity in which the transition from the Wigner to the Poisson statistics is expected to start. We calculate the level density for different values of the entropic index. Our results are consistent with an analogous calculation by Tsallis and collaborators. We calculate the spacing distribution for mixed systems with and without time-reversal symmetry. Comparing the result of calculation to a numerical experiment shows that the proposed nonextensive model provides a satisfactory description for the initial stage of the transition from chaos towards the Poisson statistics.
Clustering, randomness, and regularity in cloud fields. 4: Stratocumulus cloud fields
Lee, J.; Chou, J.; Weger, R. C.; Welch, R. M.
1994-01-01
To complete the analysis of the spatial distribution of boundary layer cloudiness, the present study focuses on nine stratocumulus Landsat scenes. The results indicate many similarities between stratocumulus and cumulus spatial distributions. Most notably, at full spatial resolution all scenes exhibit a decidedly clustered distribution. The strength of the clustering signal decreases with increasing cloud size; the clusters themselves consist of a few clouds (less than 10), occupy a small percentage of the cloud field area (less than 5%), contain between 20% and 60% of the cloud field population, and are randomly located within the scene. In contrast, stratocumulus in almost every respect are more strongly clustered than are cumulus cloud fields. For instance, stratocumulus clusters contain more clouds per cluster, occupy a larger percentage of the total area, and have a larger percentage of clouds participating in clusters than the corresponding cumulus examples. To investigate clustering at intermediate spatial scales, the local dimensionality statistic is introduced. Results obtained from this statistic provide the first direct evidence for regularity among large (more than 900 m in diameter) clouds in stratocumulus and cumulus cloud fields, in support of the inhibition hypothesis of Ramirez and Bras (1990). Also, the size compensated point-to-cloud cumulative distribution function statistic is found to be necessary to obtain a consistent description of stratocumulus cloud distributions. A hypothesis regarding the underlying physical mechanisms responsible for cloud clustering is presented. It is suggested that cloud clusters often arise from 4 to 10 triggering events localized within regions less than 2 km in diameter and randomly distributed within the cloud field. As the size of the cloud surpasses the scale of the triggering region, the clustering signal weakens and the larger cloud locations become more random.
REGULAR METHOD FOR SYNTHESIS OF BASIC BENT-SQUARES OF RANDOM ORDER
A. V. Sokolov
2016-01-01
Full Text Available The paper is devoted to the class construction of the most non-linear Boolean bent-functions of any length N = 2k (k = 2, 4, 6…, on the basis of their spectral representation – Agievich bent squares. These perfect algebraic constructions are used as a basis to build many new cryptographic primitives, such as generators of pseudo-random key sequences, crypto graphic S-boxes, etc. Bent-functions also find their application in the construction of C-codes in the systems with code division multiple access (CDMA to provide the lowest possible value of Peak-to-Average Power Ratio (PAPR k = 1, as well as for the construction of error-correcting codes and systems of orthogonal biphasic signals. All the numerous applications of bent-functions relate to the theory of their synthesis. However, regular methods for complete class synthesis of bent-functions of any length N = 2k are currently unknown. The paper proposes a regular synthesis method for the basic Agievich bent squares of any order n, based on a regular operator of dyadic shift. Classification for a complete set of spectral vectors of lengths (l = 8, 16, … based on a criterion of the maximum absolute value and set of absolute values of spectral components has been carried out in the paper. It has been shown that any spectral vector can be a basis for building bent squares. Results of the synthesis for the Agievich bent squares of order n = 8 have been generalized and it has been revealed that there are only 3 basic bent squares for this order, while the other 5 can be obtained with help the operation of step-cyclic shift. All the basic bent squares of order n = 16 have been synthesized that allows to construct the bent-functions of length N = 256. The obtained basic bent squares can be used either for direct synthesis of bent-functions and their practical application or for further research in order to synthesize new structures of bent squares of orders n = 16, 32, 64, …
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.
Measuring extremal dependencies in web graphs
Volkovich, Y.; Litvak, Nelli; Zwart, B.
We analyze dependencies in power law graph data (Web sample, Wikipedia sample and a preferential attachment graph) using statistical inference for multivariate regular variation. The well developed theory of regular variation is widely applied in extreme value theory, telecommunications and
Naparstek, Oshri; Leshem, Amir
2013-01-01
In this paper we analyze the expected time complexity of the auction algorithm for the matching problem on random bipartite graphs. We prove that the expected time complexity of the auction algorithm for bipartite matching is $O\\left(\\frac{N\\log^2(N)}{\\log\\left(Np\\right)}\\right)$ on sequential machines. This is equivalent to other augmenting path algorithms such as the HK algorithm. Furthermore, we show that the algorithm can be implemented on parallel machines with $O(\\log(N))$ processors an...
Sumedha; Jana, Nabin Kumar
2017-01-01
In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, P≤ft({{ Δ }i}\\right)=pδ ≤ft({{ Δ }i}- Δ \\right)+(1-p)δ ≤ft({{ Δ }i}+ Δ \\right) , using large deviation techniques. We find that the first order transition of the pure system is destroyed for 0.046 0.954) even at zero temperature.
ON BIPOLAR SINGLE VALUED NEUTROSOPHIC GRAPHS
Said Broumi; Mohamed Talea; Assia Bakali; Florentin Smarandache
2016-01-01
In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties.
ON BIPOLAR SINGLE VALUED NEUTROSOPHIC GRAPHS
Said Broumi; Mohamed Talea; Assia Bakali; Florentin Smarandache
2016-01-01
In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties.
On Bipolar Single Valued Neutrosophic Graphs
SAID BROUMI; MOHAMED TALEA; ASSIA BAKALI; FLORENTIN SMARANDACHE
2016-01-01
In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties.
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.
Brémaud, Pierre
2017-01-01
The emphasis in this book is placed on general models (Markov chains, random fields, random graphs), universal methods (the probabilistic method, the coupling method, the Stein-Chen method, martingale methods, the method of types) and versatile tools (Chernoff's bound, Hoeffding's inequality, Holley's inequality) whose domain of application extends far beyond the present text. Although the examples treated in the book relate to the possible applications, in the communication and computing sciences, in operations research and in physics, this book is in the first instance concerned with theory. The level of the book is that of a beginning graduate course. It is self-contained, the prerequisites consisting merely of basic calculus (series) and basic linear algebra (matrices). The reader is not assumed to be trained in probability since the first chapters give in considerable detail the background necessary to understand the rest of the book. .
From random sphere packings to regular pillar arrays: analysis of transverse dispersion.
Daneyko, Anton; Hlushkou, Dzmitry; Khirevich, Siarhei; Tallarek, Ulrich
2012-09-28
We study the impact of microscopic order on transverse dispersion in the interstitial void space of bulk (unconfined) chromatographic beds by numerical simulations of incompressible fluid flow and mass transport of a passive tracer. Our study includes polydisperse random sphere packings (computer-generated with particle size distributions of modern core-shell and sub-2 μm particles), the macropore space morphology of a physically reconstructed silica monolith, and computer-generated regular pillar arrays. These bed morphologies are analyzed by their velocity probability density distributions, transient dispersion behavior, and the dependence of asymptotic transverse dispersion coefficients on the mobile phase velocity. In our work, the spherical particles, the monolith skeleton, and the cylindrical pillars are all treated as impermeable solid phase (nonporous) and the tracer is unretained, to focus on the impact of microscopic order on flow and (particularly transverse) hydrodynamic dispersion in the interstitial void space. The microscopic order of the pillar arrays causes their velocity probability density distributions to start and end abruptly, their transient dispersion coefficients to oscillate, and the asymptotic transverse dispersion coefficients to plateau out of initial power law behavior. The microscopically disordered beds, by contrast, follow power law behavior over the whole investigated velocity range, for which we present refined equations (i.e., Eq.(13) and the data in Table 2 for the polydisperse sphere packings; Eq.(17) for the silica monolith). The bulk bed morphologies and their intrinsic differences addressed in this work determine how efficient a bed can relax the transverse concentration gradients caused by wall effects, which exist in all confined separation media used in chromatographic practice. Whereas the effect of diffusion on transverse dispersion decreases and ultimately disappears at increasing velocity with the microscopically
Yoshinaga, Masahiko
2015-01-01
Finite graphs that have a common chromatic polynomial have the same number of regular $n$-colorings. A natural question is whether there exists a natural bijection between regular $n$-colorings. We address this question using a functorial formulation. Let $G$ be a simple graph. Then for each set $X$ we can associate a set of $X$-colorings. This defines a functor, "chromatic functor" from the category of sets with injections to itself. The first main result verifies that two finite graphs dete...
David A Rolls
Full Text Available We compare two broad types of empirically grounded random network models in terms of their abilities to capture both network features and simulated Susceptible-Infected-Recovered (SIR epidemic dynamics. The types of network models are exponential random graph models (ERGMs and extensions of the configuration model. We use three kinds of empirical contact networks, chosen to provide both variety and realistic patterns of human contact: a highly clustered network, a bipartite network and a snowball sampled network of a "hidden population". In the case of the snowball sampled network we present a novel method for fitting an edge-triangle model. In our results, ERGMs consistently capture clustering as well or better than configuration-type models, but the latter models better capture the node degree distribution. Despite the additional computational requirements to fit ERGMs to empirical networks, the use of ERGMs provides only a slight improvement in the ability of the models to recreate epidemic features of the empirical network in simulated SIR epidemics. Generally, SIR epidemic results from using configuration-type models fall between those from a random network model (i.e., an Erdős-Rényi model and an ERGM. The addition of subgraphs of size four to edge-triangle type models does improve agreement with the empirical network for smaller densities in clustered networks. Additional subgraphs do not make a noticeable difference in our example, although we would expect the ability to model cliques to be helpful for contact networks exhibiting household structure.
Spectroscopic studies of regio-regular and regio-random polythiophene films
Valy Vardeny, Z.
2002-03-01
Poly(3 hexyl thiophene) [P3HT] can be synthesized with regio-regular (RR-) order in which the side groups are arranged head to tail, and regio-random (RRa-) order in which the side groups are not arranged in a particular form. It has been recently discovered that films cast from RR-P3HT form two-dimensional (2D) lamellae perpendicular to the substrate, whereas RRa-P3HT forms lamellae to a lesser extend [1,2]. The interchain interplane separation in the lamellae is of order 4 Angstr. causing a strong interchain interaction. This has a profound influence on the charged and neutral photoexcitations in RR-P3HT films compared to those of RRa-P3HT. We have employed a variety of steady state and ps transient spectroscopies to study and compare the photoexcitations in RR- and RRa- P3HT films. In the ps time domain we found [3] in RRa-P3HT films that intrachain excitons with correlated photoinduced absorption (PA) and stimulated emission (SE) bands are the primary excitations; they give rise to a moderately strong photoluminescence (PL) band. In RR-P3HT films, on the contrary the primary excitations are excitons with a much larger interchain component; this results in lack of a SE band, vanishing small intersystem crossing and very weak PL band [3]. We also measured in RR-P3HT films photogenerated polaron pairs with ultrafast dynamics that are precursor to long-lived polaron excitations. In the steady state we measured long-lived polaron excitations in both RR- and RRa- P3HT films, however with different relaxation energies [2]. Whereas the polaron relaxation energy in RRa-P3HT is of the order of 0.5 eV, it is only about 50 meV in RR-P3HT. This shows that the polarons are delocalized in the 2D lamellae, consistent with the carrier relative high mobility [1] and superconductivity [4] found in RR-P3HT films. As a result of the very low relaxation energy in RR-P3HT we found that the polaron optical transition in the mid ir spectral range overlaps with several photoinduced ir
Cycle-maximal triangle-free graphs
Durocher, Stephane; Gunderson, David S.; Li, Pak Ching
2015-01-01
Abstract We conjecture that the balanced complete bipartite graph K ⌊ n / 2 ⌋ , ⌈ n / 2 ⌉ contains more cycles than any other n -vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds...... on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing...
Zhang, Yali; Wang, Jun
2017-09-01
In an attempt to investigate the nonlinear complex evolution of financial dynamics, a new financial price model - the multitype range-intensity contact (MRIC) financial model, is developed based on the multitype range-intensity interacting contact system, in which the interaction and transmission of different types of investment attitudes in a stock market are simulated by viruses spreading. Two new random visibility graph (VG) based analyses and Lempel-Ziv complexity (LZC) are applied to study the complex behaviors of return time series and the corresponding random sorted series. The VG method is the complex network theory, and the LZC is a non-parametric measure of complexity reflecting the rate of new pattern generation of a series. In this work, the real stock market indices are considered to be comparatively studied with the simulation data of the proposed model. Further, the numerical empirical study shows the similar complexity behaviors between the model and the real markets, the research confirms that the financial model is reasonable to some extent.
Laura Hindersin
2015-11-01
Full Text Available We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-death updating, where an individual is selected to reproduce with probability proportional to its fitness and one of its neighbors is replaced by that offspring at random. If we instead focus on death-Birth updating, in which a random individual is removed and its neighbors compete for the empty spot, then the same ensemble of graphs consists of almost only suppressors of selection for which the fixation probability is decreased for advantageous mutants and increased for disadvantageous mutants. Thus, the impact of population structure on evolutionary dynamics is a subtle issue that will depend on seemingly minor details of the underlying evolutionary process.
Towards Optimal Degree-distributions for Left-perfect Matchings in Random Bipartite Graphs
Dietzfelbinger, Martin
2012-01-01
Consider a random bipartite multigraph $G$ with $n$ left nodes and $m \\geq n \\geq 2$ right nodes. Each left node $x$ has $d_x \\geq 1$ random right neighbors. The average left degree $\\Delta$ is fixed, $\\Delta \\geq 2$. We ask whether for the probability that $G$ has a left-perfect matching it is advantageous not to fix $d_x$ for each left node $x$ but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If $\\Delta$ is an integer then it is optimal to use a fixed degree of $\\Delta$ for all left nodes. If $\\Delta$ is non-integral then an optimal degree-distribution has the property that each left node $x$ has two possible degrees, $\\floor{\\Delta}$ and $\\ceil{\\Delta}$, with probability $p_x$ and $1-p_x$, respectively, where $p_x$ is from the closed interval $[0,1]$ and the average over all $p_x$ equals $\\ceil{\\Delta}-\\Delta$. Furthermore, if $n=c\\cdot m$ and $\\Delta>2$ is constant, then each distribution of...
王长群; 熊胜利
2004-01-01
A Cayley graph X=Cay(G,S) of group G is said to be normal if R(G),the group of right multiplications,is normal in Aut(X).An infinite family of normal one-regular Cayley graphs Cay(G,S) of quasi-dihedral groups G=〈x,y|x2m=y2=1,xy=xm+1〉 is obtained,where S={x,x-1,xs+1y,xs-1y},m=2s,and s is an even greater than 4.In addition,the normal and one-regular and 4-valent Cayley graphs of quasi-dihedral groups of order 2r are classified.It is proved that any 4-valent normal and one-regular Cayley graphs of quasi-dihedral ghoups G of order 2r are isomorphic to Cay(G,{x,x-1,xs+1 y,xs-1y}) where s=2r-2,r＞3.%群G的一个Cayley图X=Cay(G,S)称为正规的,如果右乘变换群R(G)在Aut(X)中正规.得到了拟二面体群G＝〈x,y|x2m=y2=1,xy=xm+1〉(其中m=2s,s为大于4的偶数)的一个无限类4度正规1-正则Cayley图 Cay(G,S),其中S={x,x-1,xs+1y,xs-1y},并且对2r阶拟二面体群的正规1-正则4度Cayley图进行了分类,其中r＞3.证明了2r阶拟二面体群的任意4度正规1-正则Cayley图同构于Cay(G,{x,x-1,xs+1y,xs-1y}),其中s=2r-2.
李建湘
2004-01-01
图被称为 K1,n-free图,如果它不含有导出子图K1,n. 设G是一个具有顶点集V(G)的图, 并设g和f是两个定义在V(G)的函数,使得g(x)≤f(x)对所有V(G)中的点x都成立.设a=max{g(x)|x∈V(G)}, b=min{f(x)|x∈V(G)}, 并有b, a≥2, n≥b/(a-1)+1(如果存在点v∈V(G)使得f(v)≡1(mod 2), 假定b≥n-1). 证明了:每个连通的使得∑x∈V(G)f(x)为偶数的K1,n-free图G有(g,f)-因子,如果它的最小度至少是((n-1)(a+1))/(b)+1「(b+a(n-1))/(2(n-1))-(n-1)/(b)「(b+a(n-1))/(2(n-1))2+n-3.这个结果是K.Ota 和T.Tokuda(J. Graph Theory. 1996, 22:59-64.)关于在K1,n-free 图中存在正则因子度条件的推广.%A graph is called K1,n-free if it contains no K1,n as an induced subgraph. Let G be a graph with vertex set V(G), and let g and f be two integer-valued functions defined on V(G) such that g(x)≤f(x) for all x∈V(G). Let a =max {g(x)|x∈V(G)}, b=min {f(x)|x∈V(G)}, and b, a≥2, n≥b/(a-1)+1(if there exists a vertex v∈V(G) such that f(v)≡1 (mod 2), b≥n-1). We prove that every K1,n-free connected graph G with ∑x∈V(G) f(x) even has a (g, f)-factor if its minimum degree is at leastThis result is the generalization for the existence theorem of regular factors in K1,n-free graphs, which is due to K. Ota and T. Tokuda (J. Graph Theory. 1996, 22:59-64).
Dynamics of influence and social balance in spatially-embedded regular and random networks
Singh, P.; Sreenivasan, S.; Szymanski, B.; Korniss, G.
2015-03-01
Structural balance - the tendency of social relationship triads to prefer specific states of polarity - can be a fundamental driver of beliefs, behavior, and attitudes on social networks. Here we study how structural balance affects deradicalization in an otherwise polarized population of leftists and rightists constituting the nodes of a low-dimensional social network. Specifically, assuming an externally moderating influence that converts leftists or rightists to centrists with probability p, we study the critical value p =pc , below which the presence of metastable mixed population states exponentially delay the achievement of centrist consensus. Above the critical value, centrist consensus is the only fixed point. Complementing our previously shown results for complete graphs, we present results for the process on low-dimensional networks, and show that the low-dimensional embedding of the underlying network significantly affects the critical value of probability p. Intriguingly, on low-dimensional networks, the critical value pc can show non-monotonicity as the dimensionality of the network is varied. We conclude by analyzing the scaling behavior of temporal variation of unbalanced triad density in the network for different low-dimensional network topologies. Supported in part by ARL NS-CTA, ONR, and ARO.
Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations
Castrillon, Julio
2016-03-02
In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped onto a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and variance of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.
Non-extensive random matrix theory--a bridge connecting chaotic and regular dynamics
Abul-Magd, A.Y. [Faculty of Science, Zagazig University, Zagazig (Egypt)]. E-mail: a_y_abul_magd@hotmail.com
2004-11-29
We consider a possible generalization of the random matrix theory, which involves the maximization of Tsallis' q-parametrized entropy. We discuss the dependence of the spacing distribution on q using a non-extensive generalization of Wigner's surmises for ensembles belonging to the orthogonal, unitary and symplectic symmetry universal classes.
Growing Random Geometric Graph Models of Super-linear Scaling Law
Zhang, Jiang
2012-01-01
Recent researches on complex systems highlighted the so-called super-linear growth phenomenon. As the system size $P$ measured as population in cities or active users in online communities increases, the total activities $X$ measured as GDP or number of new patents, crimes in cities generated by these people also increases but in a faster rate. This accelerating growth phenomenon can be well described by a super-linear power law $X \\propto P^{\\gamma}$($\\gamma>1$). However, the explanation on this phenomenon is still lack. In this paper, we propose a modeling framework called growing random geometric models to explain the super-linear relationship. A growing network is constructed on an abstract geometric space. The new coming node can only survive if it just locates on an appropriate place in the space where other nodes exist, then new edges are connected with the adjacent nodes whose number is determined by the density of existing nodes. Thus the total number of edges can grow with the number of nodes in a f...
陈海亮; 雷琳; 周石琳
2012-01-01
Ship team is an important group target of ships which cruise and battle in the sea. Focusing on the complex problem of team form and order changing, this paper summarizes the spatial regularity existing in the ship team,and establishes the fuzzy reasoning rules based on it to find the spatial regularity quantitatively, then extracts the ship team which has high spatial regularity using spectral graph partitioning. Emulational and real data show that the algorithm can extract the group target which has a team form in the case of disturbance and the team spatial relationship changes.%舰船编队是海上舰船巡航和作战中的重要群目标.针对编队组成和队形变化的复杂情况,本文总结了编队存在的空问规则性,由此制定模糊推理规则得到定量的空间规则度,最后用谱图划分方法提取具有较高空间规则度的编队目标.仿真和实测的数据表明,该方法可以在有干扰和编队空间关系适当变化的情况下提取出具有编队形式的群目标.
Moore, Jeremy; Martin, Leopoldo L; Maayani, Shai; Kim, Kyu Hyun; Chandrahalim, Hengky; Eichenfield, Matt; Martin, Inocencio R; Carmon, Tal
2016-02-01
We experimentally report on optical binding of many glass particles in air that levitate in a single optical beam. A diversity of particle sizes and shapes interact at long range in a single Gaussian beam. Our system dynamics span from oscillatory to random and dimensionality ranges from 1 to 3D. The low loss for the center of mass motion of the beads could allow this system to serve as a standard many body testbed, similar to what is done today with atoms, but at the mesoscopic scale.
傅育熙
1998-01-01
The paper proposes reaction graphs as graphical representations of computational objects.A reaction graph is a directed graph with all its arrows and some of its nodes labeled.Computations are modled by graph rewriting of a simple nature.The basic rewriting rules embody the essence of both the communications among processes and cut-eliminations in proofs.Calculi of graphs are ideentified to give a formal and algebraic account of reaction graphs in the spirit of process algebra.With the help of the calculi,it is demonstrated that reaction graphs capture many interesting aspects of computations.
Igarashi, Yutaka; Nogami, Yoshie
2017-01-01
Background No meta-analysis has examined the effect of regular aquatic exercise on blood pressure. The purpose of this study was to perform a meta-analysis to evaluate the effects of regular aquatic exercise on blood pressure. Design A meta-analysis of randomized controlled trials. Methods Databases were searched for literature published up to April 2017. The randomized controlled trials analysed involved healthy adults, an intervention group that only performed aquatic exercise and a control group that did not exercise, no other intervention, and trials indicated mean systolic blood pressure or diastolic blood pressure. The net change in blood pressure was calculated from each trial, and the changes in blood pressure were pooled by a random effects model, and the risk of heterogeneity was evaluated. Subgroup analysis of subjects with hypertension, subjects who performed endurance exercise (or not), and subjects who only swam (or not) was performed, and the net changes in blood pressure were pooled. Results The meta-analysis examined 14 trials involving 452 subjects. Pooled net changes in blood pressure improved significantly (systolic blood pressure -8.4 mmHg; diastolic blood pressure -3.3 mmHg) and the changes in systolic blood pressure contained significant heterogeneity. When subjects were limited to those with hypertension, those who performed endurance exercise and subjects who did not swim, pooled net changes in systolic and diastolic blood pressure decreased significantly, but the heterogeneity of systolic blood pressure did not improve. Conclusion Like exercise on land, aquatic exercise should have a beneficial effect by lowering blood pressure. In addition, aquatic exercise should lower the blood pressure of subjects with hypertension, and other forms of aquatic exercise besides swimming should also lower blood pressure.
Regularities in Many-body Systems Interacting by a Two-body Random Ensemble
Zhao, Y M; Yoshinaga, N
2003-01-01
The even-even nuclei always have zero ground state angular momenta $I$ and positive parities $\\pi$. This feature was believed to be just a consequence of the attractive short-range interactions between nucleons. However, in the presence of two-body random interactions, the predominance of $I^{\\pi}=0^+$ ground states (0 g.s.) was found to be robust both for bosons and for an even number of fermions. For simple systems, such as $d$ bosons, $sp$ bosons, $sd$ bosons, and a few fermions in single-$j$ shells for small $j$, there are a few approaches to predict and/or explain the distribution of angular momentum $I$ ground state probabilities. An empirical recipe to predict the $I$ g.s. probabilities is available for general cases, but a more fundamental understanding of the robustness of 0 g.s. dominance is still out of reach. Other interesting results are also reviewed concerning other robust phenomena of many-body systems in the presence of random interactions, such as odd-even staggering of binding energies, gen...
Nonlinear propagation and decay of intense regular and random waves in relaxing media
Gurbatov, S. N.; Rudenko, O. V.; Demin, I. Yu.
2015-10-01
An integro-differential equation is written down that contains terms responsible for nonlinear absorption, visco-heat-conducting dissipation, and relaxation processes in a medium. A general integral expression is obtained for calculating energy losses of the wave with arbitrary characteristics—intensity, profile (frequency spectrum), and kernel describing the internal dynamics of the medium. Profiles of stationary solutions are constructed both for an exponential relaxation kernel and for other types of kernels. Energy losses at the front of week shock waves are calculated. General integral formulas are obtained for energy losses of intense noise, which are determined by the form of the kernel, the structure of the noise correlation function, and the mean square of the derivative of realization of a random process.
Limkumnerd, Surachate
2014-03-01
Interest in thin-film fabrication for industrial applications have driven both theoretical and computational aspects of modeling its growth. One of the earliest attempts toward understanding the morphological structure of a film's surface is through a class of solid-on-solid limited-mobility growth models such as the Family, Wolf-Villain, or Das Sarma-Tamborenea models, which have produced fascinating surface roughening behaviors. These models, however, restrict the motion of an incidence atom to be within the neighborhood of its landing site, which renders them inept for simulating long-distance surface diffusion such as that observed in thin-film growth using a molecular-beam epitaxy technique. Naive extension of these models by repeatedly applying the local diffusion rules for each hop to simulate large diffusion length can be computationally very costly when certain statistical aspects are demanded. We present a graph-theoretic approach to simulating a long-range diffusion-attachment growth model. Using the Markovian assumption and given a local diffusion bias, we derive the transition probabilities for a random walker to traverse from one lattice site to the others after a large, possibly infinite, number of steps. Only computation with linear-time complexity is required for the surface morphology calculation without other probabilistic measures. The formalism is applied, as illustrations, to simulate surface growth on a two-dimensional flat substrate and around a screw dislocation under the modified Wolf-Villain diffusion rule. A rectangular spiral ridge is observed in the latter case with a smooth front feature similar to that obtained from simulations using the well-known multiple registration technique. An algorithm for computing the inverse of a class of substochastic matrices is derived as a corollary.
Wilson, D.B.; Propp, J.G.
1996-12-31
This paper shows how to obtain unbiased samples from an unknown Markov chain by observing it for O(T{sub c}) steps, where T{sub c} is the cover time. This algorithm improves on several previous algorithms, and there is a matching lower bound. Using the techniques from the sampling algorithm, we also show how to sample random directed spanning trees from a weighted directed graph, with arcs directed to a root, and probability proportional to the product of the edge weights. This tree sampling algorithm runs within 18 cover times of the associated random walk, and is more generally applicable than the algorithm of Broder and Aldous.
Thimble regularization at work: from toy models to chiral random matrix theories
Di Renzo, Francesco
2015-01-01
We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very ...
Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models
Tomasz Kajdanowicz
2016-09-01
Full Text Available Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: the Erdős–Renyi random graph model, Watts–Strogatz small world model, Albert–Barabási preferential attachment model, Price citation model, and many more. Often, researchers working with real-world data are interested in understanding the generative phenomena underlying their empirical graphs. They want to know which of the theoretical graph generation models would most probably generate a particular empirical graph. In other words, they expect some similarity assessment between the empirical graph and graphs artificially created from theoretical graph generation models. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this means comparing the empirical graph to a single realization of a theoretical graph model, where the realization is generated from the given model using an arbitrary set of parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov–Smirnov test of distances between cumulative distributions. However, this approach is both error-prone and leads to incorrect conclusions, as we show in our experiments. Therefore, we propose a new method for graph comparison and type classification by comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality. We demonstrate that our approach can help assign the empirical graph to the most similar theoretical model using a simple unsupervised learning method.
Generalized connectivity of graphs
Li, Xueliang
2016-01-01
Noteworthy results, proof techniques, open problems and conjectures in generalized (edge-) connectivity are discussed in this book. Both theoretical and practical analyses for generalized (edge-) connectivity of graphs are provided. Topics covered in this book include: generalized (edge-) connectivity of graph classes, algorithms, computational complexity, sharp bounds, Nordhaus-Gaddum-type results, maximum generalized local connectivity, extremal problems, random graphs, multigraphs, relations with the Steiner tree packing problem and generalizations of connectivity. This book enables graduate students to understand and master a segment of graph theory and combinatorial optimization. Researchers in graph theory, combinatorics, combinatorial optimization, probability, computer science, discrete algorithms, complexity analysis, network design, and the information transferring models will find this book useful in their studies.
Bizhani, Golnoosh; Grassberger, Peter; Paczuski, Maya
2011-12-01
We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [Song et al., Nature (London) 433, 392 (2005)], RSR allows a more fine-grained analysis of the renormalization group (RG) flow and unravels new features that were not discussed in the previous analyses. In particular, we find that all networks exhibit a second-order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling behavior seen there. For critical trees it happens as N/N(0) → 0 in the limit of large systems (where N(0) is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N(0) in sparse ER graphs and in the annealed model, while it happens for N/N(0) → 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a starlike structure in agreement with the results of Radicchi et al. [Phys. Rev. Lett. 101, 148701 (2008)]. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering "supernodes" as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.
Effects of Topological Randomness on Cooperation in a Deterministic Prisoner's Dilemma Game
ZHANG Mei; YANG Jun-Zhong
2011-01-01
In this work, we consider an evolutionary prisoner's dilemma game on a homogeneous random network with the richest-following strategy adoption rule. By constructing homogeneous random networks from a regular ring graph,we investigate the effects of topological randomness on cooperation. In contrast to the ordinary view that the presence of small amount of shortcuts in ring graphs favors cooperation, we find the cooperation inhibition by weak topological randomness. The explanations on the observations are presented.
Quantization of gauge fields, graph polynomials and graph homology
Kreimer, Dirk, E-mail: kreimer@physik.hu-berlin.de [Humboldt University, 10099 Berlin (Germany); Sars, Matthias [Humboldt University, 10099 Berlin (Germany); Suijlekom, Walter D. van [Radboud University Nijmegen, 6525 AJ Nijmegen (Netherlands)
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
Spectral fluctuations of quantum graphs
Pluhař, Z. [Faculty of Mathematics and Physics, Charles University, 180 00 Praha 8 (Czech Republic); Weidenmüller, H. A. [Max-Planck-Institut für Kernphysik, 69029 Heidelberg (Germany)
2014-10-15
We prove the Bohigas-Giannoni-Schmit conjecture in its most general form for completely connected simple graphs with incommensurate bond lengths. We show that for graphs that are classically mixing (i.e., graphs for which the spectrum of the classical Perron-Frobenius operator possesses a finite gap), the generating functions for all (P,Q) correlation functions for both closed and open graphs coincide (in the limit of infinite graph size) with the corresponding expressions of random-matrix theory, both for orthogonal and for unitary symmetry.
van Dam, E.R.; Haemers, W.H.
1995-01-01
A graph G has constant u = u(G) if any two vertices that are not adjacent have u common neighbours. G has constant u and u if G has constant u = u(G), and its complement G has constant u = u(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We
Measuring extremal dependencies in web graphs
Volkovich, Y.; Litvak, Nelli; Zwart, B.
2008-01-01
We analyze dependencies in power law graph data (Web sample, Wikipedia sample and a preferential attachment graph) using statistical inference for multivariate regular variation. The well developed theory of regular variation is widely applied in extreme value theory, telecommunications and mathemat
Implementing regularization implicitly via approximate eigenvector computation
Mahoney, Michael W
2010-01-01
Regularization is a powerful technique for extracting useful information from noisy data. Typically, it is implemented by adding some sort of norm constraint to an objective function and then exactly optimizing the modified objective function. This procedure typically leads to optimization problems that are computationally more expensive than the original problem, a fact that is clearly problematic if one is interested in large-scale applications. On the other hand, a large body of empirical work has demonstrated that heuristics, and in some cases approximation algorithms, developed to speed up computations sometimes have the side-effect of performing regularization implicitly. Thus, we consider the question: What is the regularized optimization objective that an approximation algorithm is exactly optimizing? We address this question in the context of computing approximations to the smallest nontrivial eigenvector of a graph Laplacian; and we consider three random-walk-based procedures: one based on the heat ...
Hukic-Markosian, Golda; Zhang, Ye; Singh, Sanjeev; Vardeny, Valy
2010-03-01
Regio-regular (RR) P3HT has been successfully used as donor polymer in organic bulk heterojunction photovoltaic cells based on blends with fullerene acceptors; with power conversion efficiencies of over 6%. However, when regio-random (RR-a) P3HT is used as donor polymer in the blend, the power conversion efficiency drops to less than 0.5%. We have used various optical and magnetic resonance techniques to elucidate the charge photogeneration in the two polymer/fullerene blends. Using tunneling electron microscopy we conclude that phase separation takes place in blends based on RR P3HT but not in blends based on RR-a P3HT. Photoluminescence spectrum shows a prominent band in RR-a P3HT blend at 1.32 eV, indicating the dominance of charge transfer exciton recombination. Photoinduced absorption shows higher localization of polarons in RRa-P3HT blend, with a distinct PA band due to negative polaron on PCBM molecules. Photoinduced absorption detected magnetic resonance resolves the contributions of RR-a P3HT and PCBM as two resonances indicating positive polarons on the polymer and negative polaron on the fullerene. A model based on our experimental results will be discussed.
The Theory of Compact Graph and Supper Compact Graph%紧图与超紧图的一些理论
陆伟成; 张宣昊
2011-01-01
研究紧图与超紧图.得出连通且正则的紧图必为超紧图.研究了正则的紧图与点可迁图的关系.%Discuss the compact graph and supper compact graph and it has been proved that the connected and regular compact graphs must be the supper compact graphs. It has also been discussed the relationship between regular compact graph and vertex-transitive graph.
图上的随机动力系统及其诱导的马尔可夫链%Markov Chain Induced by Random Dynamical System on Graph
郑洁; 刘朝阳
2008-01-01
In this paper,we define a model of random dynamical systems(RDS)on graphs and prove that they are actually homogeneous discrete-time Markov chains.Moreover,a necessary and sufficient condition is obtained for that two state vectors can communicate with each other in a random dynamical system(RDS).
Sahasranand, K R
2010-01-01
Almost all known secret sharing schemes work on numbers. Such methods will have difficulty in sharing graphs since the number of graphs increases exponentially with the number of nodes. We propose a secret sharing scheme for graphs where we use graph intersection for reconstructing the secret which is hidden as a sub graph in the shares. Our method does not rely on heavy computational operations such as modular arithmetic or polynomial interpolation but makes use of very basic operations like assignment and checking for equality, and graph intersection can also be performed visually. In certain cases, the secret could be reconstructed using just pencil and paper by authorised parties but cannot be broken by an adversary even with unbounded computational power. The method achieves perfect secrecy for (2, n) scheme and requires far fewer operations compared to Shamir's algorithm. The proposed method could be used to share objects such as matrices, sets, plain text and even a heterogeneous collection of these. S...
On cyclic orthogonal double covers of circulant graphs by special infinite graphs
R. El-Shanawany
2017-12-01
Full Text Available In this article, a technique to construct cyclic orthogonal double covers (CODCs of regular circulant graphs by certain infinite graph classes such as complete bipartite and tripartite graphs and disjoint union of butterfly and K1,2n−10 is introduced.
Paley Graphs and Their Generalizations
Elsawy, Ahmed Noubi
2012-01-01
To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We will study some important properties of the Paley graphs. In particular, we will show that the Paley graphs are connected, symmetric, and self-complementary. Also we will show that the Paley graph of order q is (q-1)/2 -regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors, which means that the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. In the first section of Chapter 3 we will see three examples of these generalizations and some of their basic properties. In the second section of Chapter 3 we will define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge if and only if there difference belongs t...
Vestergaard, Preben Dahl; Hartnell, Bert L.
2006-01-01
There are many results dealing with the problem of decomposing a fixed graph into isomorphic subgraphs. There has also been work on characterizing graphs with the property that one can delete the edges of a number of edge disjoint copies of the subgraph and, regardless of how that is done, the gr...
The XXZ Heisenberg model on random surfaces
Ambjørn, J., E-mail: ambjorn@nbi.dk [The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen (Denmark); Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radbaud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen (Netherlands); Sedrakyan, A., E-mail: sedrak@nbi.dk [The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen (Denmark); Yerevan Physics Institute, Br. Alikhanyan str. 2, Yerevan-36 (Armenia)
2013-09-21
We consider integrable models, or in general any model defined by an R-matrix, on random surfaces, which are discretized using random Manhattan lattices. The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. They can also be associated with the random graphs of multiparticle scattering nodes. As an example we formulate a random matrix model where the partition function reproduces the annealed average of the XXZ Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.
The XXZ Heisenberg model on random surfaces
Ambjorn, J
2013-01-01
We consider integrable models, or in general any model defined by an $R$-matrix, on random surfaces, which are discretized using random Manhattan lattices. The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. They can also be associated with the random graphs of multiparticle scattering nodes. As an example we formulate a random matrix model where the partition function reproduces the annealed average of the XXZ Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.
A Reduction of the Graph Reconstruction Conjecture
Monikandan S.
2014-08-01
Full Text Available A graph is said to be reconstructible if it is determined up to isomor- phism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all non-geodetic and non-interval-regular blocks G with diam(G = 2 or diam(Ḡ = diam(G = 3 are reconstructible
The Bipartite Swapping Trick on Graph Homomorphisms
Zhao, Yufei
2011-01-01
We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from $G$ to $H$ when $H$ is bipartite. We also apply our techniques to graph colorings and stable set polytopes.
On Wiener index of graph complements
Jaisankar Senbagamalar
2014-06-01
Full Text Available Let $G$ be an $(n,m$-graph. We say that $G$ has property $(ast$ if for every pair of its adjacent vertices $x$ and $y$, there exists a vertex $z$, such that $z$ is not adjacent to either $x$ or $y$. If the graph $G$ has property $(ast$, then its complement $overline G$ is connected, has diameter 2, and its Wiener index is equal to $binom{n}{2}+m$, i.e., the Wiener index is insensitive of any other structural details of the graph $G$. We characterize numerous classes of graphs possessing property $(ast$, among which are trees, regular, and unicyclic graphs.
Connectivity threshold for Bluetooth graphs
Broutin, Nicolas; Fraiman, Nicolas; Lugosi, Gábor
2011-01-01
We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius $r$ that determines the "visible neighbors" of each node and the number of edges $c$ that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters $r, c$ and completely characterize the connectivity threshold (in $c$) for values of $r$ close the critical value for connectivity in the underlying random geometric graph.
Dynamic monopolies with randomized starting configuration
Kulich, Tomas
2010-01-01
Properties of systems with majority voting rules have been exhaustingly studied. In this work we focus on the randomized case - where the system is initialized by randomized initial set of seeds. Our main aim is to give an asymptotic estimate for sampling probability, such that the initial set of seeds is (is not) a dynamic monopoly almost surely. After presenting some trivial examples, we present exhaustive results for toroidal mesh and random 4-regular graph under simple majority scenario.
Coloring geographical threshold graphs
Bradonjic, Milan [Los Alamos National Laboratory; Percus, Allon [Los Alamos National Laboratory; Muller, Tobias [EINDHOVEN UNIV. OF TECH
2008-01-01
We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e.g., wireless networks, the Internet, etc.) need to be studied by using a 'richer' stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph's clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the chromatic number is identical: {chi}1n 1n n / 1n n (1 + {omicron}(1)). Finally, we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C 1n n / (1n 1n n){sup 2}, and specify the constant C.
Cut Size Statistics of Graph Bisection Heuristics
Schreiber, G. R.; Martin, O. C.
1998-01-01
We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve approximately the graph bisection problem. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by ``local'' algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends towards a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given...
Bordenave, Charles; Salez, Justin
2011-01-01
We prove that the local weak convergence of a sequence of graphs is enough to guarantee the convergence of their normalized matching numbers. The limiting quantity is described by a local recursion defined on the weak limit of the graph sequence. However, this recursion may admit several solutions, implying non-trivial long-range dependencies between the edges of a largest matching. We overcome this lack of correlation decay by introducing a perturbative parameter called the temperature, which we let progressively go to zero. When the local weak limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation, resulting into an explicit formula that considerably extends the well-known one by Karp and Sipser for Erd\\"os-R\\'enyi random graphs.
Graph Regularized Semi-Supervised Learning on Heterogeneous Information Networks%异构信息网络上基于图正则化的半监督学习
刘钰峰; 李仁发
2015-01-01
Heterogeneous information networks ,composed of multiple types of objects and links ,are ubiquitous in real life .Therefore ,effective analysis of large‐scale heterogeneous information networks poses an interesting but critical challenge . Learning from labeled and unlabeled data via semi‐supervised classification can lead to good knowledge extraction of the hidden network structure . How ever ,although semi‐supervised learning on homogeneous netw orks has been studied for decades , classification on heterogeneous networks has not been explored until recently . In this paper , we consider the semi‐supervised classification problem on heterogeneous information networks with an arbitrary schema consisting of a number of object and link types .By applying graph regularization to preserve consistency over each relation graph corresponding to each type of links separately , we develop a classifying function which is sufficiently smooth with respect to the intrinsic structure collectively revealed by known labeled and unlabeled points .We propose an iterative framework on heterogeneous information network in which the information of labeled data can be spread to the adjacent nodes by iterative method until the steady state .We infer the class memberships of unlabeled data from those of labeled ones according to their proximities in the network .Experiments on the real DBLP data set clearly show that our approach outperforms the classic semi‐supervised Learning methods .%现实世界中存在着大量包含多种类型的对象和联系的异构信息网络，从中挖掘信息获取知识已成为当前的研究热点之一．基于图正则化的半监督学习在近年来得到了广泛的研究，然而，现有的半监督学习算法大都只能应用于同构网络．基于同构节点和异构节点的一致性假设，提出了任意结构的异构信息网络上的半监督学习的正则化分类函数，并得到分类函数的闭
Eigenvalues and expansion of bipartite graphs
Høholdt, Tom; Janwa, Heeralal
2012-01-01
We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by ...... by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c,d)-regular bipartite graphs and compare that with aclassical bound....
Areej M. Abduldaim
2013-01-01
Full Text Available We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse. An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .
Trudeau, Richard J
1994-01-01
Preface1. Pure Mathematics Introduction; Euclidean Geometry as Pure Mathematics; Games; Why Study Pure Mathematics?; What's Coming; Suggested Reading2. Graphs Introduction; Sets; Paradox; Graphs; Graph diagrams; Cautions; Common Graphs; Discovery; Complements and Subgraphs; Isomorphism; Recognizing Isomorphic Graphs; Semantics The Number of Graphs Having a Given nu; Exercises; Suggested Reading3. Planar Graphs Introduction; UG, K subscript 5, and the Jordan Curve Theorem; Are there More Nonplanar Graphs?; Expansions; Kuratowski's Theorem; Determining Whether a Graph is Planar or
SNAP: A General Purpose Network Analysis and Graph Mining Library.
Leskovec, Jure; Sosič, Rok
2016-10-01
Large networks are becoming a widely used abstraction for studying complex systems in a broad set of disciplines, ranging from social network analysis to molecular biology and neuroscience. Despite an increasing need to analyze and manipulate large networks, only a limited number of tools are available for this task. Here, we describe Stanford Network Analysis Platform (SNAP), a general-purpose, high-performance system that provides easy to use, high-level operations for analysis and manipulation of large networks. We present SNAP functionality, describe its implementational details, and give performance benchmarks. SNAP has been developed for single big-memory machines and it balances the trade-off between maximum performance, compact in-memory graph representation, and the ability to handle dynamic graphs where nodes and edges are being added or removed over time. SNAP can process massive networks with hundreds of millions of nodes and billions of edges. SNAP offers over 140 different graph algorithms that can efficiently manipulate large graphs, calculate structural properties, generate regular and random graphs, and handle attributes and meta-data on nodes and edges. Besides being able to handle large graphs, an additional strength of SNAP is that networks and their attributes are fully dynamic, they can be modified during the computation at low cost. SNAP is provided as an open source library in C++ as well as a module in Python. We also describe the Stanford Large Network Dataset, a set of social and information real-world networks and datasets, which we make publicly available. The collection is a complementary resource to our SNAP software and is widely used for development and benchmarking of graph analytics algorithms.
Seiller, Thomas
2016-01-01
Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all Geometry of Interaction (GoI) constructions introduced so far. This series of work was inspired from Girard's hyperfinite GoI, and develops a quantitative approach that should...... be understood as a dynamic version of weighted relational models. Until now, the interaction graphs framework has been shown to deal with exponentials for the constrained system ELL (Elementary Linear Logic) while keeping its quantitative aspect. Adapting older constructions by Girard, one can clearly define...... "full" exponentials, but at the cost of these quantitative features. We show here that allowing interpretations of proofs to use continuous (yet finite in a measure-theoretic sense) sets of states, as opposed to earlier Interaction Graphs constructions were these sets of states were discrete (and finite...
Diestel, Reinhard
2000-01-01
This book is a concise, yet carefully written, introduction to modern graph theory, covering all its major recent developments. It can be used both as a reliable textbook for an introductory course and as a graduate text: on each topic it covers all the basic material in full detail, and adds one or two deeper results (again with detailed proofs) to illustrate the more advanced methods of that field. This second edition extends the first in two ways. It offers a thoroughly revised and updated chapter on graph minors, which now includes full new proofs of two of the central Robertson-Seymour theorems (as well as a detailed sketch of the entire proof of their celebrated Graph Minor Theorem). Second, there is now a section of hints for all the exercises, to enhance their value for both individual study and classroom use.
Bahi, J M; Guyeux, C; Richard, A
2011-01-01
Chaotic functions are characterized by sensitivity to initial conditions, transitivity, and regularity. Providing new functions with such properties is a real challenge. This work shows that one can associate with any Boolean network a continuous function, whose discrete-time iterations are chaotic if and only if the iteration graph of the Boolean network is strongly connected. Then, sufficient conditions for this strong connectivity are expressed on the interaction graph of this network, leading to a constructive method of chaotic function computation. The whole approach is evaluated in the chaos-based pseudo-random number generation context.
Simple Markov-chain algorithms for generating bipartite graphs and tournaments
Kannan, R.; Vempala, S. [Carnegie Mellon Univ., Pittsburgh, PA (United States); Tetali, P. [Georgia Institute of Technology, Atlanta, GA (United States)
1997-06-01
We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the (near-) regular case, i.e. when all the degrees are (almost) equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near-regular. The proof techniques we use for the two problems are similar.
Beeken, Paul
2014-11-01
Graphing is an essential skill that forms the foundation of any physical science.1 Understanding the relationships between measurements ultimately determines which modeling equations are successful in predicting observations.2 Over the years, science and math teachers have approached teaching this skill with a variety of techniques. For secondary school instruction, the job of graphing skills falls heavily on physics teachers. By virtue of the nature of the topics we cover, it is our mission to develop this skill to the fine art that it is.
The Laplacian eigenvalues of graphs: a survey
Zhang, Xiao-Dong
2011-01-01
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In addition, some new unpublished results and questions are concluded. Emphasis is given on classifications of the upper and lower bounds for the Laplacian eigenvalues of graphs (including some special graphs, such as trees, bipartite graphs, triangular-free graphs, cubic graphs, etc.) as a function of other graph invariants, such as degree sequence, the average 2-degree, diameter, the maximal independence number, the maximal matching number, vertex connectivity, the domination number, the number of the spanning trees, etc.
Information Spreading in Dynamic Graphs
Clementi, Andrea; Trevisan, Luca
2011-01-01
We present a general approach to study the flooding time (a measure of how fast information spreads) in dynamic graphs (graphs whose topology changes with time according to a random process). We consider arbitrary converging Markovian dynamic graph process, that is, processes in which the topology of the graph at time $t$ depends only on its topology at time $t-1$ and which have a unique stationary distribution. The most well studied models of dynamic graphs are all Markovian and converging. Under general conditions, we bound the flooding time in terms of the mixing time of the dynamic graph process. We recover, as special cases of our result, bounds on the flooding time for the \\emph{random trip} model and the \\emph{random path} models; previous analysis techniques provided bounds only in restricted settings for such models. Our result also provides the first bound for the \\emph{random waypoint} model (which is tight for certain ranges of parameters) whose analysis had been an important open question.
Jordan, Jonathan
2011-01-01
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, $\\alpha$, $\\beta$ and $\\gamma$, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If $(1+\\alpha)(1+\\gamma)1$ then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree $d$ tends to zero. We also give some results on the number of edges and on the spectral gap.
Diestel, Reinhard
2017-01-01
This standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail. The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study. From the reviews: “This outstanding book cannot be substituted with any other book on the present textbook market. It has every chance of becoming the standard textbook for graph theory.”Acta Scientiarum Mathematiciarum “Deep, clear, wonderful. This is a serious book about the heart of graph theory. It has depth and integrity. ”Persi Diaconis & Ron Graham, SIAM Review “The book has received a very enthusiastic reception, which it amply deserves. A masterly elucidation of modern graph theo...
Graphs in Practical Situations
刘晓玫; 任心玥
2008-01-01
<正>Linear graphs are often used to depict conversion graphs and travel graphs. Example: The following graph shows the conversion between the Singapore dollar (S $) and the Malay- sian ringgit (RM) in 2000.
2016-06-01
GraphBench is a benchmark suite for graph pattern mining and graph analysis systems. The benchmark suite is a significant addition to conducting apples-apples comparison of graph analysis software (databases, in-memory tools, triple stores, etc.)
Classifying Cubic Edge-Transitive Graphs of Order 8
Mehdi Alaeiyan; M K Hosseinipoor
2009-11-01
A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let be a prime. It was shown by Folkman (J. Combin. Theory 3(1967) 215--232) that a regular edge-transitive graph of order 2 or 22 is necessarily vertex-transitive. In this paper, an extension of his result in the case of cubic graphs is given. It is proved that, every cubic edge-transitive graph of order 8 is symmetric, and then all such graphs are classified.
Subsampling for graph power spectrum estimation
Chepuri, Sundeep Prabhakar
2016-10-06
In this paper we focus on subsampling stationary random signals that reside on the vertices of undirected graphs. Second-order stationary graph signals are obtained by filtering white noise and they admit a well-defined power spectrum. Estimating the graph power spectrum forms a central component of stationary graph signal processing and related inference tasks. We show that by sampling a significantly smaller subset of vertices and using simple least squares, we can reconstruct the power spectrum of the graph signal from the subsampled observations, without any spectral priors. In addition, a near-optimal greedy algorithm is developed to design the subsampling scheme.
Reddy, A Satyanarayana
2011-01-01
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the graphs which are polynomials in the pattern polynomial graph have been studied. We also identify known graph classes which are pattern polynomial graphs.
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Volume growth and stochastic completeness of graphs
Folz, Matthew
2012-01-01
Given the variable-speed random walk on a weighted graph and a metric adapted to the structure of the random walk, we construct a Brownian motion on a closely related metric graph which behaves similarly to the VSRW and for which the associated intrinsic metric has certain desirable properties. Jump probabilities and moments of jump times for Brownian motion on metric graphs with varying edge lengths, jump conductances, and edge densities are computed. We use these results together with a theorem of Sturm for stochastic completeness, or non-explosiveness, on local Dirichlet spaces to prove sharp volume growth criteria in adapted metrics for stochastic completeness of graphs.
Gould, Ronald
2012-01-01
This introduction to graph theory focuses on well-established topics, covering primary techniques and including both algorithmic and theoretical problems. The algorithms are presented with a minimum of advanced data structures and programming details. This thoroughly corrected 1988 edition provides insights to computer scientists as well as advanced undergraduates and graduate students of topology, algebra, and matrix theory. Fundamental concepts and notation and elementary properties and operations are the first subjects, followed by examinations of paths and searching, trees, and networks. S
Weinzierl, Stefan
2013-01-01
In these lectures I discuss Feynman graphs and the associated Feynman integrals. Of particular interest are the classes functions, which appear in the evaluation of Feynman integrals. The most prominent class of functions is given by multiple polylogarithms. The algebraic properties of multiple polylogarithms are reviewed in the second part of these lectures. The final part of these lectures is devoted to Feynman integrals, which cannot be expressed in terms of multiple polylogarithms. Methods from algebraic geometry provide tools to tackle these integrals.
Roman Bondage Numbers of Some Graphs
Hu, Fu-Tao
2011-01-01
A Roman dominating function on a graph $G=(V,E)$ is a function $f: V\\to \\{0,1,2\\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The weight of a Roman dominating function is the value $f(G)=\\sum_{u\\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. This paper determines the exact value of the Roman bondage numbers of two classes of graphs, complete $t$-partite graphs and $(n-3)$-regular graphs with order $n$ for any $n\\ge 5$.
Parallel Graph Partitioning for Complex Networks
Meyerhenke, Henning; Schulz, Christian
2014-01-01
Processing large complex networks like social networks or web graphs has recently attracted considerable interest. In order to do this in parallel, we need to partition them into pieces of about equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for these networks. This paper addresses this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsened graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. For example, our algorithm can partition a web graph with 3.3 billion edges ...
Tóth, L Fejes; Ulam, S; Stark, M
1964-01-01
Regular Figures concerns the systematology and genetics of regular figures. The first part of the book deals with the classical theory of the regular figures. This topic includes description of plane ornaments, spherical arrangements, hyperbolic tessellations, polyhedral, and regular polytopes. The problem of geometry of the sphere and the two-dimensional hyperbolic space are considered. Classical theory is explained as describing all possible symmetrical groupings in different spaces of constant curvature. The second part deals with the genetics of the regular figures and the inequalities fo
MAP Estimation, Message Passing, and Perfect Graphs
Jebara, Tony S
2012-01-01
Efficiently finding the maximum a posteriori (MAP) configuration of a graphical model is an important problem which is often implemented using message passing algorithms. The optimality of such algorithms is only well established for singly-connected graphs and other limited settings. This article extends the set of graphs where MAP estimation is in P and where message passing recovers the exact solution to so-called perfect graphs. This result leverages recent progress in defining perfect graphs (the strong perfect graph theorem), linear programming relaxations of MAP estimation and recent convergent message passing schemes. The article converts graphical models into nand Markov random fields which are straightforward to relax into linear programs. Therein, integrality can be established in general by testing for graph perfection. This perfection test is performed efficiently using a polynomial time algorithm. Alternatively, known decomposition tools from perfect graph theory may be used to prove perfection ...
ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES
无
2007-01-01
In this paper, an equivalent condition of a graph G with t (2 ≤ t ≤ n) distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular (necessarily be strongly regular), an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.
Recursively-regular subdivisions and applications
Rafel Jaume
2016-05-01
Full Text Available We generalize regular subdivisions (polyhedral complexes resulting from the projection of the lower faces of a polyhedron introducing the class of recursively-regular subdivisions. Informally speaking, a recursively-regular subdivision is a subdivision that can be obtained by splitting some faces of a regular subdivision by other regular subdivisions (and continue recursively. We also define the finest regular coarsening and the regularity tree of a polyhedral complex. We prove that recursively-regular subdivisions are not necessarily connected by flips and that they are acyclic with respect to the in-front relation. We show that the finest regular coarsening of a subdivision can be efficiently computed, and that whether a subdivision is recursively regular can be efficiently decided. As an application, we also extend a theorem known since 1981 on illuminating space by cones and present connections of recursive regularity to tensegrity theory and graph-embedding problems.
CUDA Enabled Graph Subset Examiner
2016-12-22
Finding Godsil-McKay switching sets in graphs is one way to demonstrate that a specific graph is not determined by its spectrum--the eigenvalues of its adjacency matrix. An important area of active research in pure mathematics is determining which graphs are determined by their spectra, i.e. when the spectrum of the adjacency matrix uniquely determines the underlying graph. We are interested in exploring the spectra of graphs in the Johnson scheme and specifically seek to determine which of these graphs are determined by their spectra. Given a graph G, a Godsil-McKay switching set is an induced subgraph H on 2k vertices with the following properties: I) H is regular, ii) every vertex in G/H is adjacent to either 0, k, or 2k vertices of H, and iii) at least one vertex in G/H is adjacent to k vertices in H. The software package examines each subset of a user specified size to determine whether or not it satisfies those 3 conditions. The software makes use of the massive parallel processing power of CUDA enabled GPUs. It also exploits the vertex transitivity of graphs in the Johnson scheme by reasoning that if G has a Godsil-McKay switching set, then it has a switching set which includes vertex 1. While the code (in its current state) is tuned to this specific problem, the method of examining each induced subgraph of G can be easily re-written to check for any user specified conditions on the subgraphs and can therefore be used much more broadly.
Diestel, Reinhard
2012-01-01
HauptbeschreibungThis standard textbook of modern graph theory, now in its fourth edition, combinesthe authority of a classic with the engaging freshness of style that is the hallmarkof active mathematics. It covers the core material of the subject with concise yetreliably complete proofs, while offering glimpses of more advanced methodsin each field by one or two deeper results, again with proofs given in full detail.The book can be used as a reliable text for an introductory course, as a graduatetext, and for self-study. Rezension"Deep, clear, wonderful. This is a serious book about the
Merris, Russell
2001-01-01
A lively invitation to the flavor, elegance, and power of graph theoryThis mathematically rigorous introduction is tempered and enlivened by numerous illustrations, revealing examples, seductive applications, and historical references. An award-winning teacher, Russ Merris has crafted a book designed to attract and engage through its spirited exposition, a rich assortment of well-chosen exercises, and a selection of topics that emphasizes the kinds of things that can be manipulated, counted, and pictured. Intended neither to be a comprehensive overview nor an encyclopedic reference, th
Salvat, I; Zaldivar, P; Monterde, S; Montull, S; Miralles, I; Castel, A
2017-03-01
Multidisciplinary treatments have shown to be effective for fibromyalgia. We report detailed functional outcomes of patients with fibromyalgia who attended a 3-month Multidisciplinary treatment program. The hypothesis was that patients would have increased functional status, physical activity level, and exercise regularity after attending this program. We performed a retrospective analysis of a randomized, simple blinded clinical trial. The inclusion criteria consisted of female sex, a diagnosis of fibromyalgia, age 18-60 and 3-8 years of schooling. Measures from the Fibromyalgia Impact Questionnaire (FIQ) and the COOP/WONCA Functional Health Assessment Charts (WONCA) were obtained before and at the end of the treatment and at 3-, 6-, and 12-month follow-ups. Patients recorded their number of steps per day with pedometers. They performed the six-minute walk test (6 MW) before and after treatment. In total, 155 women participated in the study. Their median (interquartile interval) FIQ score was 68.0 (53.0-77.0) at the beginning of the treatment, and the difference between the Multidisciplinary and Control groups was statistically and clinically significant in all of the measures (except the 6-month follow-up). The WONCA charts showed significant clinical improvements in the Multidisciplinary group, with physical fitness in the normal range across almost all values. In that group, steps/day showed more regularity, and the 6 MW results showed improvement of -33.00 (-59.8 to -8.25) m, and the differences from the Control group were statistically significant. The patients who underwent the Multidisciplinary treatment had improved functional status, physical activity level, and exercise regularity. The functional improvements were maintained 1 year after treatment completion.
Understanding Graphs & Charts.
Cleary, John J.; Gravely, Mary Liles
Developed by educators from the Emily Griffith Opportunity School, this teacher's guide was developed for a 4-hour workshop to teach employees how to read the charts and graphs they need in the workplace. The unit covers four types of graphs: pictographs, bar graphs, line graphs, and circle graphs. The guide is divided into four sections: reading…
On the Monotonicity of (k; g, h)-graphs
Bao-guang Xu; Ping Wang; Jian-fang Wang
2002-01-01
A (k;g)-graph is a k-regular graph with girth g. A (k;g)-cage is a (k;g)-graph with the least possible number of vertices. Let f(k;g) denote the number of vertices in a (k;g)-cage. The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. A k-regular graph with girth pair (g,h) is called a (k;g,h)-graph.A (k;g,h)-cage is a (k;g,h)-graph with the least possible number of vertices. Let f(k;g,h) denote the number of vertices in a (k;g,h)-cage. In this paper, we prove the following strict inequality f(k;h-1,h)＜f(k,h), k≥3, h≥4.
From the Coxeter graph to the Klein graph
Dejter, Italo J
2010-01-01
We show that the 56-vertex Klein cubic graph $\\G'=F_{056}B$ (so denoted in the Foster census) can be obtained from the 28-vertex Coxeter graph $\\G=F_{028}A$ by 'zipping' adequately the squares of the 24 7-cycles of $\\G$ endowed with an orientation obtained by considering $\\G$ as a $\\mathcal C$-ultrahomogeneous digraph, where $\\mathcal C$ is the set of oriented 7-cycles $\\vec{C}_7$ and $2$-paths $\\vec{P}_3$, that tightly fasten those $\\vec{C}_7$ in $\\G$. In the process, it is seen that $\\G'$ is a ${\\mathcal C}'$-ultrahomogeneous graph, where ${\\mathcal C}'$ is the set of 7-cycles $C_7$ and $1$-paths $P_2$, that tightly fasten those $C_7$ in $\\G'$; this yields an embedding of $\\G'$ into a 3-torus $T_3$, which forms the Klein map of Coxeter notation $(7,3)_8$. The dual graph of $\\G'$ in $T_3$ is the distance regular Klein quartic graph, with corresponding dual map of Coxeter notation $(3,7)_8$.
General inverse problems for regular variation
Damek, Ewa; Mikosch, Thomas Valentin; Rosinski, Jan
2014-01-01
Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components ...
Radio Channel Modelling Using Stochastic Propagation Graphs
Pedersen, Troels; Fleury, Bernard Henri
2007-01-01
In this contribution the radio channel model proposed in [1] is extended to include multiple transmitters and receivers. The propagation environment is modelled using random graphs where vertices of a graph represent scatterers and edges model the wave propagation between scatterers. Furthermore...
Tan, Yong
2013-01-01
In this paper, author uses set theory to construct a logic model of abstract figure from binary relation. Based on the uniform quantified structure, author gives two logic system for graph traversal and graph coloring respectively, moreover shows a new method of cutting graph. Around this model, there are six algorithms in this paper including exact graph traversal, Algebra calculation of natural number, graph partition and graph coloring.
Heiestad, Hege; Rustaden, Anne Mette; Bø, Kari; Haakstad, Lene A H
2016-01-01
Objectives. The aim was to investigate the effects of three different types of resistance training implementation. Design. Randomized controlled trial. Methods. Inactive, overweight women (n = 143), mean BMI 31.3 ± 5.2 kg/m(2), mean age 39.9 ± 10.5 years, were randomized to one of the following groups: A (BodyPump group training), B (individual follow-up by a personal trainer), C (nonsupervised exercise), or D (controls). The intervention included 12 weeks of 45-60 minutes' full-body resistance training three sessions per week. The outcomes in this paper are all secondary outcome measures: exercise motivation, self-perceived health, and quality of life. Results. Adherence averaged 26.1 ± 10.3 of 36 prescribed sessions. After the intervention period, all three training groups (A-C) had better scores on exercise motivation (A = 43.9 ± 19.8, B = 47.6 ± 15.4, C = 48.4 ± 17.8) compared to the control group (D) (26.5 ± 18.2) (p training contributed to higher scores in important variables related to exercise motivation and self-perceived health. Low adherence showed that it was difficult to motivate previously inactive, overweight women to participate in regular strength training.
Regularized Laplacian Estimation and Fast Eigenvector Approximation
Perry, Patrick O
2011-01-01
Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \\emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \\emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistical interpretation of their approximation procedure. Our interpretation will be analogous to the manner in which $\\ell_2$-regularized or $\\ell_1$-regularized $\\ell_2$-regression (often called Ridge regression and Lasso regression, respectively) can be interpreted in terms of a Gaussian prior or a Laplace prior, respectively, on the coefficient vector of the regression problem. Our framework will imply that the solutions to the Mahoney-Orecchia regularized SDP can be interpreted as regularized estimates of the pseudoinverse of the graph Laplacian. Conversely, it will imply that the solution to this regularized estimation problem can be computed very quickly by running, e.g., the fast diffusion-base...
Generalized Cayley Graphs and Cellular Automata over them
Arrighi, Pablo; Nesme, Vincent
2012-01-01
Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their equality; to name all vertices relative to a point; the fact that they have a well-defined notion of translation, and that they can be endowed with a compact metric. We propose a notion of graph associated to a language, which conserves or generalizes these features. Whereas Cayley graphs are regular; associated graphs are arbitrary, although of a bounded degree. Moreover, it is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions for a distance on the set of configurations that makes it a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Similarly, we extend their definition to these arbitrary, bounded degree, time-varying graphs. KEYWORDS: Causal Graph Dynamics, Curtis-Hedlund-Lynden, Dynamical networks, Boolean networks, Generative networks automata, Graph Autom...
SAR image regularization with fast approximate discrete minimization.
Denis, Loïc; Tupin, Florence; Darbon, Jérôme; Sigelle, Marc
2009-07-01
Synthetic aperture radar (SAR) images, like other coherent imaging modalities, suffer from speckle noise. The presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms. Numerous approaches have been proposed to filter speckle noise. Markov random field (MRF) modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image. In this context, total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges. Speckle noise follows heavy-tailed distributions, and the MRF formulation leads to a minimization problem involving nonconvex log-likelihood terms. Such a minimization can be performed efficiently by computing minimum cuts on weighted graphs. Due to memory constraints, exact minimization, although theoretically possible, is not achievable on large images required by remote sensing applications. The computational burden of the state-of-the-art algorithm for approximate minimization (namely the alpha -expansion) is too heavy specially when considering joint regularization of several images. We show that a satisfying solution can be reached, in few iterations, by performing a graph-cut-based combinatorial exploration of large trial moves. This algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area SAR images.
Algorithms for Omega-Regular Games with Imperfect Information
Chatterjee, Krishnendu; Henzinger, Thomas A; Raskin, Jean-Francois
2006-01-01
We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Buechi objective. This set is of interest because in the absence of perfect information, randomized...
Lawes, Jonathan F.
2013-01-01
Graphing polar curves typically involves a combination of three traditional techniques, all of which can be time-consuming and tedious. However, an alternative method--graphing the polar function on a rectangular plane--simplifies graphing, increases student understanding of the polar coordinate system, and reinforces graphing techniques learned…
2014-01-01
© 2015 Elsevier B.V. Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of a similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are determined.
Efficient dynamic graph construction for inductive semi-supervised learning.
Dornaika, F; Dahbi, R; Bosaghzadeh, A; Ruichek, Y
2017-10-01
Most of graph construction techniques assume a transductive setting in which the whole data collection is available at construction time. Addressing graph construction for inductive setting, in which data are coming sequentially, has received much less attention. For inductive settings, constructing the graph from scratch can be very time consuming. This paper introduces a generic framework that is able to make any graph construction method incremental. This framework yields an efficient and dynamic graph construction method that adds new samples (labeled or unlabeled) to a previously constructed graph. As a case study, we use the recently proposed Two Phase Weighted Regularized Least Square (TPWRLS) graph construction method. The paper has two main contributions. First, we use the TPWRLS coding scheme to represent new sample(s) with respect to an existing database. The representative coefficients are then used to update the graph affinity matrix. The proposed method not only appends the new samples to the graph but also updates the whole graph structure by discovering which nodes are affected by the introduction of new samples and by updating their edge weights. The second contribution of the article is the application of the proposed framework to the problem of graph-based label propagation using multiple observations for vision-based recognition tasks. Experiments on several image databases show that, without any significant loss in the accuracy of the final classification, the proposed dynamic graph construction is more efficient than the batch graph construction. Copyright © 2017 Elsevier Ltd. All rights reserved.
Hege Heiestad
2016-01-01
Full Text Available Objectives. The aim was to investigate the effects of three different types of resistance training implementation. Design. Randomized controlled trial. Methods. Inactive, overweight women (n=143, mean BMI 31.3±5.2 kg/m2, mean age 39.9±10.5 years, were randomized to one of the following groups: A (BodyPump group training, B (individual follow-up by a personal trainer, C (nonsupervised exercise, or D (controls. The intervention included 12 weeks of 45–60 minutes’ full-body resistance training three sessions per week. The outcomes in this paper are all secondary outcome measures: exercise motivation, self-perceived health, and quality of life. Results. Adherence averaged 26.1±10.3 of 36 prescribed sessions. After the intervention period, all three training groups (A–C had better scores on exercise motivation (A=43.9±19.8, B=47.6±15.4, C=48.4±17.8 compared to the control group (D (26.5±18.2 (p<0.001. Groups B and C scored better on self-perceived health (B=1.9±0.8, C=2.3±0.8, compared to group D (3.0±0.6 (p<0.001. For quality of life measurement, there was no statistically significant difference between either intervention groups or the control. Conclusions. Resistance training contributed to higher scores in important variables related to exercise motivation and self-perceived health. Low adherence showed that it was difficult to motivate previously inactive, overweight women to participate in regular strength training.
Ramirez, Daniel Perez; Whiteman, David N.; Veselovskii, Igor; Kolgotin, Alexei; Korenskiy, Michael; Alados-Arboledas, Lucas
2013-01-01
In this work we study the effects of systematic and random errors on the inversion of multiwavelength (MW) lidar data using the well-known regularization technique to obtain vertically resolved aerosol microphysical properties. The software implementation used here was developed at the Physics Instrumentation Center (PIC) in Troitsk (Russia) in conjunction with the NASA/Goddard Space Flight Center. Its applicability to Raman lidar systems based on backscattering measurements at three wavelengths (355, 532 and 1064 nm) and extinction measurements at two wavelengths (355 and 532 nm) has been demonstrated widely. The systematic error sensitivity is quantified by first determining the retrieved parameters for a given set of optical input data consistent with three different sets of aerosol physical parameters. Then each optical input is perturbed by varying amounts and the inversion is repeated. Using bimodal aerosol size distributions, we find a generally linear dependence of the retrieved errors in the microphysical properties on the induced systematic errors in the optical data. For the retrievals of effective radius, number/surface/volume concentrations and fine-mode radius and volume, we find that these results are not significantly affected by the range of the constraints used in inversions. But significant sensitivity was found to the allowed range of the imaginary part of the particle refractive index. Our results also indicate that there exists an additive property for the deviations induced by the biases present in the individual optical data. This property permits the results here to be used to predict deviations in retrieved parameters when multiple input optical data are biased simultaneously as well as to study the influence of random errors on the retrievals. The above results are applied to questions regarding lidar design, in particular for the spaceborne multiwavelength lidar under consideration for the upcoming ACE mission.
Heitmann, Janika; van Hemel-Ruiter, Madelon E; Vermeulen, Karin M; Ostafin, Brian D; MacLeod, Colin; Wiers, Reinout W; DeFuentes-Merillas, Laura; Fledderus, Martine; Markus, Wiebren; de Jong, Peter J
2017-05-23
The automatic tendency to attend to and focus on substance-related cues in the environment (attentional bias), has been found to contribute to the persistence of addiction. Attentional bias modification (ABM) interventions might, therefore, contribute to treatment outcome and the reduction of relapse rates. Based on some promising research findings, we designed a study to test the clinical relevance of ABM as an add-on component of regular intervention for alcohol and cannabis patients. The current protocol describes a study which will investigate the effectiveness and cost-effectiveness of a newly developed home-delivered, multi-session, internet-based ABM (iABM) intervention as an add-on to treatment as usual (TAU). TAU consists of cognitive behavioural therapy-based treatment according to the Dutch guidelines for the treatment of addiction. Participants (N = 213) will be outpatients from specialized addiction care institutions diagnosed with alcohol or cannabis dependency who will be randomly assigned to one of three conditions: TAU + iABM; TAU + placebo condition; TAU-only. Primary outcome measures are substance use, craving, and rates of relapse. Changes in attentional bias will be measured to investigate whether changes in primary outcome measures can be attributed to the modification of attentional bias. Indices of cost-effectiveness and secondary physical and psychological complaints (depression, anxiety, and stress) are assessed as secondary outcome measures. This randomized control trial will be the first to investigate whether a home-delivered, multi-session iABM intervention is (cost-) effective in reducing relapse rates in alcohol and cannabis dependency as an add-on to TAU, compared with an active and a waiting list control group. If proven effective, this ABM intervention could be easily implemented as a home-delivered component of current TAU. Netherlands Trial Register, NTR5497 , registered on 18th September 2015.
An empirical study of dynamic graph algorithms
Alberts, D. [Freie Universitaet Berlin (Germany); Cattaneo, G. [Universita di Salerno (Italy); Italiano, G.F. [Universita Ca Forscari di Venezia (Italy)
1996-12-31
We conduct an empirical study on some dynamic graph algorithms which where developed recently. The following implementations were tested and compared with simple algorithms: dynamic connectivity, and dynamic minimum 1 spanning tree based on sparsification by Eppstein et al.; dynamic connectivity based on a very recent paper by Henzinger and King. In our experiments, we considered both random and non-random inputs. Moreover, we present a simplified variant of the algorithm by Henzinger and King, which for random inputs was always faster than the original implementation. Indeed, this variant was among the fastest implementations for random inputs. For non-random inputs, sparsification was the fastest algorithm for small sequences of updates; for medium and large sequences of updates, the original algorithm by Henzinger and King was faster. Perhaps one of the main practical results of this paper is that our implementations of the sophisticated dynamic graph algorithms were faster than simpler algorithms for most practical values of the graph parameters, and competitive with simpler algorithms even in case of very small graphs (say graphs with less than a dozen vertices and edges). From the theoretical point of view, we analyze the average case running time of sparsification and prove that the logarithmic overhead for simple sparsification vanishes for dynamic random graphs.
Consensus on Moving Neighborhood Model of Peterson Graph
Arendt, Hannah
2012-01-01
In this paper, we study the consensus problem of multiple agents on a kind of famous graph, Peterson graph. It is an undirected graph with 10 vertices and 15 edges. Each agent randomly walks on this graph and communicates with each other if and only if they coincide on a node at the same time. We conduct numerical study on the consensus problem in this framework and show that global consensus can be achieved.
Nyasha Mureriwa
2016-02-01
Full Text Available The invasive taxa of Prosopis is rated the world’s top 100 unwanted species, and a lack of spatial data about the invasion dynamics has made the current control and monitoring methods unsuccessful. This study thus tests the use of in situ spectroscopy data with a newly-developed algorithm, guided regularized random forest (GRRF, to spectrally discriminate Prosopis from coexistent acacia species (Acacia karroo, Acacia mellifera and Ziziphus mucronata in the arid environment of South Africa. Results show that GRRF was able to reduce the high dimensionality of the spectroscopy data and select key wavelengths (n = 11 for discriminating amongst the species. These wavelengths are located at 356.3 nm, 468.5 nm, 531.1 nm, 665.2 nm, 1262.3 nm, 1354.1 nm, 1361.7 nm, 1376.9 nm, 1407.1 nm, 1410.9 nm and 1414.6 nm. The use of these selected wavelengths increases the overall classification accuracy from 79.19% and a Kappa value of 0.7201 when using all wavelengths to 88.59% and a Kappa of 0.8524 when the selected wavelengths were used. Based on our relatively high accuracies and ease of use, it is worth considering the GRRF method for reducing the high dimensionality of spectroscopy data. However, this assertion should receive considerable additional testing and comparison before it is accepted as a substitute for reliable high dimensionality reduction.
Betweenness-based algorithm for a partition scale-free graph
Zhang Bai-Da; Wu Jun-Jie; Tang Yu-Hua; Zhou Jing
2011-01-01
Many real-world networks are found to be scale-free.However,graph partition technology,as a technology capable of parallel computing,performs poorly when scale-free graphs are provided.The reason for this is that traditional partitioning algorithms are designed for random networks and regular networks,rather than for scale-free networks.Multilevel graph-partitioning algorithms are currently considered to be the state of the art and are used extensively.In this paper,we analyse the reasons why traditional multilevel graph-partitioning algorithms perform poorly and present a new multilevel graph-partitioning paradigm,top down partitioning,which derives its name from the comparison with the traditional bottom-up partitioning.A new multilevel partitioning algorithm,named betweenness-bnsed partitioning algorithm,is also presented as an implementation of top-down partitioning paradigm.An experimental evaluation of seven different real-world scale-free networks shows that the betweenness-based partitioning algorithm significantly outperforms the existing state-of-the-art approaches.
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T. (University of New Mexico)
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Spectral clustering and biclustering learning large graphs and contingency tables
Bolla, Marianna
2013-01-01
Explores regular structures in graphs and contingency tables by spectral theory and statistical methods This book bridges the gap between graph theory and statistics by giving answers to the demanding questions which arise when statisticians are confronted with large weighted graphs or rectangular arrays. Classical and modern statistical methods applicable to biological, social, communication networks, or microarrays are presented together with the theoretical background and proofs. This book is suitable for a one-semester course for graduate students in data mining, mult
Visibility graph analysis on heartbeat dynamics of meditation training
Jiang, Sen; Bian, Chunhua; Ning, Xinbao; Ma, Qianli D. Y.
2013-06-01
We apply the visibility graph analysis to human heartbeat dynamics by constructing the complex networks of heartbeat interval time series and investigating the statistical properties of the network before and during chi and yoga meditation. The experiment results show that visibility graph analysis can reveal the dynamical changes caused by meditation training manifested as regular heartbeat, which is closely related to the adjustment of autonomous neural system, and visibility graph analysis is effective to evaluate the effect of meditation.
Hansen, Lars Kai; Rasmussen, Carl Edward; Svarer, C.
1994-01-01
Regularization, e.g., in the form of weight decay, is important for training and optimization of neural network architectures. In this work the authors provide a tool based on asymptotic sampling theory, for iterative estimation of weight decay parameters. The basic idea is to do a gradient descent...... in the estimated generalization error with respect to the regularization parameters. The scheme is implemented in the authors' Designer Net framework for network training and pruning, i.e., is based on the diagonal Hessian approximation. The scheme does not require essential computational overhead in addition...... to what is needed for training and pruning. The viability of the approach is demonstrated in an experiment concerning prediction of the chaotic Mackey-Glass series. The authors find that the optimized weight decays are relatively large for densely connected networks in the initial pruning phase, while...
Kirkpatrick, Bonnie; Reshef, Yakir; Finucane, Hilary; Jiang, Haitao; Zhu, Binhai; Karp, Richard M
2012-09-01
Pedigree graphs, or family trees, are typically constructed by an expensive process of examining genealogical records to determine which pairs of individuals are parent and child. New methods to automate this process take as input genetic data from a set of extant individuals and reconstruct ancestral individuals. There is a great need to evaluate the quality of these methods by comparing the estimated pedigree to the true pedigree. In this article, we consider two main pedigree comparison problems. The first is the pedigree isomorphism problem, for which we present a linear-time algorithm for leaf-labeled pedigrees. The second is the pedigree edit distance problem, for which we present (1) several algorithms that are fast and exact in various special cases, and (2) a general, randomized heuristic algorithm. In the negative direction, we first prove that the pedigree isomorphism problem is as hard as the general graph isomorphism problem, and that the sub-pedigree isomorphism problem is NP-hard. We then show that the pedigree edit distance problem is APX-hard in general and NP-hard on leaf-labeled pedigrees. We use simulated pedigrees to compare our edit-distance algorithms to each other as well as to a branch-and-bound algorithm that always finds an optimal solution.
Spectral recognition of graphs
Cvetković Dragoš
2012-01-01
Full Text Available At some time, in the childhood of spectral graph theory, it was conjectured that non-isomorphic graphs have different spectra, i.e. that graphs are characterized by their spectra. Very quickly this conjecture was refuted and numerous examples and families of non-isomorphic graphs with the same spectrum (cospectral graphs were found. Still some graphs are characterized by their spectra and several mathematical papers are devoted to this topic. In applications to computer sciences, spectral graph theory is considered as very strong. The benefit of using graph spectra in treating graphs is that eigenvalues and eigenvectors of several graph matrices can be quickly computed. Spectral graph parameters contain a lot of information on the graph structure (both global and local including some information on graph parameters that, in general, are computed by exponential algorithms. Moreover, in some applications in data mining, graph spectra are used to encode graphs themselves. The Euclidean distance between the eigenvalue sequences of two graphs on the same number of vertices is called the spectral distance of graphs. Some other spectral distances (also based on various graph matrices have been considered as well. Two graphs are considered as similar if their spectral distance is small. If two graphs are at zero distance, they are cospectral. In this sense, cospectral graphs are similar. Other spectrally based measures of similarity between networks (not necessarily having the same number of vertices have been used in Internet topology analysis, and in other areas. The notion of spectral distance enables the design of various meta-heuristic (e.g., tabu search, variable neighbourhood search algorithms for constructing graphs with a given spectrum (spectral graph reconstruction. Several spectrally based pattern recognition problems appear in many areas (e.g., image segmentation in computer vision, alignment of protein-protein interaction networks in bio
Pristine transfinite graphs and permissive electrical networks
Zemanian, Armen H
2001-01-01
A transfinite graph or electrical network of the first rank is obtained conceptually by connecting conventionally infinite graphs and networks together at their infinite extremities. This process can be repeated to obtain a hierarchy of transfiniteness whose ranks increase through the countable ordinals. This idea, which is of recent origin, has enriched the theories of graphs and networks with radically new constructs and research problems. The book provides a more accessible introduction to the subject that, though sacrificing some generality, captures the essential ideas of transfiniteness for graphs and networks. Thus, for example, some results concerning discrete potentials and random walks on transfinite networks can now be presented more concisely. Conversely, the simplifications enable the development of many new results that were previously unavailable. Topics and features: *A simplified exposition provides an introduction to transfiniteness for graphs and networks.*Various results for conventional g...
Probability distributions with summary graph structure
Wermuth, Nanny
2010-01-01
A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class are multivariate regression chain graphs. They describe the independences of stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that then result after possible marginalising or conditioning, we use summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences which remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to de...
Pancyclic and bipancyclic graphs
George, John C; Wallis, W D
2016-01-01
This book is focused on pancyclic and bipancyclic graphs and is geared toward researchers and graduate students in graph theory. Readers should be familiar with the basic concepts of graph theory, the definitions of a graph and of a cycle. Pancyclic graphs contain cycles of all possible lengths from three up to the number of vertices in the graph. Bipartite graphs contain only cycles of even lengths, a bipancyclic graph is defined to be a bipartite graph with cycles of every even size from 4 vertices up to the number of vertices in the graph. Cutting edge research and fundamental results on pancyclic and bipartite graphs from a wide range of journal articles and conference proceedings are composed in this book to create a standalone presentation. The following questions are highlighted through the book: - What is the smallest possible number of edges in a pancyclic graph with v vertices? - When do pancyclic graphs exist with exactly one cycle of every possible length? - What is the smallest possible number of...
Assessing statistical significance in causal graphs
Chindelevitch Leonid
2012-02-01
Full Text Available Abstract Background Causal graphs are an increasingly popular tool for the analysis of biological datasets. In particular, signed causal graphs--directed graphs whose edges additionally have a sign denoting upregulation or downregulation--can be used to model regulatory networks within a cell. Such models allow prediction of downstream effects of regulation of biological entities; conversely, they also enable inference of causative agents behind observed expression changes. However, due to their complex nature, signed causal graph models present special challenges with respect to assessing statistical significance. In this paper we frame and solve two fundamental computational problems that arise in practice when computing appropriate null distributions for hypothesis testing. Results First, we show how to compute a p-value for agreement between observed and model-predicted classifications of gene transcripts as upregulated, downregulated, or neither. Specifically, how likely are the classifications to agree to the same extent under the null distribution of the observed classification being randomized? This problem, which we call "Ternary Dot Product Distribution" owing to its mathematical form, can be viewed as a generalization of Fisher's exact test to ternary variables. We present two computationally efficient algorithms for computing the Ternary Dot Product Distribution and investigate its combinatorial structure analytically and numerically to establish computational complexity bounds. Second, we develop an algorithm for efficiently performing random sampling of causal graphs. This enables p-value computation under a different, equally important null distribution obtained by randomizing the graph topology but keeping fixed its basic structure: connectedness and the positive and negative in- and out-degrees of each vertex. We provide an algorithm for sampling a graph from this distribution uniformly at random. We also highlight theoretical
Frobenius circulant graphs of valency six
Thomson, Alison
2012-01-01
A Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. It is well known that such a group is a semidirect product $G = K \\rtimes H$, where $K$ is a nilpotent normal subgroup. A first kind $G$-Frobenius graph is a Cayley graph on $K$ whose connection set is an $H$-orbit $a^H$ for some $a \\in K$ with $ = K$, where $H$ is of even order or $a$ is an involution. Such graphs admit 'perfect' routing and gossiping schemes in some sense. Because of this and the importance of circulant graphs in network design, it is desirable to classify first kind Frobenius circulant graphs. In this paper we classify all 6-valent first kind Frobenius circulant graphs. We give optimal gossiping, routing and broadcasting schemes for such graphs and compute their forwarding indices, Wiener indices and minimum gossip time. We also prove that the broadcasting time of any 6-valent first kind Frobenius circulant is equal to its diameter plus two or three, indicati...
Compeau, Phillip E.C
2011-01-01
We consider four families of pancake graphs, which are Cayley graphs, whose vertex sets are either the symmetric group on n objects or the hyperoctahedral group on n objects and whose generating sets...
2013-01-01
on Facebook , one would like to detect tightly connected communities, which is useful for subsequent tasks like customized recommendation and... advertisement . Graphs in modern applications have several characteristics that complicate graph clustering: • Small density gap: the edge density across
Shuai, Hong-Han; Yu, Philip S; Shen, Chih-Ya; Chen, Ming-Syan
2013-01-01
The importance of graph mining has been widely recognized thanks to a large variety of applications in many areas, while real datasets always play important roles to examine the solution quality and efficiency of a graph mining algorithm. Nevertheless, the size of a real dataset is usually fixed and constrained according to the available resources, such as the efforts to crawl an on-line social network. In this case, employing a synthetic graph generator is a possible way to generate a massive graph (e.g., billions nodes) for evaluating the scalability of an algorithm, and current popular statistical graph generators are properly designed to maintain statistical metrics such as total node degree, degree distribution, diameter, and clustering coefficient of the original social graphs. Nevertheless, in addition to the above metrics, recent studies on graph mining point out that graph frequent patterns are also important to provide useful implications for the corresponding social networking applications, but thi...
Baillie, C F; Johnston, D A; Plechác, P
1995-01-01
In a recent paper we found strong evidence from simulations that the Ising antiferromagnet on ``thin'' random graphs - Feynman diagrams - displayed a mean-field spin glass transition. The intrinsic interest of considering such random graphs is that they give mean field results without long range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice, or in previous replica calculations. We then investigate numerically spin glasses with a plus or minus J bond distribution fo rthe Ising and Q=3,3,10,50 state Potts models, paying particular attention t...
Coxeter, H S M
1973-01-01
Polytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H. S. M. Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information
WANG; Wei
2001-01-01
［1］ Nagaev, A. V., Integral limit theorems for large deviations when Cramer's condition is not fulfilled I, II, Theory Prob. Appl., 1969, 14: 51-64, 193-208.［2］ Nagaev, A. V., Limit theorems for large deviations where Cramer's conditions are violated (In Russian), Izv. Akad. Nauk USSR Ser., Fiz-Mat Nauk., 1969, 7: 17.［3］ Heyde, C. C., A contribution to the theory of large deviations for sums of independent random variables, Z. Wahrscheinlichkeitsth, 1967, 7: 303.［4］ Heyde, C. C., On large deviation probabilities for sums of random variables which are not attracted to the normal law, Ann. Math. Statist., 1967, 38: 1575.［5］ Heyde, C. C., On large deviation probabilities in the case of attraction to a nonnormal stable law, Sanky, 1968, 30: 253.［6］ Nagaev, S. V., Large deviations for sums of independent random variables, in Sixth Prague Conf. on Information Theory, Random Processes and Statistical Decision Functions, Prague: Academic, 1973, 657674.［7］ Nagaev, S. V., Large deviations of sums of independent random variables, Ann. Prob., 1979, 7: 745.［8］ Embrechts, P., Klüppelberg, C., Mikosch, T., Modelling Extremal Events for Insurance and Finance, Berlin-Heidelberg: Springer-Verlag, 1997.［9］ Cline, D. B. H., Hsing, T., Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, Preprint, Texas A&M University, 1991.［10］ Klüppelberg, C., Mikosch, T., Large deviations of heavy-tailed random sums with applications to insurance and finance, J. Appl. Prob., 1997, 34: 293.
On prime non-primitive von Neumann regular algebras
Abrams, Gene; Rangaswamy, Kulumani M
2011-01-01
Let $E$ be any directed graph, and $K$ any field. We classify those graphs $E$ for which the Leavitt path algebra $L_K(E)$ is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.
Evolutionary Graph Drawing Algorithms
Huang Jing-wei; Wei Wen-fang
2003-01-01
In this paper, graph drawing algorithms based on genetic algorithms are designed for general undirected graphs and directed graphs. As being shown, graph drawing algorithms designed by genetic algorithms have the following advantages: the frames of the algorithms are unified, the method is simple, different algorithms may be attained by designing different objective functions, therefore enhance the reuse of the algorithms. Also, aesthetics or constrains may be added to satisfy different requirements.
On molecular graph comparison.
Melo, Jenny A; Daza, Edgar
2011-06-01
Since the last half of the nineteenth century, molecular graphs have been present in several branches of chemistry. When used for molecular structure representation, they have been compared after mapping the corresponding graphs into mathematical objects. However, direct molecular comparison of molecular graphs is a research field less explored. The goal of this mini-review is to show some distance and similarity coefficients which were proposed to directly compare molecular graphs or which could be useful to do so.
Ellens, W.; Spieksma, F.M.; Mieghem, P. van; Jamakovic, A.; Kooij, R.E.
2011-01-01
This paper studies an interesting graph measure that we call the effective graph resistance. The notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices. The objective of the
Graphing Inequalities, Connecting Meaning
Switzer, J. Matt
2014-01-01
Students often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and J. Matt Switzer's students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an…
Charles Suffel
1982-01-01
Full Text Available A graph is subeulerian if it is spanned by an eulerian supergraph. Boesch, Suffel and Tindell have characterized the class of subeulerian graphs and determined the minimum number of additional lines required to make a subeulerian graph eulerian.
Loukas, A.
2015-01-01
We have recently seen a surge of research focusing on the processing of graph data. The emerging field of signal processing on graphs focuses on the extension of classical discrete signal processing techniques to the graph setting. Arguably, the greatest breakthrough of the field has been the extens
Graphs of groups on surfaces interactions and models
White, AT
2001-01-01
The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on sufaces. Automorphism groups of both graphs and maps are studied. In addition connections are made to other areas of mathematics, such as hypergraphs, block designs, finite geometries, and finite fields. There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English
Multiple Structure-View Learning for Graph Classification.
Wu, Jia; Pan, Shirui; Zhu, Xingquan; Zhang, Chengqi; Yu, Philip S
2017-09-20
Many applications involve objects containing structure and rich content information, each describing different feature aspects of the object. Graph learning and classification is a common tool for handling such objects. To date, existing graph classification has been limited to the single-graph setting with each object being represented as one graph from a single structure-view. This inherently limits its use to the classification of complicated objects containing complex structures and uncertain labels. In this paper, we advance graph classification to handle multigraph learning for complicated objects from multiple structure views, where each object is represented as a bag containing several graphs and the label is only available for each graph bag but not individual graphs inside the bag. To learn such graph classification models, we propose a multistructure-view bag constrained learning (MSVBL) algorithm, which aims to explore substructure features across multiple structure views for learning. By enabling joint regularization across multiple structure views and enforcing labeling constraints at the bag and graph levels, MSVBL is able to discover the most effective substructure features across all structure views. Experiments and comparisons on real-world data sets validate and demonstrate the superior performance of MSVBL in representing complicated objects as multigraph for classification, e.g., MSVBL outperforms the state-of-the-art multiview graph classification and multiview multi-instance learning approaches.
Yoder, Sharon K.
This book discusses four kinds of graphs that are taught in mathematics at the middle school level: pictographs, bar graphs, line graphs, and circle graphs. The chapters on each of these types of graphs contain information such as starting, scaling, drawing, labeling, and finishing the graphs using "LogoWriter." The final chapter of the book…
Shab-Bidar Sakineh
2011-11-01
Full Text Available Abstract Background Endothelial dysfunction has been proposed as the underlying cause of diabetic angiopathy that eventually leads to cardiovascular disease, the major cause of death in diabetes. We recently demonstrated the ameliorating effect of regular vitamin D intake on the glycemic status of patients with type 2 diabetes (T2D. In this study, the effects of improvement of vitamin D status on glycemic status, lipid profile and endothelial biomarkers in T2D subjects were investigated. Methods Subjects with T2D were randomly allocated to one of the two groups to receive either plain yogurt drink (PYD; containing 170 mg calcium and no vitamin D/250 mL, n1 = 50 or vitamin D3-fortified yogurt drink (FYD; containing 170 mg calcium and 500 IU/250 mL, n2 = 50 twice a day for 12 weeks. Anthropometric measures, glycemic status, lipid profile, body fat mass (FM and endothelial biomarkers including serum endothelin-1, E-selectin and matrix metalloproteinase (MMP-9 were evaluated at the beginning and after the 12-week intervention period. Results The intervention resulted in a significant improvement in fasting glucose, the Quantitative Insulin Check Index (QUICKI, glycated hemoglobin (HbA1c, triacylglycerols, high-density lipoprotein cholesterol (HDL-C, endothelin-1, E-selectin and MMP-9 in FYD compared to PYD (P P = 0.028; -3.8 ± 7.3 versus 0.95 ± 8.3, P = 0.003 and -2.3 ± 3.7 versus 0.44 ± 7.1 ng/mL, respectively, P P = 0.009 and P = 0.005, respectively but disappeared for E-selectin (P = 0.092. On the contrary, after controlling for serum 25(OHD, the differences disappeared for endothelin-1(P = 0.066 and MMP-9 (P = 0.277 but still remained significant for E-selectin (P = 0.011. Conclusions Ameliorated vitamin D status was accompanied by improved glycemic status, lipid profile and endothelial biomarkers in T2D subjects. Our findings suggest both direct and indirect ameliorating effects of vitamin D on the endothelial biomarkers. Trial registration
Gross, Jonathan L
2003-01-01
The Handbook of Graph Theory is the most comprehensive single-source guide to graph theory ever published. Best-selling authors Jonathan Gross and Jay Yellen assembled an outstanding team of experts to contribute overviews of more than 50 of the most significant topics in graph theory-including those related to algorithmic and optimization approaches as well as ""pure"" graph theory. They then carefully edited the compilation to produce a unified, authoritative work ideal for ready reference.Designed and edited with non-experts in mind, the Handbook of Graph Theory makes information easy to fi
Wong, Pak C.; Mackey, Patrick S.; Perrine, Kenneth A.; Foote, Harlan P.; Thomas, James J.
2008-12-23
Methods for visualizing a graph by automatically drawing elements of the graph as labels are disclosed. In one embodiment, the method comprises receiving node information and edge information from an input device and/or communication interface, constructing a graph layout based at least in part on that information, wherein the edges are automatically drawn as labels, and displaying the graph on a display device according to the graph layout. In some embodiments, the nodes are automatically drawn as labels instead of, or in addition to, the label-edges.
Perfect state transfer, integral circulants and join of graphs
Angeles-Canul, Ricardo Javier; Opperman, Michael C; Paribello, Christopher C; Russell, Matthew C; Tamon, Christino
2009-01-01
We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\\v{s}i\\'{c} et al \\cite{bps09,bp09} and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\\textsc{ICG}_{n}(\\{2,n/2^{b}\\} \\cup Q)$ has perfect state transfer, where $b \\in \\{1,2\\}$, $n$ is a multiple of 16 and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil \\cite{godsil08}.
Caetano, Tiberio S; Cheng, Li; Le, Quoc V; Smola, Alex J
2008-01-01
As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the `labels' are ma...
Harrison, JM; Robbins, JM; 10.1098/rspa.2010.0254
2011-01-01
Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molec...
Simplicial complexes of graphs
Jonsson, Jakob
2008-01-01
A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this volume also provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees; it is successfully applied to a large number of graph and digraph complexes.
The genus distributions for a certain type of permutation graphs in orientable surfaces
Rong-xia; HAO; Wei-li; HE; Yan-pei; LIU; Er-ling; WEI
2007-01-01
A circuit is a connected nontrivial 2-regular graph. A graph G is a permutation graph over a circuit C, if G can be obtained from two copies of C by joining these two copies with a perfect matching. In this paper, based on the joint tree method introduced by Liu, the genus polynomials for a certain type of permutation graphs in orientable surfaces are given.
胡细; 王汉兴; 赵飞
2007-01-01
The flooding distance is an important parameter in the design and evaluation of a routing protocol, which is related not only to the delay time in the route discovery, but also to the stability and reliability of the route. In this paper,the average flooding distance (AFD) for a mobile ad hoc network (MANET) in a random graph model was given based on the dynamic source routing (DSR) protocol. The influence of spatial reuse on the AFD was also studied. Compared with that in the model without the spatial reuse, the AFD in the model with the spatial reuse has much smaller value, when the connetivity probability between nodes in the network is small and when the number of reused times is large. This means that the route discovery with the spatial reuse is much more effective.
Clustering with Multi-Layer Graphs: A Spectral Perspective
Dong, Xiaowen; Vandergheynst, Pierre; Nefedov, Nikolai
2011-01-01
Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (users) with different edges (pairwise relationships). In this paper, we address the problem of combining different layers of the multi-layer graph for improved clustering of the vertices compared to using layers independently. We propose two novel methods, which are based on joint matrix factorization and graph regularization framework respectively, to efficiently combine the spectrum of the multiple graph layers, namely the eigenvectors of the graph Laplacian matrices. In each case, the resulting combination, which we call a "joint spectrum" of multiple graphs, is used for clustering the vertices. We evaluate our approaches by simulations with several real world social network datasets. Results demonstrate the superior or competitive performance of the proposed methods over state-of-the-art technique and common baseline methods, such a...
Neural Population Dynamics Modeled by Mean-Field Graphs
Kozma, Robert; Puljic, Marko
2011-09-01
In this work we apply random graph theory approach to describe neural population dynamics. There are important advantages of using random graph theory approach in addition to ordinary and partial differential equations. The mathematical theory of large-scale random graphs provides an efficient tool to describe transitions between high- and low-dimensional spaces. Recent advances in studying neural correlates of higher cognition indicate the significance of sudden changes in space-time neurodynamics, which can be efficiently described as phase transitions in the neuropil medium. Phase transitions are rigorously defined mathematically on random graph sequences and they can be naturally generalized to a class of percolation processes called neuropercolation. In this work we employ mean-field graphs with given vertex degree distribution and edge strength distribution. We demonstrate the emergence of collective oscillations in the style of brains.
Jampani, Krishnam Raju
2010-01-01
In a recent paper, we introduced the simultaneous representation problem (defined for any graph class C) and studied the problem for chordal, comparability and permutation graphs. For interval graphs, the problem is defined as follows. Two interval graphs G_1 and G_2, sharing some vertices I (and the corresponding induced edges), are said to be `simultaneous interval graphs' if there exist interval representations R_1 and R_2 of G_1 and G_2, such that any vertex of I is mapped to the same interval in both R_1 and R_2. Equivalently, G_1 and G_2 are simultaneous interval graphs if there exist edges E' between G_1-I and G_2-I such that G_1 \\cup G_2 \\cup E' is an interval graph. Simultaneous representation problems are related to simultaneous planar embeddings, and have applications in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated versi...
Sparse graphs are not flammable
Prałat, Paweł
2012-01-01
In this paper, we consider the following \\emph{$k$-many firefighter problem} on a finite graph $G=(V,E)$. Suppose that a fire breaks out at a given vertex $v \\in V$. In each subsequent time unit, a firefighter protects $k$ vertices which are not yet on fire, and then the fire spreads to all unprotected neighbours of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate $\\rho(G)$ of $G$ is defined as the expected percentage of vertices that can be saved when a fire breaks out at a random vertex of $G$. Let $\\tau_k = k+2-\\frac {1}{k+2}$. We show that for any $\\eps >0$ and $k \\ge 2$, each graph $G$ on $n$ vertices with at most $(\\tau_k-\\eps)n$ edges is not flammable; that is, $\\rho(G) > \\frac {2\\eps}{5\\tau_k} > 0$. Moreover, a construction of a family of flammable random graphs is proposed to show that the constant $\\tau_k$ cannot be improved.
Classifying cubic symmetric graphs of order 10p or 10p2
FENG; Yanquan; KWAK; Jin; Ho
2006-01-01
A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥ 1 are classified when the fibre-preserving automorphism groups act arc-transitively.As an application of these results, all s-regular cubic graphs of order 10p or 10p2 are also classified for each s ≥ 1 and each prime p, of which the proof depends on the classification of finite simple groups.
Kernel-Based Reconstruction of Graph Signals
Romero, Daniel; Ma, Meng; Giannakis, Georgios B.
2017-02-01
A number of applications in engineering, social sciences, physics, and biology involve inference over networks. In this context, graph signals are widely encountered as descriptors of vertex attributes or features in graph-structured data. Estimating such signals in all vertices given noisy observations of their values on a subset of vertices has been extensively analyzed in the literature of signal processing on graphs (SPoG). This paper advocates kernel regression as a framework generalizing popular SPoG modeling and reconstruction and expanding their capabilities. Formulating signal reconstruction as a regression task on reproducing kernel Hilbert spaces of graph signals permeates benefits from statistical learning, offers fresh insights, and allows for estimators to leverage richer forms of prior information than existing alternatives. A number of SPoG notions such as bandlimitedness, graph filters, and the graph Fourier transform are naturally accommodated in the kernel framework. Additionally, this paper capitalizes on the so-called representer theorem to devise simpler versions of existing Thikhonov regularized estimators, and offers a novel probabilistic interpretation of kernel methods on graphs based on graphical models. Motivated by the challenges of selecting the bandwidth parameter in SPoG estimators or the kernel map in kernel-based methods, the present paper further proposes two multi-kernel approaches with complementary strengths. Whereas the first enables estimation of the unknown bandwidth of bandlimited signals, the second allows for efficient graph filter selection. Numerical tests with synthetic as well as real data demonstrate the merits of the proposed methods relative to state-of-the-art alternatives.
Analysis of Graphs for Digital Preservation Suitability
Cartledge, Charles L
2010-01-01
We investigate the use of autonomically created small-world graphs as a framework for the long term storage of digital objects on the Web in a potentially hostile environment. We attack the classic Erdos - Renyi random, Barab'asi and Albert power law, Watts - Strogatz small world and our Unsupervised Small-World (USW) graphs using different attacker strategies and report their respective robustness. Using different attacker profiles, we construct a game where the attacker is allowed to use a strategy of his choice to remove a percentage of each graph's elements. The graph is then allowed to repair some portion of its self. We report on the number of alternating attack and repair turns until either the graph is disconnected, or the game exceeds the number of permitted turns. Based on our analysis, an attack strategy that focuses on removing the vertices with the highest betweenness value is most advantageous to the attacker. Power law graphs can become disconnected with the removal of a single edge; random gra...
Volkers, Karin M.; Scherder, Erik J. A.
2011-01-01
Background: Physical activity has proven to be beneficial for physical functioning, cognition, depression, anxiety, rest-activity rhythm, quality of life (QoL), activities of daily living (ADL) and pain in older people. The aim of this study is to investigate the effect of walking regularly on physi
Fujie, Futaba
2014-01-01
Covering Walks in Graphs is aimed at researchers and graduate students in the graph theory community and provides a comprehensive treatment on measures of two well studied graphical properties, namely Hamiltonicity and traversability in graphs. This text looks into the famous Kӧnigsberg Bridge Problem, the Chinese Postman Problem, the Icosian Game and the Traveling Salesman Problem as well as well-known mathematicians who were involved in these problems. The concepts of different spanning walks with examples and present classical results on Hamiltonian numbers and upper Hamiltonian numbers of graphs are described; in some cases, the authors provide proofs of these results to illustrate the beauty and complexity of this area of research. Two new concepts of traceable numbers of graphs and traceable numbers of vertices of a graph which were inspired by and closely related to Hamiltonian numbers are introduced. Results are illustrated on these two concepts and the relationship between traceable concepts and...
Dosen, K
2011-01-01
Plural (or multiple-conclusion) cuts are inferences made by applying a structural rule introduced by Gentzen for his sequent formulation of classical logic. As singular (single-conclusion) cuts yield trees, which underlie ordinary natural deduction derivations, so plural cuts yield graphs of a more complicated kind, related to trees, which this paper defines. Besides the inductive definition of these oriented graphs, which is based on sequent systems, a non-inductive, graph-theoretical, combinatorial, definition is given, and to reach that other definition is the main goal of the paper. As trees underlie multicategories, so the graphs of plural cuts underlie polycategories. The graphs of plural cuts are interesting in particular when the plural cuts are appropriate for sequent systems without the structural rule of permutation, and the main body of the paper deals with that matter. It gives a combinatorial characterization of the planarity of the graphs involved.
Velasco, Pedro Pablo Perez
2008-01-01
This book objective is to develop an algebraization of graph grammars. Equivalently, we study graph dynamics. From the point of view of a computer scientist, graph grammars are a natural generalization of Chomsky grammars for which a purely algebraic approach does not exist up to now. A Chomsky (or string) grammar is, roughly speaking, a precise description of a formal language (which in essence is a set of strings). On a more discrete mathematical style, it can be said that graph grammars -- Matrix Graph Grammars in particular -- study dynamics of graphs. Ideally, this algebraization would enforce our understanding of grammars in general, providing new analysis techniques and generalizations of concepts, problems and results known so far.
Arrighi, Pablo
2012-01-01
We generalize the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time - but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalized. The definition we provide for these `causal graph dynamics' is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of Cellular Automata theory carry through: causal graph dynamics admit a characterization as continuous functions and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics. Keywords: Dynamical networks, Boolean network...
D' Azevedo, Ed F [ORNL; Imam, Neena [ORNL
2015-01-01
This document describes the effort to implement the Graph 500 benchmark using OpenSHMEM based on the MPI-2 one-side version. The Graph 500 benchmark performs a breadth-first search in parallel on a large randomly generated undirected graph and can be implemented using basic MPI-1 and MPI-2 one-sided communication. Graph 500 requires atomic bit-wise operations on unsigned long integers but neither atomic bit-wise operations nor OpenSHMEM for unsigned long are available in OpenSHEM. Such needed bit-wise atomic operations and support for unsigned long are implemented using atomic condition swap (CSWAP) on signed long integers. Preliminary results on comparing the OpenSHMEM and MPI-2 one-sided implementations on a Silicon Graphics Incorporated (SGI) cluster and the Cray XK7 are presented.
Buczyńska, Weronika
2010-01-01
We define toric projective model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. Generators of the projective coordinate ring of the models of graphs with one cycle are explicitly described. The models of graphs with the same topological invariants are deformation equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function.
Chartrand, Gary
1984-01-01
Graph theory is used today in the physical sciences, social sciences, computer science, and other areas. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style. Author Gary Chartrand covers the important elementary topics of graph theory and its applications. In addition, he presents a large variety of proofs designed to strengthen mathematical techniques and offers challenging opportunities to have fun with mathematics. Ten major topics - profusely illustrated - include: Mathematical Models, Elementary Concepts of Grap
Creating more effective graphs
Robbins, Naomi B
2012-01-01
A succinct and highly readable guide to creating effective graphs The right graph can be a powerful tool for communicating information, improving a presentation, or conveying your point in print. If your professional endeavors call for you to present data graphically, here's a book that can help you do it more effectively. Creating More Effective Graphs gives you the basic knowledge and techniques required to choose and create appropriate graphs for a broad range of applications. Using real-world examples everyone can relate to, the author draws on her years of experience in gr
Lothian, Josh [ORNL; Powers, Sarah S [ORNL; Sullivan, Blair D [ORNL; Baker, Matthew B [ORNL; Schrock, Jonathan [ORNL; Poole, Stephen W [ORNL
2013-12-01
The benchmarking effort within the Extreme Scale Systems Center at Oak Ridge National Laboratory seeks to provide High Performance Computing benchmarks and test suites of interest to the DoD sponsor. The work described in this report is a part of the effort focusing on graph generation. A previously developed benchmark, SystemBurn, allowed the emulation of dierent application behavior profiles within a single framework. To complement this effort, similar capabilities are desired for graph-centric problems. This report examines existing synthetic graph generator implementations in preparation for further study on the properties of their generated synthetic graphs.
Thomassen, Carsten
2014-01-01
We prove a general result on graph factors modulo k . A special case says that, for each natural number k , every (12k−7)-edge-connected graph with an even number of vertices contains a spanning subgraph in which each vertex has degree congruent to k modulo 2k.......We prove a general result on graph factors modulo k . A special case says that, for each natural number k , every (12k−7)-edge-connected graph with an even number of vertices contains a spanning subgraph in which each vertex has degree congruent to k modulo 2k....
Gelfand, I M; Shnol, E E
2002-01-01
The second in a series of systematic studies by a celebrated mathematician I. M. Gelfand and colleagues, this volume presents students with a well-illustrated sequence of problems and exercises designed to illuminate the properties of functions and graphs. Since readers do not have the benefit of a blackboard on which a teacher constructs a graph, the authors abandoned the customary use of diagrams in which only the final form of the graph appears; instead, the book's margins feature step-by-step diagrams for the complete construction of each graph. The first part of the book employs simple fu
Bradford, Robert; Chmutov, Sergei
2011-01-01
We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ribbon graph from a virtual link whose extended Bollob\\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.
Alberto Apostolico
2009-08-01
Full Text Available The Web Graph is a large-scale graph that does not fit in main memory, so that lossless compression methods have been proposed for it. This paper introduces a compression scheme that combines efficient storage with fast retrieval for the information in a node. The scheme exploits the properties of the Web Graph without assuming an ordering of the URLs, so that it may be applied to more general graphs. Tests on some datasets of use achieve space savings of about 10% over existing methods.
Framings for graph hypersurfaces
Brown, Francis
2013-01-01
We present a method for computing the framing on the cohomology of graph hypersurfaces defined by the Feynman differential form. This answers a question of Bloch, Esnault and Kreimer in the affirmative for an infinite class of graphs for which the framings are Tate motives. Applying this method to the modular graphs of Brown and Schnetz, we find that the Feynman differential form is not of Tate type in general. This finally disproves a folklore conjecture stating that the periods of Feynman integrals of primitive graphs in phi^4 theory factorise through a category of mixed Tate motives.
The many faces of graph dynamics
Pignolet, Yvonne Anne; Roy, Matthieu; Schmid, Stefan; Tredan, Gilles
2017-06-01
The topological structure of complex networks has fascinated researchers for several decades, resulting in the discovery of many universal properties and reoccurring characteristics of different kinds of networks. However, much less is known today about the network dynamics: indeed, complex networks in reality are not static, but rather dynamically evolve over time. Our paper is motivated by the empirical observation that network evolution patterns seem far from random, but exhibit structure. Moreover, the specific patterns appear to depend on the network type, contradicting the existence of a ‘one fits it all’ model. However, we still lack observables to quantify these intuitions, as well as metrics to compare graph evolutions. Such observables and metrics are needed for extrapolating or predicting evolutions, as well as for interpolating graph evolutions. To explore the many faces of graph dynamics and to quantify temporal changes, this paper suggests to build upon the concept of centrality, a measure of node importance in a network. In particular, we introduce the notion of centrality distance, a natural similarity measure for two graphs which depends on a given centrality, characterizing the graph type. Intuitively, centrality distances reflect the extent to which (non-anonymous) node roles are different or, in case of dynamic graphs, have changed over time, between two graphs. We evaluate the centrality distance approach for five evolutionary models and seven real-world social and physical networks. Our results empirically show the usefulness of centrality distances for characterizing graph dynamics compared to a null-model of random evolution, and highlight the differences between the considered scenarios. Interestingly, our approach allows us to compare the dynamics of very different networks, in terms of scale and evolution speed.
USE OF EIGENVECTOR CENTRALITY TO DETECT GRAPH ISOMORPHISM
Natarajan Meghanathan
2015-11-01
Full Text Available Graph Isomorphism is one of the classical problems of graph theory for which no deterministic polynomial-time algorithm is currently known, but has been neither proven to be NP-complete. Several heuristic algorithms have been proposed to determine whether or not two graphs are isomorphic (i.e., structurally the same. In this research, we propose to use the sequence (either the non-decreasing or nonincreasing order of eigenvector centrality (EVC values of the vertices of two graphs as a precursor step to decide whether or not to further conduct tests for graph isomorphism. The eigenvector centrality of a vertex in a graph is a measure of the degree of the vertex as well as the degrees of its neighbors. We hypothesize that if the non-increasing (or non-decreasing order of listings of the EVC values of the vertices of two test graphs are not the same, then the two graphs are not isomorphic. If two test graphs have an identical non-increasing order of the EVC sequence, then they are declared to be potentially isomorphic and confirmed through additional heuristics. We test our hypothesis on random graphs (generated according to the Erdos-Renyi model and we observe the hypothesis to be indeed true: graph pairs that have the same sequence of non-increasing order of EVC values have been confirmed to be isomorphic using the well-known Nauty software.
信怀义
2016-01-01
Web topics are noisy.Users can boost the topic by two ways when they browsing the Internet - add related web pages into the topic and delete unrelated contents from the topic,this process is called web topic boosting.In this paper,we proposed a heterogeneous graph based random walk model to simulate web topic boosting.In this model,heterogeneous graph simulates relationships among web contents and random walk simulates the behavior of web browsing.The random walking produces a probability ranking of objects to a given noisy topic,buy which we can de-termine the boosted topic.The results demonstrate that our model simulates web topic boosting process correctly and completely.In addition,the user studies also demonstrate the effectiveness of this model.%网络话题充满噪声,用户在浏览网络的过程中,逐步添加关联性高的网页到话题中,并从话题中删除关联性低的网页,从而形成纯净话题,这就是话题优化的过程.基于此,本文提出一种基于异质图随机游走的模型来模拟用户优化话题的过程,异质图模拟网络内容的关联性,而随机游走模拟用户浏览网络的过程.对于一个网络话题,该模型能够计算出所有网页属于该话题的概率,根据概率分布就能够判断真正属于该话题的网页,从而模拟网络话题优化的过程.仿真结果证实,本文提出的模型可以准确、完整的模拟话题的优化.而通过用户对优化结果的主观评价,同样证实了模型的有效性.
Caetano, Tibério S; McAuley, Julian J; Cheng, Li; Le, Quoc V; Smola, Alex J
2009-06-01
As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm.
Rensink, Arend; Distefano, Dino
2005-01-01
Graphs may be used as representations of system states in operational semantics and model checking; in the latter context, they are being investigated as an alternative to bit vectors. The corresponding transitions are obtained as derivations from graph production rules. In this paper we propose an
Rensink, Arend; Distefano, Dino; Mukhopadhyay, S.; Roychoudhury, A.; Yang, Z.
2006-01-01
Graphs may be used as representations of system states in operational semantics and model checking; in the latter context, they are being investigated as an alternative to bit vectors. The corresponding transitions are obtained as derivations from graph production rules. In this paper we propose an
Moment graphs and representations
Jantzen, Jens Carsten
2012-01-01
Moment graphs and sheaves on moment graphs are basically combinatorial objects that have be used to describe equivariant intersectiion cohomology. In these lectures we are going to show that they can be used to provide a direct link from this cohomology to the representation theory of simple Lie...
Husfeldt, Thore
2015-01-01
This chapter presents an introduction to graph colouring algorithms. The focus is on vertex-colouring algorithms that work for general classes of graphs with worst-case performance guarantees in a sequential model of computation. The presentation aims to demonstrate the breadth of available...
Behnaz Tolue
2018-07-01
Full Text Available In this paper we introduce stable subgroup graph associated to the group $G$. It is a graph with vertex set all subgroups of $G$ and two distinct subgroups $H_1$ and $H_2$ are adjacent if $St_{G}(H_1\\cap H_2\
Mol, de Maarten; Rensink, Arend; Hunt, James J.
2012-01-01
This paper introduces an approach for adding graph transformation-based functionality to existing JAVA programs. The approach relies on a set of annotations to identify the intended graph structure, as well as on user methods to manipulate that structure, within the user’s own JAVA class declaration
Husfeldt, Thore
2015-01-01
This chapter presents an introduction to graph colouring algorithms. The focus is on vertex-colouring algorithms that work for general classes of graphs with worst-case performance guarantees in a sequential model of computation. The presentation aims to demonstrate the breadth of available techniques and is organized by algorithmic paradigm.
Perepelitsa, VA; Sergienko, [No Value; Kochkarov, AM
1999-01-01
Definitions of prefractal and fractal graphs are introduced, and they are used to formulate mathematical models in different fields of knowledge. The topicality of fractal-graph recognition from the point of view, of fundamental improvement in the efficiency of the solution of algorithmic problems i
Belkhechine, Houmem; Elayech, Mohamed Baka
2010-01-01
Given a (directed) graph G=(V,A), a subset X of V is an interval of G provided that for any a, b\\in X and x\\in V-X, (a,x)\\in A if and only if (b,x)\\in A and (x,a)\\in A if and only if (x,b)\\in A. For example, \\emptyset, \\{x\\} (x \\in V) and V are intervals of G, called trivial intervals. A graph, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A vertex x of an indecomposable graph is critical if G-x is decomposable. In 1993, J.H. Schmerl and W.T. Trotter characterized the indecomposable graphs, all the vertices of which are critical, called critical graphs. In this article, we characterize the indecomposable graphs which admit a single non critical vertex, that we call (-1)-critical graphs.} This gives an answer to a question asked by Y. Boudabbous and P. Ille in a recent article studying the critical vertices in an indecomposable graph.
A. Assari
2016-01-01
Full Text Available In this paper, a graph is assigned to any probability measure on the σ-algebra of Borel sets of a topological space. Using this construction, it is proved that given any number n (finite or infinite there exists a nonregular graph such that its clique, chromatic, and dominating number equals n.
Moment graphs and representations
Jantzen, Jens Carsten
2012-01-01
Moment graphs and sheaves on moment graphs are basically combinatorial objects that have be used to describe equivariant intersectiion cohomology. In these lectures we are going to show that they can be used to provide a direct link from this cohomology to the representation theory of simple Lie...... algebras and of simple algebraic groups. The first section contains some background on equivariant cohomology....
Graphs: Associated Markov Chains
2012-01-01
In this research paper, weighted / unweighted, directed / undirected graphs are associated with interesting Discrete Time Markov Chains (DTMCs) as well as Continuous Time Markov Chains (CTMCs). The equilibrium / transient behaviour of such Markov chains is studied. Also entropy dynamics (Shannon entropy) of certain structured Markov chains is investigated. Finally certain structured graphs and the associated Markov chains are studied.
Kim, Suh-Ryung; Park, Boram; Sano, Yoshio
2011-01-01
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of $G$ is the smallest number of such isolated vertices. In general, it is hard to compute the competition number $k(G)$ for a graph $G$ and it has been one of the important research problems in the study of competition graphs. Opsut~[1982] suggested that the edge clique cover number $\\theta_E(G)$ should be closely related to $k(G)$ by showing $\\theta_E(G)-|V(G)|+2 \\leq k(G) \\leq \\theta_E(G)$. In this note, we study on these inequalities. We first show that for any positive integer $m$ satisfying $2 \\leq m \\leq |V(G)|$, there is a graph $G$ satisfying $k(G)=\\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\\...
Derandomization of Online Assignment Algorithms for Dynamic Graphs
Sahai, Ankur
2011-01-01
This paper analyzes different online algorithms for the problem of assigning weights to edges in a fully-connected bipartite graph that minimizes the overall cost while satisfying constraints. Edges in this graph may disappear and reappear over time. Performance of these algorithms is measured using simulations. This paper also attempts to derandomize the randomized online algorithm for this problem.
Topological structure of dictionary graphs
Fukś, Henryk; Krzemiński, Mark
2009-09-01
We investigate the topological structure of the subgraphs of dictionary graphs constructed from WordNet and Moby thesaurus data. In the process of learning a foreign language, the learner knows only a subset of all words of the language, corresponding to a subgraph of a dictionary graph. When this subgraph grows with time, its topological properties change. We introduce the notion of the pseudocore and argue that the growth of the vocabulary roughly follows decreasing pseudocore numbers—that is, one first learns words with a high pseudocore number followed by smaller pseudocores. We also propose an alternative strategy for vocabulary growth, involving decreasing core numbers as opposed to pseudocore numbers. We find that as the core or pseudocore grows in size, the clustering coefficient first decreases, then reaches a minimum and starts increasing again. The minimum occurs when the vocabulary reaches a size between 103 and 104. A simple model exhibiting similar behavior is proposed. The model is based on a generalized geometric random graph. Possible implications for language learning are discussed.
Safe cooperative robot dynamics on graphs
Ghrist, Robert; Koditschek, Daniel
2000-01-01
This paper initiates the use of vector fields to design, optimize, and implement reactive schedules for safe cooperative robot patterns on planar graphs. We consider Automated Guided Vehicles (AGV's) operating upon a predefined network of pathways. In contrast to the case of locally Euclidean configuration spaces, regularization of collisions is no longer a local procedure, and issues concerning the global topology of configuration spaces must be addressed. The focus of the present inquiry is...
Horyniak, Danielle; Dietze, Paul; Lenton, Simon; Alati, Rosa; Bruno, Raimondo; Matthews, Allison; Breen, Courtney; Burns, Lucy
2017-07-01
Driving following illicit drug consumption ('drug-driving') is a potential road safety risk. Roadside drug testing (RDT) is conducted across Australia with the dual aims of prosecuting drivers with drugs in their system and deterring drug-driving. We examined trends over time in self-reported past six-month drug-driving among sentinel samples of regular drug users and assessed the impact of experiences of RDT on drug-driving among these participants. Data from 1913 people who inject drugs (PWID) and 3140 regular psychostimulant users (RPU) who were first-time participants in a series of repeat cross-sectional sentinel studies conducted in Australian capital cities from 2007 to 2013 and reported driving in the past six months were analysed. Trends over time were assessed using the χ(2) test for trend. Multivariable logistic regressions assessed the relationship between experiences of RDT and recent drug-driving, adjusting for survey year, jurisdiction of residence and socio-demographic and drug use characteristics. The percentage of participants reporting recent (past six months) drug-driving decreased significantly over time among both samples (PWID: 83% [2007] vs. 74% [2013], pdrug-driving remained prevalent. Lifetime experience of RDT increased significantly over time (PWID: 6% [2007] vs. 32% [2013], pdrug-driving among either PWID or RPU. Although there is some evidence that drug-driving among key risk groups of regular drug users is declining in Australia, possibly reflecting a general deterrent effect of RDT, experiencing RDT appears to have no specific deterrent effect on drug-driving. Further intervention, with a particular focus on changing attitudes towards drug-driving, may be needed to further reduce this practice among these groups. Copyright © 2017 Elsevier Ltd. All rights reserved.
A Graph Search Heuristic for Shortest Distance Paths
Chow, E
2005-03-24
This paper presents a heuristic for guiding A* search for finding the shortest distance path between two vertices in a connected, undirected, and explicitly stored graph. The heuristic requires a small amount of data to be stored at each vertex. The heuristic has application to quickly detecting relationships between two vertices in a large information or knowledge network. We compare the performance of this heuristic with breadth-first search on graphs with various topological properties. The results show that one or more orders of magnitude improvement in the number of vertices expanded is possible for large graphs, including Poisson random graphs.
Subgraph detection using graph signals
Chepuri, Sundeep Prabhakar
2017-03-06
In this paper we develop statistical detection theory for graph signals. In particular, given two graphs, namely, a background graph that represents an usual activity and an alternative graph that represents some unusual activity, we are interested in answering the following question: To which of the two graphs does the observed graph signal fit the best? To begin with, we assume both the graphs are known, and derive an optimal Neyman-Pearson detector. Next, we derive a suboptimal detector for the case when the alternative graph is not known. The developed theory is illustrated with numerical experiments.
Information Spreading in Stationary Markovian Evolving Graphs
Clementi, Andrea; Pasquale, Francesco; Silvestri, Riccardo
2011-01-01
Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the "stationary phase" by analyzing the completion time of the "flooding mechanism". We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs. "Geometric Markovian evolving graphs" where the Markovian behaviour is yielded by "n" mobile radio stations, with fixed transmission radius, that perform independent random walks over a square region of the plane. "Edge-Markovian evolving graphs" where the probability of existence of any edge at time "t" depends on the existence (or not) of the same edge at time "t-1". In both cases, the obtained upper...
QAPV: a polynomial invariant for graph isomorphism testing
Valdir Agustinho de Melo
2013-08-01
Full Text Available To each instance of the Quadratic Assignment Problem (QAP a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.
Undirected Graph Languages%无向图语言
师海忠
2011-01-01
无向图是图论中的基本概念,图半群是1991年提出的一个概念,形式语言与自动机理论是计算机科学与技术科学的重要基础理论.借助无向图和图半群,提出了无向图语言的概念,并研究了无向图语言的一个子类--平面图语言,给出了如下结论:一个无向图语言是平面图语言当且仅当它不包含K5语言或K3.3语言的剖分图语言.另外提出了几个开问题,其中之一是无向图语言与正则语言、上下文无关语言、上下文有关语言以及短语结构语言有何关系?%Undirected graph is a elementary concept of graph theory. Graph semigroup is a concept proposed in 1991. A concept of undirected graph language was proposed with undirected graph and graph semigroup. Furthermore, a subclass of undirected graph language, plane graph language was studied. It is proved that a graph language is a plane graph language if and only if it contains no subdivision graph language of Ks language or K3.3 language. In addition, four open problems were proposed. One of them: what is the relations between undirected graph language and regular language,context free language, context sensitive language, phrase structure language, respectively.
On two energy-like invariants of line graphs and related graph operations
Xiaodan Chen
2016-02-01
Full Text Available Abstract For a simple graph G of order n, let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 $\\mu_{1}\\geq\\mu_{2}\\geq\\cdots\\geq\\mu_{n}=0$ be its Laplacian eigenvalues, and let q 1 ≥ q 2 ≥ ⋯ ≥ q n ≥ 0 $q_{1}\\geq q_{2}\\geq\\cdots\\geq q_{n}\\geq0$ be its signless Laplacian eigenvalues. The Laplacian-energy-like invariant and incidence energy of G are defined as, respectively, LEL ( G = ∑ i = 1 n − 1 μ i and IE ( G = ∑ i = 1 n q i . $$\\mathit{LEL}(G=\\sum_{i=1}^{n-1}\\sqrt{ \\mu_{i}} \\quad\\mbox{and}\\quad \\mathit {IE}(G=\\sum_{i=1}^{n} \\sqrt{q_{i}}. $$ In this paper, we present some new upper and lower bounds on LEL and IE of line graph, subdivision graph, para-line graph and total graph of a regular graph, some of which improve previously known results. The main tools we use here are the Cauchy-Schwarz inequality and the Ozeki inequality.
Modeling and Analysis of Time-Varying Graphs
Basu, Prithwish; Ramanathan, Ram; Johnson, Matthew P
2010-01-01
We live in a world increasingly dominated by networks -- communications, social, information, biological etc. A central attribute of many of these networks is that they are dynamic, that is, they exhibit structural changes over time. While the practice of dynamic networks has proliferated, we lag behind in the fundamental, mathematical understanding of network dynamism. Existing research on time-varying graphs ranges from preliminary algorithmic studies (e.g., Ferreira's work on evolving graphs) to analysis of specific properties such as flooding time in dynamic random graphs. A popular model for studying dynamic graphs is a sequence of graphs arranged by increasing snapshots of time. In this paper, we study the fundamental property of reachability in a time-varying graph over time and characterize the latency with respect to two metrics, namely store-or-advance latency and cut-through latency. Instead of expected value analysis, we concentrate on characterizing the exact probability distribution of routing l...
Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs.
Yilun Shang
Full Text Available Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.
Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs.
Shang, Yilun
2015-01-01
Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.
Model Selection Framework for Graph-based data
Caceres, Rajmonda S; Schmidt, Matthew C; Miller, Benjamin A; Campbell, William M
2016-01-01
Graphs are powerful abstractions for capturing complex relationships in diverse application settings. An active area of research focuses on theoretical models that define the generative mechanism of a graph. Yet given the complexity and inherent noise in real datasets, it is still very challenging to identify the best model for a given observed graph. We discuss a framework for graph model selection that leverages a long list of graph topological properties and a random forest classifier to learn and classify different graph instances. We fully characterize the discriminative power of our approach as we sweep through the parameter space of two generative models, the Erdos-Renyi and the stochastic block model. We show that our approach gets very close to known theoretical bounds and we provide insight on which topological features play a critical discriminating role.
Niedzialomski Amanda
2016-11-01
Full Text Available For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u − f(v| ≥ k + 1 − d(u, v. We consider k-radio labelings of G when k = diam(G. In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful Hamming graphs. The main result shows that the Cartesian product of t copies of a complete graph is radio graceful for certain t. Graphs of this form provide infinitely many examples of radio graceful graphs of arbitrary diameter. We also show that these graphs are not radio graceful for large t.
Bidimensionality and Geometric Graphs
Fomin, Fedor V; Saurabh, Saket
2011-01-01
In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and...
Force-Directed Method in Mirror Frames for Graph Drawing
Jing Lee
2010-11-01
Full Text Available The most widely used algorithms for graph drawing are force-directed algorithms. We should modify a hybrid force model that is coupling a traditional spring force model and a novel repulsive force model will be proposed to solve the graph drawing problems in 2-D space. Especially, regular triangle drawing frame can be applied to binary tree drawing problems that on an important contribution to computer science. And apply circle drawing frame to normal graph drawing problems, we get satisfactory and aesthetic criteria graphics.
Gross, Jonathan L; Zhang, Ping
2013-01-01
In the ten years since the publication of the best-selling first edition, more than 1,000 graph theory papers have been published each year. Reflecting these advances, Handbook of Graph Theory, Second Edition provides comprehensive coverage of the main topics in pure and applied graph theory. This second edition-over 400 pages longer than its predecessor-incorporates 14 new sections. Each chapter includes lists of essential definitions and facts, accompanied by examples, tables, remarks, and, in some cases, conjectures and open problems. A bibliography at the end of each chapter provides an ex
Bollobas, Bela
2004-01-01
The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory.Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. A
Asymptote Misconception on Graphing Functions: Does Graphing Software Resolve It?
Öçal, Mehmet Fatih
2017-01-01
Graphing function is an important issue in mathematics education due to its use in various areas of mathematics and its potential roles for students to enhance learning mathematics. The use of some graphing software assists students' learning during graphing functions. However, the display of graphs of functions that students sketched by hand may…
Constructing regular ultrafilters from a model-theoretic point of view
Malliaris, M
2012-01-01
This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\\lcf(\\aleph_0, \\de)$ of $\\aleph_0$ modulo $\\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected by non-low theories. Assuming $\\kappa > \\aleph_0$ is measurable, we construct a regular ultrafilter on $\\lambda \\geq 2^\\kappa$ which is flexible (thus: ok) but not good, and which moreover has large $\\lcf(\\aleph_0)$ but does not even saturate models of the random graph. We prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultr...
Saboori, Shadi; Greene, Leslie E; Moe, Christine L; Freeman, Matthew C; Caruso, Bethany A; Akoko, Daniel; Rheingans, Richard D
2013-10-01
We assessed whether supplying soap to primary schools on a regular basis increased pupil hand washing and decreased Escherichia coli hand contamination. Multiple rounds of structured observations of hand washing events after latrine use were conducted in 60 Kenyan schools, and hand rinse samples were collected one time in a subset of schools. The proportion of pupils observed practicing hand washing with soap (HWWS) events was significantly higher in schools that received a soap provision intervention (32%) and schools that received soap and latrine cleaning materials (38%) compared with controls (3%). Girls and boys had similar hand washing rates. There were non-significant reductions in E. coli contamination among intervention school pupils compared with controls. Removing the barrier of soap procurement can significantly increase availability of soap and hand washing among pupils; however, we discuss limitations in the enabling policy and institutional environment that may have prevented reaching desired levels of HWWS.
Sarriá Beatriz
2012-04-01
Full Text Available Abstract Background Dietary fibre is both preventive and therapeutic for bowel functional diseases. Soluble cocoa products are good sources of dietary fibre that may be supplemented with this dietary component. This study assessed the effects of regularly consuming two soluble cocoa products (A and B with different non-starch polysaccharides levels (NSP, 15.1 and 22.0% w/w, respectively on bowel habits using subjective intestinal function and symptom questionnaires, a daily diary and a faecal marker in healthy individuals. Methods A free-living, two-stage, randomized, crossover, single-blind intervention was carried out in 44 healthy men and women, between 18-55 y old, who had not taken dietary supplements, laxatives, or antibiotics six months before the start of the study. In the four-week-long intervention stages, separated by a three-week-wash-out stage, two servings of A and B, that provided 2.26 vs. 6.60 g/day of NSP respectively, were taken. In each stage, volunteers' diet was recorded using a 72-h food intake report. Results Regularly consuming cocoa A and B increased fibre intake, although only cocoa B significantly increased fibre intake (p Conclusions Regular consumption of the cocoa products increases dietary fibre intake to recommended levels and product B improves bowel habits. The use of both objective and subjective assessments to evaluate the effects of food on bowel habits is recommended.
The Interval Graph Completion Problem on Split Graphs
ZHANG Zhen-kun; YU Min
2015-01-01
The interval graph completion problem on a graph G is to find an added edge set F such that G+F is an interval supergraph with the smallest possible number of edges. The problem has important applications to numerical algebra, V LSI-layout and algorithm graph theory etc; And it has been known to be N P-complete on general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the interval graph completion problem on split graphs is investigated.
Graph Operations on Clique-Width Bounded Graphs
Gurski, Frank
2007-01-01
Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width and NLC-width under various graph operations and graph transformations. We give upper and lower bounds for the clique-width and NLC-width of the modified graphs in terms of the clique-width and NLC-width of the involved graphs.
GraphState - a tool for graph identification and labelling
Batkovich, D; Kompaniets, M; Novikov, S
2014-01-01
We present python libraries for Feynman graphs manipulation. The key feature of these libraries is usage of generalization of graph representation offered by B. G. Nickel et al. In this approach graph is represented in some unique 'canonical' form that depends only on its combinatorial type. The uniqueness of graph representation gives an efficient way for isomorphism finding, searching for subgraphs and other graph manipulation tasks. Though offered libraries were originally designed for Feynman graphs, they might be useful for more general graph problems.
Tikhonov, K. Â. S.; Mirlin, A. Â. D.
2016-11-01
We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree). We show that the wave-function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally treelike structure without a boundary (random regular graph) where extended wave functions are ergodic.
Graph theoretical analysis of brain connectivity in phantom sound perception.
Mohan, Anusha; De Ridder, Dirk; Vanneste, Sven
2016-02-02
Tinnitus is a phantom sound commonly thought of to be produced by the brain related to auditory deafferentation. The current study applies concepts from graph theory to investigate the differences in lagged phase functional connectivity using the average resting state EEG of 311 tinnitus patients and 256 healthy controls. The primary finding of the study was a significant increase in connectivity in beta and gamma oscillations and a significant reduction in connectivity in the lower frequencies for the tinnitus group. There also seems to be parallel processing of long-distance information between delta, theta, alpha1 and gamma frequency bands that is significantly stronger in the tinnitus group. While the network reorganizes into a more regular topology in the low frequency carrier oscillations, development of a more random topology is witnessed in the high frequency oscillations. In summary, tinnitus can be regarded as a maladaptive 'disconnection' syndrome, which tries to both stabilize into a regular topology and broadcast the presence of a deafferentation-based bottom-up prediction error as a result of a top-down prediction.
A CLASS OF LDPC CODE'S CONSTRUCTION BASED ON AN ITERATIVE RANDOM METHOD
Huang Zhonghu; Shen Lianfeng
2006-01-01
This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code's parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.
Anderson Localization for a Multi-Particle Quantum Graph
Sabri, Mostafa
2012-01-01
We study a multi-particle quantum graph with random potential. Taking the approach of multiscale analysis we prove exponential and strong dynamical localization of any order in the Hilbert-Schmidt norm near the spectral edge. Apart from the results on multi-particle systems, we also prove Lifshitz-type asymptotics for single-particle systems. This shows in particular that localization for single-particle quantum graphs holds under a weaker assumption on the random potential than previously kn...
Wilson, Robin J
1985-01-01
Graph Theory has recently emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. This book provides a comprehensive introduction to the subject.
Alspach, BR
1985-01-01
This volume deals with a variety of problems involving cycles in graphs and circuits in digraphs. Leading researchers in this area present here 3 survey papers and 42 papers containing new results. There is also a collection of unsolved problems.
Haynes Teresa W.
2014-08-01
Full Text Available A path π = (v1, v2, . . . , vk+1 in a graph G = (V,E is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi ≥ deg(vi+1, where deg(vi denotes the degree of vertex vi ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is at most half its order, and that the downhill domination number of a tree is at most one third its order. We characterize the graphs obtaining each of these bounds
Categorical constructions in graph theory
Richard T. Bumby
1986-01-01
Full Text Available This paper presents some graph-theoretic questions from the viewpoint of the portion of category theory which has become common knowledge. In particular, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related.
A Semantic Graph Query Language
Kaplan, I L
2006-10-16
Semantic graphs can be used to organize large amounts of information from a number of sources into one unified structure. A semantic query language provides a foundation for extracting information from the semantic graph. The graph query language described here provides a simple, powerful method for querying semantic graphs.
The Least Eigenvalue of Graphs
Guidong YU; Yizheng FAN; Yi WANG
2012-01-01
In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.
Solsolitons associated with graphs
Lafuente, Ramiro A
2010-01-01
We show how to associate with each graph with a certain property (positivity) a family of simply connected solvable Lie groups endowed with left-invariant Riemannian metrics that are Ricci solitons (called solsolitons). We classify them up to isometry, obtaining families depending on many parameters of explicit examples of Ricci solitons. A classification of graphs with up to 3 coherent components according to positivity is also given.
Graph Embedding for Pattern Analysis
Ma, Yunqian
2013-01-01
Graph Embedding for Pattern Analysis covers theory methods, computation, and applications widely used in statistics, machine learning, image processing, and computer vision. This book presents the latest advances in graph embedding theories, such as nonlinear manifold graph, linearization method, graph based subspace analysis, L1 graph, hypergraph, undirected graph, and graph in vector spaces. Real-world applications of these theories are spanned broadly in dimensionality reduction, subspace learning, manifold learning, clustering, classification, and feature selection. A selective group of experts contribute to different chapters of this book which provides a comprehensive perspective of this field.
Bollobás, Béla
1998-01-01
The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed ...
Arrighi, Pablo
2016-01-01
Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the evolution, and may be driven to be in a quantum superposition of graphs---in accordance to the superposition principle. We show that these unitary causal operators must decompose as a finite-depth circuit of local unitary gates. This unifies a result on Quantum Cellular Automata with another on Reversible Causal Graph Dynamics. Along the way we formalize a notion of causality which is valid in the context of quantum superpositions of time-varying graphs, and has a number of good properties. Keywords: Quantum Lattice Gas Automata, Block-representation, Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum Graphity, Causal Dynamical Triangula...
Commuting projections on graphs
Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing; Zikatanov, Ludmil T. [Pennsylvania State Univ., University Park, PA (United States). Dept. of Mathematics
2013-02-19
For a given (connected) graph, we consider vector spaces of (discrete) functions defined on its vertices and its edges. These two spaces are related by a discrete gradient operator, Grad and its adjoint, ₋Div, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ℓ_{2}-projection Q_{H} onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π _{H} from the original edge-space onto a properly constructed coarse edge-space associated with the edges of the coarse graph. The projections π _{H} and Q_{H} commute with the discrete divergence operator, i.e., we have div π _{H} = Q_{H} div. The respective pair of coarse edge-space and coarse vertexspace offer the potential to construct two-level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian which utilizes the discrete divergence operator. The performance of one two-level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples.
Clique graphs and overlapping communities
Evans, T. S.
2010-12-01
It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.
A Transformation-Based Approach to Implication of GSTE Assertion Graphs
Guowu Yang
2013-01-01
its powerful capacity in formal verification of VLSI systems. GSTE is an extension of symbolic trajectory evaluation (STE to the model checking of ω-regular properties. It is an alternative to classical model checking algorithms where properties are specified as finite-state automata. In GSTE, properties are specified as assertion graphs, which are labeled directed graphs where each edge is labeled with two labeling functions: antecedent and consequent. In this paper, we show the complement relation between GSTE assertion graphs and finite-state automata with the expressiveness of regular languages and ω-regular languages. We present an algorithm that transforms a GSTE assertion graph to a finite-state automaton and vice versa. By applying this algorithm, we transform the problem of GSTE assertion graphs implication to the problem of automata language containment. We demonstrate our approach with its application to verification of an FIFO circuit.
Learning Potential Energy Landscapes using Graph Kernels
Ferré, G; Barros, K
2016-01-01
Recent machine learning methods make it possible to model potential energy of atomic configurations with chemical-level accuracy (as calculated from ab-initio calculations) and at speeds suitable for molecular dynamics simulation. Best performance is achieved when the known physical constraints are encoded in the machine learning models. For example, the atomic energy is invariant under global translations and rotations; it is also invariant to permutations of same-species atoms. Although simple to state, these symmetries are complicated to encode into machine learning algorithms. In this paper, we present a machine learning approach based on graph theory that naturally incorporates translation, rotation, and permutation symmetries. Specifically, we use a random walk graph kernel to measure the similarity of two adjacency matrices, each of which represents a local atomic environment. We show on a standard benchmark that our Graph Approximated Energy (GRAPE) method is competitive with state of the art kernel m...
The Many Faces of Graph Dynamics
Pignolet, Yvonne Anne; Schmid, Stefan; Tredan, Gilles
2016-01-01
The topological structure of complex networks has fascinated researchers for several decades, resulting in the discovery of many universal properties and reoccurring characteristics of different kinds of networks. However, much less is known today about the network dynamics: indeed, complex networks in reality are not static, but rather dynamically evolve over time. Our paper is motivated by the empirical observation that network evolution patterns seem far from random, but exhibit structure. Moreover, the specific patterns appear to depend on the network type, contradicting the existence of a "one fits it all" model. However, we still lack observables to quantify these intuitions, as well as metrics to compare graph evolutions. Such observables and metrics are needed for extrapolating or predicting evolutions, as well as for interpolating graph evolutions. To explore the many faces of graph dynamics and to quantify temporal changes, this paper suggests to build upon the concept of centrality, a measure of no...
Zu, Baokai; Xia, Kewen; Pan, Yongke; Niu, Wenjia
2017-01-01
Semisupervised Discriminant Analysis (SDA) is a semisupervised dimensionality reduction algorithm, which can easily resolve the out-of-sample problem. Relative works usually focus on the geometric relationships of data points, which are not obvious, to enhance the performance of SDA. Different from these relative works, the regularized graph construction is researched here, which is important in the graph-based semisupervised learning methods. In this paper, we propose a novel graph for Semisupervised Discriminant Analysis, which is called combined low-rank and k-nearest neighbor (LRKNN) graph. In our LRKNN graph, we map the data to the LR feature space and then the kNN is adopted to satisfy the algorithmic requirements of SDA. Since the low-rank representation can capture the global structure and the k-nearest neighbor algorithm can maximally preserve the local geometrical structure of the data, the LRKNN graph can significantly improve the performance of SDA. Extensive experiments on several real-world databases show that the proposed LRKNN graph is an efficient graph constructor, which can largely outperform other commonly used baselines.
Baokai Zu
2017-01-01
Full Text Available Semisupervised Discriminant Analysis (SDA is a semisupervised dimensionality reduction algorithm, which can easily resolve the out-of-sample problem. Relative works usually focus on the geometric relationships of data points, which are not obvious, to enhance the performance of SDA. Different from these relative works, the regularized graph construction is researched here, which is important in the graph-based semisupervised learning methods. In this paper, we propose a novel graph for Semisupervised Discriminant Analysis, which is called combined low-rank and k-nearest neighbor (LRKNN graph. In our LRKNN graph, we map the data to the LR feature space and then the kNN is adopted to satisfy the algorithmic requirements of SDA. Since the low-rank representation can capture the global structure and the k-nearest neighbor algorithm can maximally preserve the local geometrical structure of the data, the LRKNN graph can significantly improve the performance of SDA. Extensive experiments on several real-world databases show that the proposed LRKNN graph is an efficient graph constructor, which can largely outperform other commonly used baselines.
Higher-order graph wavelets and sparsity on circulant graphs
Kotzagiannidis, Madeleine S.; Dragotti, Pier Luigi
2015-08-01
The notion of a graph wavelet gives rise to more advanced processing of data on graphs due to its ability to operate in a localized manner, across newly arising data-dependency structures, with respect to the graph signal and underlying graph structure, thereby taking into consideration the inherent geometry of the data. In this work, we tackle the problem of creating graph wavelet filterbanks on circulant graphs for a sparse representation of certain classes of graph signals. The underlying graph can hereby be data-driven as well as fixed, for applications including image processing and social network theory, whereby clusters can be modelled as circulant graphs, respectively. We present a set of novel graph wavelet filter-bank constructions, which annihilate higher-order polynomial graph signals (up to a border effect) defined on the vertices of undirected, circulant graphs, and are localised in the vertex domain. We give preliminary results on their performance for non-linear graph signal approximation and denoising. Furthermore, we provide extensions to our previously developed segmentation-inspired graph wavelet framework for non-linear image approximation, by incorporating notions of smoothness and vanishing moments, which further improve performance compared to traditional methods.
Wishart distributions for decomposable covariance graph models
Khare, Kshitij; 10.1214/10-AOS841
2011-01-01
Gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph $G$. These models are distinctly different from the traditional concentration graph models (often also referred to as Gaussian graphical models or covariance selection models) since the zeros in the parameter are now reflected in the covariance matrix $\\Sigma$, as compared to the concentration matrix $\\Omega =\\Sigma^{-1}$. The parameter space of interest for covariance graph models is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. As in Letac and Massam [Ann. Statist. 35 (2007) 1278--1323], we consider the case where $G$ is decomposable. In this paper, we construct on the cone $P_G$ a family of Wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by Letac and Massam [Ann. Statist. 35 (2007) 1278--1323] and Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317] do in ...
Logical complexity of graphs: a survey
Pikhurko, Oleg
2010-01-01
We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number of variables occurring in such a sentence. The logical length $L(G)$ is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, or the contribution of Frank Ramsey to the research on Hilbert's Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible (after powering with counting quantifiers).
On the Minimum Number of Spanning Trees in k-Edge-Connected Graphs
Ok, Seongmin; Thomassen, Carsten
2016-01-01
>2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples show that c3≤ √2+√3≈1.93. However, we have no examples of 5-edge-connected graphs with fewer spanning trees than the n-cycle with all edge multiplicities (except one) equal to 3, which is almost 6-regular. We have...... no examples of 5-regular 5-edge-connected graphs with fewer than 3.09n-1 spanning trees, which is more than the corresponding number for 6-regular 6-edge-connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity....
The STAPL Parallel Graph Library
Harshvardhan,
2013-01-01
This paper describes the stapl Parallel Graph Library, a high-level framework that abstracts the user from data-distribution and parallelism details and allows them to concentrate on parallel graph algorithm development. It includes a customizable distributed graph container and a collection of commonly used parallel graph algorithms. The library introduces pGraph pViews that separate algorithm design from the container implementation. It supports three graph processing algorithmic paradigms, level-synchronous, asynchronous and coarse-grained, and provides common graph algorithms based on them. Experimental results demonstrate improved scalability in performance and data size over existing graph libraries on more than 16,000 cores and on internet-scale graphs containing over 16 billion vertices and 250 billion edges. © Springer-Verlag Berlin Heidelberg 2013.
Fundamentals of algebraic graph transformation
Ehrig, Hartmut; Prange, Ulrike; Taentzer, Gabriele
2006-01-01
Graphs are widely used to represent structural information in the form of objects and connections between them. Graph transformation is the rule-based manipulation of graphs, an increasingly important concept in computer science and related fields. This is the first textbook treatment of the algebraic approach to graph transformation, based on algebraic structures and category theory. Part I is an introduction to the classical case of graph and typed graph transformation. In Part II basic and advanced results are first shown for an abstract form of replacement systems, so-called adhesive high-level replacement systems based on category theory, and are then instantiated to several forms of graph and Petri net transformation systems. Part III develops typed attributed graph transformation, a technique of key relevance in the modeling of visual languages and in model transformation. Part IV contains a practical case study on model transformation and a presentation of the AGG (attributed graph grammar) tool envir...
Vassilis Giakoumakis
1997-12-01
Full Text Available We study the P 4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P 4-domination in perfect graphs. This class strictly contains the P 4-extendible graphs and the P 4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P 4-tidy graphs and P 4-lite graphs are closely related. Note that the class of P 4-lite graphs is a class of brittle graphs strictly containing the P 4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P 4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22], Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].
Random holographic "large worlds" with emergent dimensions
Trugenberger, Carlo A.
2016-11-01
I propose a random network model governed by a Gaussian weight corresponding to Ising link antiferromagnetism as a model for emergent quantum space-time. In this model, discrete space is fundamental, not a regularization; its spectral dimension ds is not a model input but is, rather, completely determined by the antiferromagnetic coupling constant. Perturbative terms suppressing triangles and favoring squares lead to locally Euclidean ground states that are Ricci flat "large worlds" with power-law extension. I then consider the quenched graphs of lowest energy for ds=2 and ds=3 , and I show how quenching leads to the spontaneous emergence of embedding spaces of Hausdorff dimension dH=4 and dH=5 , respectively. One of the additional, spontaneous dimensions can be interpreted as time, causality being an emergent property that arises in the large N limit (with N the number of vertices). For ds=2 , the quenched graphs constitute a discrete version of a 5D-space-filling surface with a number of fundamental degrees of freedom scaling like N2 /5, a graph version of the holographic principle. These holographic degrees of freedom can be identified with the squares of the quenched graphs, which, being triangle-free, are the fundamental area (or loop) quanta.
Yong Hua LI; Hai Bin KAN; Bing Jun YU
2004-01-01
In this paper, a special kind of partial algebras called projective partial groupoids is defined.It is proved that the inverse image of all projections of a fundamental weak regular *-semigroup under the homomorphism induced by the maximum idempotent-separating congruence of a weak regular *-semigroup has a projective partial groupoid structure. Moreover, a weak regular *-product which connects a fundamental weak regular *-semigroup with corresponding projective partial groupoid is defined and characterized. It is finally proved that every weak regular *-product is in fact a weak regular *-semigroup and any weak regular *-semigroup is constructed in this way.
Juvenile zebra finches learn the underlying structural regularities of their fathers’ song
Otilia eMenyhart
2015-05-01
Full Text Available Natural behaviors, such as foraging, tool use, social interaction, birdsong, and language, exhibit branching sequential structure. Such structure should be learnable if it can be inferred from the statistics of early experience. We report that juvenile zebra finches learn such sequential structure in song. Song learning in finches has been extensively studied, and it is generally believed that young males acquire song by imitating tutors (Zann, 1996. Variability in the order of elements in an individual’s mature song occurs, but the degree to which variation in a zebra finch’s song follows statistical regularities has not been quantified, as it has typically been dismissed as production error (Sturdy et al., 1999. Allowing for the possibility that such variation in song is non-random and learnable, we applied a novel analytical approach, based on graph-structured finite-state grammars, to each individual’s full corpus of renditions of songs. This method does not assume syllable-level correspondence between individuals. We find that song variation can be described by probabilistic finite-state graph grammars that are individually distinct, and that the graphs of juveniles are more similar to those of their fathers than to those of other adult males. This grammatical learning is a new parallel between birdsong and language. Our method can be applied across species and contexts to analyze complex variable learned behaviors, as distinct as foraging, tool use, and language.
Optimized Graph Search Using Multi-Level Graph Clustering
Kala, Rahul; Shukla, Anupam; Tiwari, Ritu
Graphs find a variety of use in numerous domains especially because of their capability to model common problems. The social networking graphs that are used for social networking analysis, a feature given by various social networking sites are an example of this. Graphs can also be visualized in the search engines to carry search operations and provide results. Various searching algorithms have been developed for searching in graphs. In this paper we propose that the entire network graph be clustered. The larger graphs are clustered to make smaller graphs. These smaller graphs can again be clustered to further reduce the size of graph. The search is performed on the smallest graph to identify the general path, which may be further build up to actual nodes by working on the individual clusters involved. Since many searches are carried out on the same graph, clustering may be done once and the data may be used for multiple searches over the time. If the graph changes considerably, only then we may re-cluster the graph.
Figure-Ground Segmentation Using Factor Graphs.
Shen, Huiying; Coughlan, James; Ivanchenko, Volodymyr
2009-06-04
Foreground-background segmentation has recently been applied [26,12] to the detection and segmentation of specific objects or structures of interest from the background as an efficient alternative to techniques such as deformable templates [27]. We introduce a graphical model (i.e. Markov random field)-based formulation of structure-specific figure-ground segmentation based on simple geometric features extracted from an image, such as local configurations of linear features, that are characteristic of the desired figure structure. Our formulation is novel in that it is based on factor graphs, which are graphical models that encode interactions among arbitrary numbers of random variables. The ability of factor graphs to express interactions higher than pairwise order (the highest order encountered in most graphical models used in computer vision) is useful for modeling a variety of pattern recognition problems. In particular, we show how this property makes factor graphs a natural framework for performing grouping and segmentation, and demonstrate that the factor graph framework emerges naturally from a simple maximum entropy model of figure-ground segmentation.We cast our approach in a learning framework, in which the contributions of multiple grouping cues are learned from training data, and apply our framework to the problem of finding printed text in natural scenes. Experimental results are described, including a performance analysis that demonstrates the feasibility of the approach.