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.
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...
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...
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...
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...
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...
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 \
The Clique Problem in Ray Intersection Graphs
Cabello, Sergio; Langerman, Stefan
2011-01-01
Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochv\\'il and Ne\\v{s}et\\v{r}il.
The clique problem in ray intersection graphs
Langerman, Stefan; Cardinal, Jean; Cabello, Sergio
2015-01-01
Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochvíl and Nešetřil (Comment Math Univ Carolinae 31(1):85-93, 1990).
Spanners for geometric intersection graphs with applications
Martin Fürer
2012-05-01
Full Text Available A ball graph is an intersection graph of a set of balls with arbitrary radii. Given a real numbert>1, we say that a subgraph G' of a graph G is a t-spanner of G, if for every pair of verticesu,v in G, there exists a path in G' of length at most t times the distance between u and v inG. In this paper, we consider the problem of efficiently constructing sparse spanners of ball graphs which supports fast shortest path distance queries.We present the first algorithm for constructing spanners of ball graphs. For a ball graph in Rk, we construct a (1+ε-spanner for any ε>0 with O(nε-k+1 edges in O(n2ℓ+δε-k logℓ S time, using an efficient partitioning of space into hypercubes and solving intersection problems. Here ℓ=1-1/(⌊k/2⌋+2, δ is any positive constant, and S is the ratio between the largest and smallest radius. For the special case when the balls all have unit size, we show that the complexity of constructing a (1+ε-spanner is almost equal to the complexity of constructing a Euclidean minimum spanning tree. The algorithm extends naturally to other disk-likeobjects, also in higher dimensions.The algorithm uses an efficient subdivision of space to construct a sparse graph having many of the same distance properties as the input ball graph. Additionally, the constructed spanners have a small vertex separator decomposition (hereditary. In dimension k=2, the disk graph spanner has an O(n1/2ε-3/2+ε-3log S separator. The presence of a small separator is then exploited to obtain very efficient data structures for approximate distance queries. The results on geometric graph separators might be of independent interest. For example, since complete Euclidean graphs are just a special case of (unit ball graphs, our results also provide a new approach for constructing spanners with small separators in these graphs.
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.
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.
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.
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...
Tur\\'an and Ramsey Properties of Subcube Intersection Graphs
Johnson, J Robert
2011-01-01
The discrete cube $\\{0,1\\}^d$ is a fundamental combinatorial structure. A subcube of $\\{0,1\\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no $k$ which have non-empty intersection and no $l$ which are pairwise disjoint? These questions are naturally expressed as Tur\\'an and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Tur\\...
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...
Coloring intersection graphs of x-monotone curves in the plane
Suk, Andrew
2012-01-01
A class of graphs G is \\chi-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple x-monotone curves in the plane intersecting a vertical line is \\chi-bounded. As a corollary, the class of intersection graphs of rays in the plane is \\chi-bounded.
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.
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.
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...
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...
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.
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...
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.
Metelsky Yury
2017-02-01
Full Text Available We characterize the class L32$L_3^2 $ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n-time algorithm for the recognition of graphs from L32$L_3^2 $ in the class of threshold graphs, where n is the number of vertices of a tested graph.
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.
(3, 1)*-Choosability of graphs of nonnegative characteristic without intersecting short cycles
Haihui Zhang
2016-05-01
A graph is called (, )*-choosable if for every list assignment satisfying $|L(v)|\\geq k$ for all $v \\in V (G)$, there is an -coloring of such that each vertex of has at most neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting -cycles for all = 3, 4, 5 is (3, 1)*-choosable.
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.
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...
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} ...
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.
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.
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.
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).
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.
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 ...
Intersection of a Sure Ellipsoid and a Random Ellipsoid
Arjun K. Gupta
2012-07-01
Full Text Available Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} An Expression for the expected value of the intersection of a sure sphere and a random sphere has been derived by Laurent (1974. In the present paper we derive the expression for the expected intersection volume of a sure ellipsoid and a random ellipsoid
Intersection of a Sure Ellipsoid and a Random Ellipsoid
Arjun K. Gupta
2012-07-01
Full Text Available Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} An Expression for the expected value of the intersection of a sure sphere and a random sphere has been derived by Laurent (1974. In the present paper we derive the expression for the expected intersection volume of a sure ellipsoid and a random ellipsoid
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.
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...
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...
Kobylkin, Konstantin
2016-10-01
Computational complexity and approximability are studied for the problem of intersecting of a set of straight line segments with the smallest cardinality set of disks of fixed radii r > 0 where the set of segments forms straight line embedding of possibly non-planar geometric graph. This problem arises in physical network security analysis for telecommunication, wireless and road networks represented by specific geometric graphs defined by Euclidean distances between their vertices (proximity graphs). It can be formulated in a form of known Hitting Set problem over a set of Euclidean r-neighbourhoods of segments. Being of interest computational complexity and approximability of Hitting Set over so structured sets of geometric objects did not get much focus in the literature. Strong NP-hardness of the problem is reported over special classes of proximity graphs namely of Delaunay triangulations, some of their connected subgraphs, half-θ6 graphs and non-planar unit disk graphs as well as APX-hardness is given for non-planar geometric graphs at different scales of r with respect to the longest graph edge length. Simple constant factor approximation algorithm is presented for the case where r is at the same scale as the longest edge length.
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.
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.
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...
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
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...
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.
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.
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.
The intersection numbers of the p-spin curves from random matrix theory
Brezin, E
2012-01-01
The intersection numbers of p-spin curves are computed through correlation functions of Gaussian ensembles of random matrices in an external matrix source. The p-dependence of intersection numbers is determined as polynomial in p; the large p behavior is also considered. The analytic continuation of intersection numbers to negative values of p is discussed in relation to SL(2,R)/U(1) black hole sigma model.
The intersection numbers of the p-spin curves from random matrix theory
Brézin, E.; Hikami, S.
2013-02-01
The intersection numbers of p-spin curves are computed through correlation functions of Gaussian ensembles of random matrices in an external matrix source. The p-dependence of intersection numbers is determined as polynomial in p; the large p behavior is also considered. The analytic continuation of intersection numbers to negative values of p is discussed in relation to SL(2,R)/U(1) black hole sigma model.
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...
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...
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...
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...
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.
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
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.
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 ...
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...
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.
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
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.
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...
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...
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 .
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.
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
Upper tails of self-intersection local times of random walks: survey of proof techniques
König, Wolfgang
2010-01-01
The asymptotics of the probability that the self-intersection local time of a random walk on $\\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on {\\it Excess self-intersection local times, and related topics} in Luminy, 6-10 Dec., 2010.
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.
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.
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.
Large deviations for self-intersection local times of stable random walks
Laurent, Clément
2010-01-01
Let $(X_t,t\\geq 0)$ be a random walk on $\\mathbb{Z}^d$. Let $ l_T(x)= \\int_0^T \\delta_x(X_s)ds$ the local time at the state $x$ and $ I_T= \\sum\\limits_{x\\in\\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \\cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and M\\"orters obtain in \\cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \\cite{Asselah5} a large deviations principle for the SILT with $q=2$. We extend these results to an $\\alpha$-stable process (i.e. $\\alpha\\in]0,2]$) in the case where $q(d-\\alpha)\\geq d$.
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...
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 ...
Castell, Fabienne; Mélot, Clothilde
2012-01-01
Let $(X_t, t \\geq 0)$ be an $\\alpha$-stable random walk with values in $\\Z^d$. Let $l_t(x) = \\int_0^t \\delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \\sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \\geq r_t)$ for all scales $r_t \\gg \\E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and K\\"onig 2010, and Laurent 2012.
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...
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...
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},...
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.
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.
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.
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.
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.
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.
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.
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...
Asteroidal Quadruples in non Rooted Path Graphs
Gutierrez Marisa
2015-11-01
Full Text Available A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. For this purpose, we are studying in this paper directed path graphs that are non rooted path graphs. We prove that such graphs always contain an asteroidal quadruple.
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.
Boxicity of Circular Arc Graphs
Bhowmick, Diptendu; Chandran, L. Sunil
2008-01-01
A $k$-dimensional box is the cartesian product $R_1 \\times R_2 \\times ... \\times R_k$ where each $R_i$ is a closed interval on the real line. The {\\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of ...
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.
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...
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...
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.
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)) .
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.
马小龙; 李成名; 赵占杰
2014-01-01
针对三维道路场景中多条道路交叉时出现的单纯重叠问题，基于开源三维图形渲染引擎 OSG (OpenSceneGraph)，提出一种三维道路交叉口的自动检测与建模方法，该方法通过分析道路矢量数据的存储特点及道路交叉口的设计规范，快速获取交叉口区域并识别其所属类型，结合计算机图形学和OSG场景渲染技术实现道路交叉口的自动建模。试验表明，该方法可有效解决三叉型、四叉型、环岛型和分离式立体交叉口的一键式生产，无须人工干预，为大数据三维道路场景的快速构建提供了有效方法。%Targeting the simple overlap arising when multiple roads intersect in a threeGdimensional road scene,an automatic detection and modeling method is provided for 3D road intersections on the basis of openGsource 3D rendering engine OSG (OpenSceneGraph).In this method,by analyzing road vectorGdata storage characteristics and intersection design specifications,an intersection area can be quickly captured or formed,with its type being identified simultaneously,so as to achieve its automatic modeling by means of computer graphics and OSG scene rendering technique.The experiment proves that this method has effectually solved oneGbutton generation of threeGway,fourGway,roundabout,and separate 3D intersecG tions,without the need of manual intervention,serving as a foundation for the quick construction of future bigGdata 3DGroad scenes.
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.
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.
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...
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
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
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.
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.
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. .
一个n-可扩图的局部交条件%A LOCAL INTERSECTION CONDITION FOR n-EXTENDABLE GRAPHS
潘林强; 陈旭谨; 张克民
2000-01-01
设G是一个有偶数个顶点的连通图,α2v表示由N2(v)导出的子图的独立数,其中 Nk(v)＝{u|,∈V(G)并且d(u,v)＝k}, 对任意uv∈E(G),我们定义λuv＝|N(u)∩N(u)|,T2u(v)＝N2(u)\\N(v),t2uv＝min{T2u(v)|,|T2v(u)|}和α2uv=min{α2u,α2v}.本文证明如果对任意d(u,v)=2的u,v,有λuv≥{α2uv,t2uv}+2n,则G是n-可扩的.%Let G be a connected graph with even order. Let al denote independence number of the subgraph induced by N2(v) in G, where Nk(v)= {v|v∈V(G) and d(u,v)=k}. For any uv∈ E(G),we define λuv= |N(u)∩N(v)|,T2u(v)=N2(u)\\N(v),t2uv=min{|T2u(v)|,|T2v.(u)|} andα2uv=min{α2u,α2v}. It is proved that if ,λuv≥min{α2uv,t2uv}+2n for any two vertices u and v with d(u,v)= 2, then G is n-extendable.
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.
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.
Marfelt, Mikkel Mouritz
2016-01-01
oriented but still emphasizes stable concepts. Moreover, it does not give primacy to oppression. Finally, it adopts a critical stance on the nature of the macro, meso, and micro levels as dominant analytical perspectives. As a result, this paper focusses on the importance of intersectionality...
Barbara Harbin Cobb
2009-01-01
<正>I’m a few years older than the People’s Re- public of China, but hardly an infant compared to China’s vast history and culture. China and I have intersected at many points, and I want to tell you about a few of them.
On a conjecture concerning helly circle graphs
Durán Guillermo
2003-01-01
Full Text Available We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle, and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000 states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges. Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.
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...
Ferrara, S; Morales, J F; Samtleben, H
2009-01-01
We apply the entropy formalism to the study of the near-horizon geometry of extremal black p-brane intersections in D>5 dimensional supergravities. The scalar flow towards the horizon is described in terms an effective potential given by the superposition of the kinetic energies of all the forms under which the brane is charged. At the horizon active scalars get fixed to the minima of the effective potential and the entropy function is given in terms of U-duality invariants built entirely out of the black p-brane charges. The resulting entropy function reproduces the central charges of the dual boundary CFT and gives rise to a Bekenstein-Hawking like area law. The results are illustrated in the case of black holes and black string intersections in D=6, 7, 8 supergravities where the effective potentials, attractor equations, moduli spaces and entropy/central charges are worked out in full detail.
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...
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.
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.
Vempala, Santosh S
2005-01-01
Random projection is a simple geometric technique for reducing the dimensionality of a set of points in Euclidean space while preserving pairwise distances approximately. The technique plays a key role in several breakthrough developments in the field of algorithms. In other cases, it provides elegant alternative proofs. The book begins with an elementary description of the technique and its basic properties. Then it develops the method in the context of applications, which are divided into three groups. The first group consists of combinatorial optimization problems such as maxcut, graph coloring, minimum multicut, graph bandwidth and VLSI layout. Presented in this context is the theory of Euclidean embeddings of graphs. The next group is machine learning problems, specifically, learning intersections of halfspaces and learning large margin hypotheses. The projection method is further refined for the latter application. The last set consists of problems inspired by information retrieval, namely, nearest neig...
Junction trees of general graphs
Xiaofei WANG; Jianhua GUO
2008-01-01
In this paper,we study the maximal prime subgraphs and their corresponding structure for any undirected graph.We introduce the notion of junction trees and investigate their structural characteristics,including junction properties,induced-subtree properties,running-intersection properties and maximum-weight spanning tree properties.Furthermore,the characters of leaves and edges on junction trees are discussed.
Borel, Armand
1984-01-01
This book is a publication in Swiss Seminars, a subseries of Progress in Mathematics. It is an expanded version of the notes from a seminar on intersection cohomology theory, which met at the University of Bern, Switzerland, in the spring of 1983. This volume supplies an introduction to the piecewise linear and sheaf-theoretic versions of that theory as developed by M. Goresky and R. MacPherson in Topology 19 (1980), and in Inventiones Mathematicae 72 (1983). While some familiarity with algebraic topology and sheaf theory is assumed, the notes include a self-contained account of further material on constructibility, derived categories, Verdier duality, biduality, and on stratified spaces, which is used in the second paper but not found in standard texts. "The volume should be useful to someone interested in acquiring some basic knowledge about the field..." —Mathematical Reviews.
Triangle-intersecting families on eight vertices
Filmus, Yuval
2011-01-01
Simonovits and S\\'{o}s conjectured that the maximal size of a triangle-intersecting family of graphs on $n$ vertices is $2^{\\binom{n}{2}-3}$. Their conjecture has recently been proved using spectral methods. We provide an elementary proof of the special case of $8$ vertices using a partition argument.
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.
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.
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).
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...
Complexity Results on Graphs with Few Cliques
Bill Rosgen
2007-01-01
Full Text Available A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.
Notes on large angle crossing graphs
Dujmovic, Vida; Morin, Pat; Wolle, Thomas
2009-01-01
A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2.
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...
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.
INTERSECTIONAL DISCRIMINATION AGAINST CHILDREN
Ravnbøl, Camilla Ida
This paper adds a perspective to existing research on child protection by engaging in a debate on intersectional discrimination and its relationship to child protection. The paper has a twofold objective, (1) to further establish intersectionality as a concept to address discrimination against ch...... children, and (2) to illustrate the importance of addressing intersectionality within rights-based programmes of child protection....
Deburring small intersecting holes
Gillespie, L.K.
1980-08-01
Deburring intersecting holes is one of the most difficult deburring tasks faced by many industries. Only 14 of the 37 major deburring processes are applicable to most intersecting hole applications. Only five of these are normally applicable to small or miniature holes. Basic process capabilities and techniques used as a function of hole sizes and intersection depths are summarized.
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.
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.
Nested cycles in large triangulations and crossing-critical graphs
Hernandez-Velez, Cesar; Thomas, Robin
2009-01-01
We show that every sufficiently large plane triangulation has a large collection of nested cycles that either are pairwise disjoint, or pairwise intersect in exactly one vertex, or pairwise intersect in exactly two vertices. We apply this result to show that for each fixed positive integer $k$, there are only finitely many $k$-crossing-critical simple graphs of average degree at least six. Combined with the recent constructions of crossing-critical graphs given by Bokal, this settles the question of for which numbers $q>0$ there is an infinite family of $k$-crossing-critical simple graphs of average degree $q$.
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
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.
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.
Buick, Otto; Falcon, Pat; Alexander, G.; Siegel, Edward Carl-Ludwig
2013-03-01
Einstein[Dover(03)] critical-slowing-down(CSD)[Pais, Subtle in The Lord; Life & Sci. of Albert Einstein(81)] is Siegel CyberWar denial-of-access(DOA) operations-research queuing theory/pinning/jamming/.../Read [Aikido, Aikibojitsu & Natural-Law(90)]/Aikido(!!!) phase-transition critical-phenomenon via Siegel DIGIT-Physics (Newcomb[Am.J.Math. 4,39(1881)]-{Planck[(1901)]-Einstein[(1905)])-Poincare[Calcul Probabilités(12)-p.313]-Weyl [Goett.Nachr.(14); Math.Ann.77,313 (16)]-{Bose[(24)-Einstein[(25)]-Fermi[(27)]-Dirac[(1927)]}-``Benford''[Proc.Am.Phil.Soc. 78,4,551 (38)]-Kac[Maths.Stat.-Reasoning(55)]-Raimi[Sci.Am. 221,109 (69)...]-Jech[preprint, PSU(95)]-Hill[Proc.AMS 123,3,887(95)]-Browne[NYT(8/98)]-Antonoff-Smith-Siegel[AMS Joint-Mtg.,S.-D.(02)] algebraic-inversion to yield ONLY BOSE-EINSTEIN QUANTUM-statistics (BEQS) with ZERO-digit Bose-Einstein CONDENSATION(BEC) ``INTERSECTION''-BECOME-UNION to Barabasi[PRL 876,5632(01); Rev.Mod.Phys.74,47(02)...] Network /Net/GRAPH(!!!)-physics BEC: Strutt/Rayleigh(1881)-Polya(21)-``Anderson''(58)-Siegel[J.Non-crystalline-Sol.40,453(80)
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.
Statistical modeling of total crash frequency at highway intersections
Arash M. Roshandeh
2016-04-01
Full Text Available Intersection-related crashes are associated with high proportion of accidents involving drivers, occupants, pedestrians, and cyclists. In general, the purpose of intersection safety analysis is to determine the impact of safety-related variables on pedestrians, cyclists and vehicles, so as to facilitate the design of effective and efficient countermeasure strategies to improve safety at intersections. This study investigates the effects of traffic, environmental, intersection geometric and pavement-related characteristics on total crash frequencies at intersections. A random-parameter Poisson model was used with crash data from 357 signalized intersections in Chicago from 2004 to 2010. The results indicate that out of the identified factors, evening peak period traffic volume, pavement condition, and unlighted intersections have the greatest effects on crash frequencies. Overall, the results seek to suggest that, in order to improve effective highway-related safety countermeasures at intersections, significant attention must be focused on ensuring that pavements are adequately maintained and intersections should be well lighted. It needs to be mentioned that, projects could be implemented at and around the study intersections during the study period (7 years, which could affect the crash frequency over the time. This is an important variable which could be a part of the future studies to investigate the impacts of safety-related works at intersections and their marginal effects on crash frequency at signalized intersections.
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...
Wu Yang YU; Dong Hua WU; Gang Song LENG
2007-01-01
The purpose of this paper is to generalize the notion of intersection bodies to that of quasi Lp-intersection bodies. The Lp-analogs of the Busemann intersection inequality and the Brunn Minkowski inequality for the quasi Lp-intersection bodies are obtained. The Aleksandrov-Fenchel inequality for the mixed quasi Lp-intersection bodies is also established.
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.)
INTERSECTIONAL DISCRIMINATION AGAINST CHILDREN
Ravnbøl, Camilla Ida
This paper adds a perspective to existing research on child protection by engaging in a debate on intersectional discrimination and its relationship to child protection. The paper has a twofold objective, (1) to further establish intersectionality as a concept to address discrimination against...
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.
On a k-clique-join of a class of partitionable graphs
Mihai Talmaciu
2005-05-01
Full Text Available We call a graph G O-graph if there is an optimal coloring of the set of vertices and an optimal (disjoint covering with cliques such that any class of colors intersects any clique. In this paper, it has been established the relation to [p,q,r]-partite graphs and the fact that the O-graphs admit a k-clique-join. American Mathematical Society (2000: 05C17.
Connectivity graphs of uncertainty regions
Chambers, Erin; Lenchner, Jonathan; Sember, Jeff; Srinivasan, Venkatesh; Stege, Ulrike; Stolpner, Svetlana; Weibel, Christophe; Whitesides, Sue
2010-01-01
We study a generalization of the well known bottleneck spanning tree problem called "Best Case Connectivity with Uncertainty": Given a family of geometric regions, choose one point per region, such that the length of the longest edge in a spanning tree of a disc intersection graph is minimized. We show that this problem is NP-hard even for very simple scenarios such as line segments and squares. We also give exact and approximation algorithms for the case of line segments and unit discs respectively.
Graphs obtained from collections of blocks
Colton Magnant
2015-03-01
Full Text Available Given a collection of $d$-dimensional rectangular solids called blocks, no two of which sharing interior points, construct a block graph by adding a vertex for each block and an edge if the faces of the two corresponding blocks intersect nontrivially. It is known that if $d \\geq 3$, such block graphs can have arbitrarily large chromatic number. We prove that the chromatic number can be bounded with only a mild restriction on the sizes of the blocks. We also show that block graphs of block configurations arising from partitions of $d$-dimensional hypercubes into sub-hypercubes are at least $d$-connected. Bounds on the diameter and the hamiltonicity of such block graphs are also discussed.
Hypergraph Partitioning through Vertex Separators on Graphs
Kayaaslan, Enver; Catalyurek, Umit V; Aykanat, Cevdet
2011-01-01
The modeling flexibility provided by hypergraphs has drawn a lot of interest from the combinatorial scientific community, leading to novel models and algorithms, their applications, and development of associated tools. Hypergraphs are now a standard tool in combinatorial scientific computing. The modeling flexibility of hypergraphs however, comes at a cost: algorithms on hypergraphs are inherently more complicated than those on graphs, which sometimes translate to nontrivial increases in processing times. Neither the modeling flexibility of hypergraphs, nor the runtime efficiency of graph algorithms can be overlooked. Therefore, the new research thrust should be how to cleverly trade-off between the two. This work addresses one method for this trade-off by solving the hypergraph partitioning problem by finding vertex separators on graphs. Specifically, we investigate how to solve the hypergraph partitioning problem by seeking a vertex separator on its net intersection graph (NIG), where each net of the hyperg...
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 ...
Aldazabal, G; Ibáñez, L E; Rabadan, Raul; Uranga, Angel M
2001-01-01
It is known that chiral fermions naturally appear at certain intersections of branes at angles. Motivated by this fact, we propose a string scenario in which different standard model gauge interactions propagate on different (intersecting) brane worlds, partially wrapped in the extra dimensions. Quarks and leptons live at brane intersections, and are thus located at different positions in the extra dimensions. Replication of families follows naturally from the fact that the branes generically intersect at several points. Gauge and Yukawa couplings can be computed in terms of the compactification radii. Hierarchical Yukawa couplings appear naturally, since amplitudes involving three different intersections are proportional to exp(-A_{ijk}), where A_{ijk} is the area of a string world-sheet extending among the intersections. The models are non-supersymmetric but the string scale may be lowered down to 1-10 TeV. The proton is however stable due to a set of discrete symmetries arising from world-sheet selection r...
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
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…
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.
CERN PhotoLab
1974-01-01
The experimental apparatus used at intersection 1 by the CERN-Bologna Collaboration (experiment R105). It consists of two almost identical magnetic spectrometers centered at 90 degrees on opposite sides of the intersection region. In each spectrometer one can see magnetostrictive wire spark chambers, a magnet, more chambers and various hodoscopes of scintillation counters. Gas Cerenkov counters (almost invisible in the picture) are located in the gap of each magnet. On the left hand side, a matrix of 119 lead glass Cerenkov counters is located behind some concrete and iron shielding.
CERN PhotoLab
1971-01-01
Intersection I-1 of the ISR in August 1971 showing the 90 degree large acceptance spectrometer of the Saclay-Strasbourg Collaboration which is studying the momentum spectra of electrons at large angles. On the left of the crossing region can be seen the track used by the CERN-Bucharest-Cracow-Tata Collaboration to bring nuclear emulsions into the ISR.
1971-01-01
Intersection I-2 of the ISR during the installation of experiments. On the left to the crossing region can be seen the massive iron plate structure of the muon detector being used by a British collaboration in a search for the intermediate vector boson. The magnet and hodoscopes on the right are part of the spectrometer arm of the Bristish-Scandinavian Collaration.
Intersectional embodiment and power
Elg, Camilla; Jensen, Sune Qvotrup
Through almost two decades the term ‘intersectionality' has gained influence in post colonial studies, gender studies, feminist theory and other research fields occupied with how social differences are distributed and how individuals are socially constructed in stratified societies. The ‘interse...... differences and discuss the implications this has for our understanding of power relations....
Families of graph-different Hamilton paths
Körner, János; Simonyi, Gábor
2011-01-01
Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M(n;D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erd\\"os, Simonovits and S\\'os. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging.
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.
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.
Crash patterns at signalized intersections
Polders, Evelien; Daniels, Stijn; HERMANS, Elke; Brijs, Tom; Wets, Geert
2015-01-01
Traffic signals are often implemented to provide for efficient movement and to improve traffic safety. Nevertheless, severe crashes still occur at signalized intersections. This study aims to improve the understanding of signalized intersection safety by identifying crash types, locations and factors associated with signalized intersections. For this purpose, 1295 police-reported crashes at 87 signalized intersections are analyzed based on detailed crash descriptions, i.e. crash data and c...
On the Connectedness and Diameter of a Geometric Johnson Graph
Bautista-Santiago, Crevel; Fabila-Monroy, Ruy; Flores-Peñaloza, David; González-Aguilar, Hernán; Lara, Dolores; Sarmiento, Eliseo; Urrutia, Jorge
2012-01-01
Let $P$ be a set of $n$ points in general position in the plane. A subset $I$ of $P$ is called an \\emph{island} if there exists a convex set $C$ such that $I = P \\cap C$. In this paper we define the \\emph{generalized island Johnson graph} of $P$ as the graph whose vertex consists of all islands of $P$ of cardinality $k$, two of which are adjacent if their intersection consists of exactly $l$ elements. We show that for large enough values of $n$, this graph is connected, and give upper and lower bounds on its diameter.
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.
Intersectional embodiment and power
Elg, Camilla; Jensen, Sune Qvotrup
. The ‘intersectional' perspective creates a focus on the coexistence and mutual formation of different kinds of social difference in the construction of the individual social position. The concept of habitus as it was coined by Pierre Bourdieu has been an inspiration for some of this research. However, the focus...... on the embodiment of social dispositions and the social formation of the individual as body inherent in Bourdieu's work have not yet gained weight in this development within ‘difference research'. It might not come to much surprise as the radical sociology of embodiment implicit in the theory of habitus does...... not seem to gain much attention in social stratification research in general. In our paper we will present our work on an embodied approach to intersectionality, which is inspired by Pierre Bourdieu and other thinkers of embodiment. We will argue for the importance of a focus on the embodiment of social...
A Fast and Simple Graph Kernel for RDF
de Vries, G.K.D.; de Rooij, S.
2013-01-01
In this paper we study a graph kernel for RDF based on constructing a tree for each instance and counting the number of paths in that tree. In our experiments this kernel shows comparable classification performance to the previously introduced intersection subtree kernel, but is significantly faster
A Note on Longest Paths in Circular Arc Graphs
Joos Felix
2015-08-01
Full Text Available As observed by Rautenbach and Sereni [SIAM J. Discrete Math. 28 (2014 335-341] there is a gap in the proof of the theorem of Balister et al. [Combin. Probab. Comput. 13 (2004 311-317], which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.
Chen, C M; Sharakin, S A; Chen, Chiang-Mei; Gal'tsov, Dmitri V.; Sharakin, Sergei A.
1999-01-01
New solution to the six-dimensional vacuum Einstein's equations is constructed as a non-linear superposition of two five-dimensional solutions representing the Melvin-Gibbons-Maeda Universe and its S-dual. Then using duality between D=8 vacuum and a certain class of D=11 supergravity configurations we generate M2 and M5 fluxbranes as well as some of their intersections also including waves and KK-monopoles.
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...
A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs
Adiga, Abhijin; Chandran, L Sunil
2011-01-01
Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - \\epsilon})$-factor, for any $\\epsilon >0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^\\epsilon$-factor approximation algorithm for computing boxicity is known, for any $\\epsilon <1$. In this paper, we study the boxicity problem on Circular Arc graphs - intersection graphs of arcs of a circle. We give a $(2+\\frac{1}{k})$-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where $k \\ge 1$ is its ...
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.
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.
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
The Simultaneous Membership Problem for Chordal, Comparability and Permutation graphs
Jampani, Krishnam Raju
2008-01-01
In this paper we introduce the 'simultaneous membership problem', defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G_1 and G_2, sharing some vertices X (and the corresponding induced edges), are said to be 'simultaneous members' of graph class C, if there exist representations R_1 and R_2 of G_1 and G_2 that are "consistent" on X. Equivalently (for the classes C that we consider) there exist edges E' between G_1-X and G_2-X such that G_1 \\cup G_2 \\cup E' belongs to class C. Simultaneous membership problems have application 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 version of the other. Simultaneous membership problems are related to simultaneous planar embeddings, graph sandwich problems and probe graph recognition problems. In this paper we give effici...
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
The probability that a complete intersection is smooth
Bucur, Alina
2010-01-01
Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two curves in projective 3-space is strictly less than the number of points on the projective line.
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...
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.
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.
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
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…
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.
On KL-convergent graphs%线团-收敛图
王艳; 钱建国
2002-01-01
The line-clique graph, KL(G), of a graph G is the intersection graph of the cliques of the line graph L(G) of G. For a natural number n, a graph G is n-KL-convergent if the nth iterated-line-clique graph KLn(G) is isomorphic to a fixed graph G′. A graph G is KL-convergent if there is a natural number n such that G is n-KL-convergent. Otherwise G is divergent.In this paper, KL-convergent graphs and divergent graphs are characterized. A method for constructing any KL-convergent graph from a graph triangle free is provided. We also discuss the KL-convergent index of KL-convergent graphs.%一个图的线团图就是这个图的线图的团图. 对于自然数n,一个图被称为n-线团-收敛的,如果它的n次线团图同构于一个固定的图. 否则称之为发散的.本文刻画了线团-收敛图与发散图,给出一个线团-收敛图的构造方法. 并且,讨论了线团-收敛图的线团-收敛指数.
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.
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.
无
2007-01-01
In this paper, by using the Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory, associated with Lp intersection body and dual mixed volume, some dual Brunn-Minkowski inequalities and their isolate forms are established for Lp intersection body about the normalized Lp radial addition and Lp radial linear combination. Some properties of operator Lp are given.
INEQUALITIES FOR MIXED INTERSECTION BODIES
YUAN SHUFENG; LENG GANGSONG
2005-01-01
In this paper, some properties of mixed intersection bodies are given, and inequalities from the dual Brunn-Minkowski theory (such as the dual Minkowski inequality, the dual Aleksandrov-Fenchel inequalities and the. dual Brunn-Minkowski inequalities) are established for mixed intersection bodies.
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.
-Mixed Intersection Bodies and Star Duality
Zhao Chang-jian; Wing-Sum Cheung
2010-09-01
The paper extends the two notions of the dual mixed volumes and -intersection body to -dual mixed volumes and -mixed intersection body, respectively. Inequalities for the star dual of -mixed intersection bodies are established.
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...
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...
Modelling of urban traffic networkof signalized intersections
2013-01-01
This report presents how traffic network of signalized intersection in a chosen urban area called Tema is synchronized. Using a modular approach, two different types of traffic intersection commonly found in an urban area were modelled i.e. a simple intersection and a complex intersection. A direct road, even though not an intersection, was also included in the modelling because it’s commonly found in an urban area plus it connects any two intersections. Each of these scenarios was modelled u...
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
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
Intersectional perspective in elderly care
Marta Cuesta
2016-05-01
Full Text Available Earlier research has shown that power relationships at workplaces are constructed by power structures. Processes related to power always influence the working conditions for (in this study in elderly care the working groups involved. Power structures are central for intersectional analysis, in the sense that the intersectional perspective highlights aspects such as gender and ethnicity (subjective dimensions and interrelates them to processes of power (objective dimension. This qualitative study aims to explore in what way an intersectional perspective could contribute to increased knowledge of power structures in a nursing home where the employees were mostly immigrants from different countries. By using reflexive dialogues related to an intersectional perspective, new knowledge which contributes to the employees’ well-being could develop. Narrative analysis was the method used to conduct this study. Through a multi-stage focus group on six occasions over 6 months, the staff were engaged in intersectional and critical reflections about power relationship with the researchers, by identifying patterns in their professional activities that could be connected to their subjectivities (gender, ethnicity, etc.. The result of this study presents three themes that express the staff's experiences and connect these experiences to structural discrimination. 1 Intersectionality, knowledge, and experiences of professionalism; 2 Intersectionality, knowledge, and experiences of collaboration; and 3 Intersectionality, knowledge, and experiences of discrimination. The result demonstrates that an intersectional perspective reinforces the involved abilities, during the conversations, into being clear about, for example, their experiences of discrimination, and consequently developing a better understanding of their professionalism and collaboration. Such deeper reflections became possible through a process of consciousness raising, strengthening the employee
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.
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.
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)=\\...
Modeling signalized intersection safety with corridor-level spatial correlations.
Guo, Feng; Wang, Xuesong; Abdel-Aty, Mohamed A
2010-01-01
Intersections in close spatial proximity along a corridor should be considered as correlated due to interacted traffic flows as well as similar road design and environmental characteristics. It is critical to incorporate this spatial correlation for assessing the true safety impacts of risk factors. In this paper, several Bayesian models were developed to model the crash data from 170 signalized intersections in the state of Florida. The safety impacts of risk factors such as geometric design features, traffic control, and traffic flow characteristics were evaluated. The Poisson and Negative Binomial Bayesian models with non-informative priors were fitted but the focus is to incorporate spatial correlations among intersections. Two alternative models were proposed to capture this correlation: (1) a mixed effect model in which the corridor-level correlation is incorporated through a corridor-specific random effect and (2) a conditional autoregressive model in which the magnitude of correlations is determined by spatial distances among intersections. The models were compared using the Deviance Information Criterion. The results indicate that the Poisson spatial model provides the best model fitting. Analysis of the posterior distributions of model parameters indicated that the size of intersection, the traffic conditions by turning movement, and the coordination of signal phase have significant impacts on intersection safety.
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.
Statistical mechanics of graph models and their implications for emergent manifolds
Chen, Si
2012-01-01
Inspired by "quantum graphity" models for spacetime, a statistical model of graphs is proposed to explore possible realizations of emergent manifolds. Graphs with given numbers of vertices and edges are considered, governed by a very general Hamiltonian that merely favors graphs with near-constant valency and local rotational symmetry. The ratio of vertices to edges controls the dimensionality of the emergent manifold. The model is simulated numerically in the canonical ensemble for a given vertex to edge ratio, where it is found that the low energy states are almost triangulations of two dimensional manifolds. The resulting manifold shows topological "handles" and surface intersections in a higher embedding space as well as non-trivial fractal dimension. The transition is first order, underlying the difficulty of graph models in describing criticality that is independent of the details of the underlying graph. Another interesting phenomenon is that the entropy of the graphs are super-extensive, a fact known ...
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...
Fractional graph theory a rational approach to the theory of graphs
Scheinerman, Edward R
2013-01-01
A unified treatment of the most important results in the study of fractional graph concepts, this volume explores the various ways in which integer-valued concepts can be modified to derive nonintegral values. It begins with the general fractional theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics. Subjects include fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, and fractional isomorphism. The final chapter examines additional topics such as fractional domination, fractional intersection numbers
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...
团图为树的图中的团横贯数%Clique-transversal number of graphs whose clique-graphs are trees
梁作松; 单而芳
2008-01-01
Given a graph G,a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and |C|≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted as TC (G), is the minimum cardinality of a clique-transversal set in G. The clique-graph of G, denoted as K (G), is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques in G have nonempty intersection. Let F be a class of graphs G such that F={G|K(G) is a tree}. In this paper the graphs in F having independent clique-transversal sets are shown and thus TC (G)/|G|≤1/2 for all G ∈ F.
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...
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…
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.
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.
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.
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.
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...
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.
Intersectionality in European Union policymaking
Lombardo, Emanuela; Agustin, Lise Rolandsen
2016-01-01
is particularly apt to deal with equality and diversity in policymaking. By analysing a selection of European Union policy documents on gender-based violence in the period 2000–2014, we attend to the question of what intersectionality can bring to policymaking in terms of strengthening inclusiveness and address...
Dhifli, Wajdi
2013-01-01
This thesis is in the intersection of two proliferating research fields, namely data mining and bioinformatics. With the emergence of graph data in the last few years, many efforts have been devoted to mining frequent subgraphs from graph databases. Yet, the number of discovered frequentsubgraphs is usually exponential, mainly because of the combinatorial nature of graphs. Many frequent subgraphs are irrelevant because they are redundant or just useless for the user. Besides, their high numbe...
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).
Reconstructing surface triangulations by their intersection matrices
2014-01-01
The intersection matrix of a finite simplicial complex has as each of its entries the rank of the intersection of its respective simplices. We prove that such matrix defines the triangulation of a closed connected surface up to isomorphism.
Using Intersectionality in Student Affairs Research
Strayhorn, Terrell L.
2017-01-01
This chapter presents intersectionality as a useful heuristic for conducting research in higher education and student affairs contexts. Much more than just another theory, intersectionality can powerfully shape student affairs research in both obvious and tacit ways.
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].
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.
Subdominant pseudoultrametric on graphs
Dovgoshei, A A; Petrov, E A [Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (Ukraine)
2013-08-31
Let (G,w) be a weighted graph. We find necessary and sufficient conditions under which the weight w:E(G)→R{sup +} can be extended to a pseudoultrametric on V(G), and establish a criterion for the uniqueness of such an extension. We demonstrate that (G,w) is a complete k-partite graph, for k≥2, if and only if for any weight that can be extended to a pseudoultrametric, among all such extensions one can find the least pseudoultrametric consistent with w. We give a structural characterization of graphs for which the subdominant pseudoultrametric is an ultrametric for any strictly positive weight that can be extended to a pseudoultrametric. Bibliography: 14 titles.
White, AT
1985-01-01
The field of topological graph theory has expanded greatly in the ten years since the first edition of this book appeared. The original nine chapters of this classic work have therefore been revised and updated. Six new chapters have been added, dealing with: voltage graphs, non-orientable imbeddings, block designs associated with graph imbeddings, hypergraph imbeddings, map automorphism groups and change ringing.Thirty-two new problems have been added to this new edition, so that there are now 181 in all; 22 of these have been designated as ``difficult'''' and 9 as ``unsolved''''. Three of the four unsolved problems from the first edition have been solved in the ten years between editions; they are now marked as ``difficult''''.
Iacovacci, Jacopo
2015-01-01
Visibility algorithms transform time series into graphs and encode dynamical information in their topology, paving the way for graph-theoretical time series analysis as well as building a bridge between nonlinear dynamics and network science. In this work we introduce and study the concept of visibility graph motifs, smaller substructures that appear with characteristic frequencies. We develop a theory to compute in an exact way the motif profiles associated to general classes of deterministic and stochastic dynamics. We find that this simple property is indeed a highly informative and computationally efficient feature capable to distinguish among different dynamics and robust against noise contamination. We finally confirm that it can be used in practice to perform unsupervised learning, by extracting motif profiles from experimental heart-rate series and being able, accordingly, to disentangle meditative from other relaxation states. Applications of this general theory include the automatic classification a...
Ribes, Luis
2017-01-01
This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open quest...
Hyperbolicity in Median Graphs
José M Sigarreta
2013-11-01
If is a geodesic metric space and $x_1,x_2,x_3\\in X$, a geodesic triangle $T=\\{x_1,x_2,x_3\\}$ is the union of the three geodesics $[x_1 x_2],[x_2 x_3]$ and $[x_3 x_1]$ in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . If is hyperbolic, we denote by () the sharp hyperbolicity constant of , i.e.,$(X)=\\inf\\{≥ 0: X \\quad\\text{is}\\quad -\\text{hyperbolic}\\}$. In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.
Erickson, Lindsay
2010-01-01
The game of Nim as played on graphs was introduced in Nim on Graphs I and extended in Nim on Graphs II by Masahiko Fukuyama. His papers detail the calculation of Grundy numbers for graphs under specific circumstances. We extend these results and introduce the strategy for even cycles. This paper examines a more general class of graphs by restricting the edge weight to one. We provide structural conditions for which there exist a winning strategy. This yields the solution for the complete graph.
Graph theory and interconnection networks
Hsu, Lih-Hsing
2008-01-01
The advancement of large scale integrated circuit technology has enabled the construction of complex interconnection networks. Graph theory provides a fundamental tool for designing and analyzing such networks. Graph Theory and Interconnection Networks provides a thorough understanding of these interrelated topics. After a brief introduction to graph terminology, the book presents well-known interconnection networks as examples of graphs, followed by in-depth coverage of Hamiltonian graphs. Different types of problems illustrate the wide range of available methods for solving such problems. The text also explores recent progress on the diagnosability of graphs under various models.
Randerath, Bert; Vestergaard, Preben D.
2010-01-01
A graph G is P3-equipackable if any sequence of successive removals of edge-disjoint copies of P3 from G always terminates with a graph having at most one edge. All P3-equipackable graphs are characterised. They belong to a small number of families listed here.......A graph G is P3-equipackable if any sequence of successive removals of edge-disjoint copies of P3 from G always terminates with a graph having at most one edge. All P3-equipackable graphs are characterised. They belong to a small number of families listed here....
Feynman motives of banana graphs
Aluffi, Paolo
2008-01-01
We consider the infinite family of Feynman graphs known as the ``banana graphs'' and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern--Schwartz--MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.
Locally identifying coloring of graphs
Esperet, Louis; Montassier, Mickael; Ochem, Pascal; Parreau, Aline
2010-01-01
A vertex-coloring of a graph G is said to be locally identifying if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct. In this paper, we give several bounds on the minimum number of colors needed in such a coloring for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether a subcubic bipartite graph with large girth has a locally identifying coloring with 3 colors is an NP-complete problem.
Macdonald Robin
2006-12-01
Full Text Available Abstract Background Australian epidemiologists have recognised that lay readers have difficulty understanding statistical graphs in reports on population health. This study aimed to provide evidence for graph design improvements that increase comprehension by non-experts. Methods This was a double-blind, randomised, controlled trial of graph-design interventions, conducted as a postal survey. Control and intervention participants were randomly selected from telephone directories of health system employees. Eligible participants were on duty at the listed location during the study period. Controls received a booklet of 12 graphs from original publications, and intervention participants received a booklet of the same graphs with design modifications. A questionnaire with 39 interpretation tasks was included with the booklet. Interventions were assessed using the ratio of the prevalence of correct responses given by the intervention group to those given by the control group for each task. Results The response rate from 543 eligible participants (261 intervention and 282 control was 67%. The prevalence of correct answers in the control group ranged from 13% for a task requiring knowledge of an acronym to 97% for a task identifying the largest category in a pie chart. Interventions producing the greatest improvement in comprehension were: changing a pie chart to a bar graph (3.6-fold increase in correct point reading, changing the y axis of a graph so that the upward direction represented an increase (2.9-fold increase in correct judgement of trend direction, a footnote to explain an acronym (2.5-fold increase in knowledge of the acronym, and matching the y axis range of two adjacent graphs (two-fold increase in correct comparison of the relative difference in prevalence between two population subgroups. Conclusion Profound population health messages can be lost through use of overly technical language and unfamiliar statistical measures. In our
Interactive exploration of large-scale time-varying data using dynamic tracking graphs
Widanagamaachchi, W.
2012-10-01
Exploring and analyzing the temporal evolution of features in large-scale time-varying datasets is a common problem in many areas of science and engineering. One natural representation of such data is tracking graphs, i.e., constrained graph layouts that use one spatial dimension to indicate time and show the "tracks" of each feature as it evolves, merges or disappears. However, for practical data sets creating the corresponding optimal graph layouts that minimize the number of intersections can take hours to compute with existing techniques. Furthermore, the resulting graphs are often unmanageably large and complex even with an ideal layout. Finally, due to the cost of the layout, changing the feature definition, e.g. by changing an iso-value, or analyzing properly adjusted sub-graphs is infeasible. To address these challenges, this paper presents a new framework that couples hierarchical feature definitions with progressive graph layout algorithms to provide an interactive exploration of dynamically constructed tracking graphs. Our system enables users to change feature definitions on-the-fly and filter features using arbitrary attributes while providing an interactive view of the resulting tracking graphs. Furthermore, the graph display is integrated into a linked view system that provides a traditional 3D view of the current set of features and allows a cross-linked selection to enable a fully flexible spatio-temporal exploration of data. We demonstrate the utility of our approach with several large-scale scientific simulations from combustion science. © 2012 IEEE.
Graph-based knowledge representation computational foundations of conceptual graphs
Chein, Michel; Chein, Michel
2008-01-01
In addressing the question of how far it is possible to go in knowledge representation and reasoning through graphs, the authors cover basic conceptual graphs, computational aspects, and kernel extensions. The basic mathematical notions are summarized.
Algorithms for Planar Graphs and Graphs in Metric Spaces
Wulff-Nilsen, Christian
Algorithms for network problems play an increasingly important role in modern society. The graph structure of a network is an abstract and very useful representation that allows classical graph algorithms, such as Dijkstra and Bellman-Ford, to be applied. Real-life networks often have additional...... preprocessing time, an O(n log n) time algorithm for the replacement paths problem, and a min st-cut oracle with nearlinear preprocessing time. We also give improved time bounds for computing various graph invariants such as diameter and girth. In the second part, we consider stretch factor problems...... for geometric graphs and graphs embedded in metric spaces. Roughly speaking, the stretch factor is a real value expressing how well a (geo-)metric graph approximates the underlying complete graph w.r.t. distances. We give improved algorithms for computing the stretch factor of a given graph and for augmenting...
SOME RESULTS ON CIRCULAR PERFECT GRAPHS AND PERFECT GRAPHS
XU Baogang
2005-01-01
An r-circular coloring of a graph G is a map f from V(G) to the set of open unit intervals of an Euclidean circle of length r,such that f(u) ∩ f(v) = φ whenever uv ∈ E(G).Circular perfect graphs are defined analogously to perfect graphs by means of two parameters,the circular chromatic number and the circular clique number.In this paper,we study the properties of circular perfect graphs.We give (1) a necessary condition for a graph to be circular perfect,(2) some circular critical imperfect graphs,and (3) a characterization of graphs with the property that each of their induced subgraphs has circular clique number the same as its clique number,and then the two conjectures that are equivalent to the perfect graph conjecture.
Optimal graph based segmentation using flow lines with application to airway wall segmentation
Petersen, Jens; Nielsen, Mads; Lo, Pechin
2011-01-01
This paper introduces a novel optimal graph construction method that is applicable to multi-dimensional, multi-surface segmentation problems. Such problems are often solved by refining an initial coarse surface within the space given by graph columns. Conventional columns are not well suited...... images. Comparison with manual annotations on 649 cross-sectional images from 15 different subjects shows significantly smaller contour distances and larger area of overlap than are obtained with recently published graph based methods. Airway abnormality measurements obtained with the method on 480 scan...... for surfaces with high curvature or complex shapes but the proposed columns, based on properly generated flow lines, which are non-intersecting, guarantee solutions that do not self-intersect and are better able to handle such surfaces. The method is applied to segment human airway walls in computed tomography...
Optimal Graph Based Segmentation using Flow Lines with Application to Airway Wall Segmentation
Petersen, Jens; Nielsen, Mads; Lo, Pechin Chien Pau;
2011-01-01
This paper introduces a novel optimal graph construction method that is applicable to multi-dimensional, multi-surface segmentation problems. Such problems are often solved by refining an initial coarse surface within the space given by graph columns. Conventional columns are not well suited...... for surfaces with high curvature or complex shapes but the proposed columns, based on properly generated flow lines, which are non-intersecting, guarantee solutions that do not self-intersect and are better able to handle such surfaces. The method is applied to segment human airway walls in computed tomography...... images. Comparison with manual annotations on 649 cross-sectional images from 15 different subjects shows significantly smaller contour distances and larger area of overlap than are obtained with recently published graph based methods. Airway abnormality measurements obtained with the method on 480 scan...
ZHANG Guoqiang; CHEN Yixiang
2001-01-01
This paper provides a concrete and simple introduction to two pillars of domain theory: (1) solving recursive domain equations, and (2) universal and saturated domains. Our exposition combines Larsen and Winskel's idea on solving domain equations using information systems with Girard's idea of stable domain theory in the form of coherence spaces, or graphs.Detailed constructions are given for universal and even homogeneous objects in two categories of graphs: one representing binary complete, prime algebraic domains with complete primes covering the bottom; the other representing ω-algebraic, prime algebraic lattices. The backand-forth argument in model theory helps to enlighten the constructions.
Cheung, King Sing
2014-01-01
Petri nets are a formal and theoretically rich model for the modelling and analysis of systems. A subclass of Petri nets, augmented marked graphs possess a structure that is especially desirable for the modelling and analysis of systems with concurrent processes and shared resources.This monograph consists of three parts: Part I provides the conceptual background for readers who have no prior knowledge on Petri nets; Part II elaborates the theory of augmented marked graphs; finally, Part III discusses the application to system integration. The book is suitable as a first self-contained volume
Haynes Teresa W.; Hedetniemi Stephen T.; Jamieson Jessie D.; Jamieson William B.
2014-01-01
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 grap...
Stevanovic, Dragan
2015-01-01
Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the
Distributed Evolutionary Graph Partitioning
Sanders, Peter
2011-01-01
We present a novel distributed evolutionary algorithm, KaFFPaE, to solve the Graph Partitioning Problem, which makes use of KaFFPa (Karlsruhe Fast Flow Partitioner). The use of our multilevel graph partitioner KaFFPa provides new effective crossover and mutation operators. By combining these with a scalable communication protocol we obtain a system that is able to improve the best known partitioning results for many inputs in a very short amount of time. For example, in Walshaw's well known benchmark tables we are able to improve or recompute 76% of entries for the tables with 1%, 3% and 5% imbalance.
Continuum Percolation in the Intrinsically Secure Communications Graph
Pinto, Pedro C
2010-01-01
The intrinsically secure communications graph (iS-graph) is a random graph which captures the connections that can be securely established over a large-scale network, in the presence of eavesdroppers. It is based on principles of information-theoretic security, widely accepted as the strictest notion of security. In this paper, we are interested in characterizing the global properties of the iS-graph in terms of percolation on the infinite plane. We prove the existence of a phase transition in the Poisson iS-graph, whereby an unbounded component of securely connected nodes suddenly arises as we increase the density of legitimate nodes. Our work shows that long-range communication in a wireless network is still possible when a secrecy constraint is present.
New Meta-Heuristic for Combinatorial Optimization Problems:Intersection Based Scaling
Peng Zou; Zhi Zhou; Ying-Yu Wan; Guo-Liang Chen; Jun Gu
2004-01-01
Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviation),is proposed and it can be applied to the combinatorial optimization problems. The main idea of IBS is to scale the size of the instance based on the intersection of some local optima, and to simplify the search space by extracting the intersection from the instance, which makes the search more efficient. The combination of IBS with some local search heuristics of different combinatorial optimization problems such as Traveling Salesman Problem (TSP) and Graph Partitioning Problem (GPP) is studied, and comparisons are made with some of the best heuristic algorithms and meta-heuristic algorithms. It is found that it has significantly improved the performance of existing local search heuristics and significantly outperforms the known best algorithms.
Handbook of graph grammars and computing by graph transformation
Engels, G; Kreowski, H J; Rozenberg, G
1999-01-01
Graph grammars originated in the late 60s, motivated by considerations about pattern recognition and compiler construction. Since then, the list of areas which have interacted with the development of graph grammars has grown quite impressively. Besides the aforementioned areas, it includes software specification and development, VLSI layout schemes, database design, modeling of concurrent systems, massively parallel computer architectures, logic programming, computer animation, developmental biology, music composition, visual languages, and many others.The area of graph grammars and graph tran
GraphXML: an XML based graph interchange format
I. Herman (Ivan); M.S. Marshall (Scott)
2000-01-01
textabstractGraphXML is a graph description language in XML that can be used as an interchange format for graph drawing and visualization packages. The generality and rich features of XML make it possible to define an interchange format that not only supports the pure, mathematical description of a
Random graphs: from static to dynamic
Van den Esker, H.
2008-01-01
Many empirical studies on real-life networks show that many networks are small worlds, meaning that typical distances in these networks are small, and many of them have power-law degree sequences, meaning that the number of nodes with degree k falls off as kˆ (-τ) for some exponent τ>
Composite fluxbranes with general intersections
Ivashchuk, V D
2002-01-01
Generalized composite fluxbrane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold which contains a product of n Ricci-flat spaces M_1 x ... x M_n with 1-dimensional M_1. They are defined up to a set of functions H_s obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H_s for intersections related to semisimple Lie algebras is suggested. This conjecture is valid for Lie algebras: A_m, C_{m+1}, m > 0. For simple Lie algebras the powers of polynomials coincide with the components of the dual Weyl vector in the basis of simple roots. Explicit formulas for A_1 + ... + A_1 (orthogonal), "block-ortogonal" and A_2 solutions are obtained. Certain examples of solutions in D = 11 and D =10 (II A) supergravities (e.g. with A_2 intersection rules) and Kaluza-Klein dyonic A_2 flux tube, are considered.
Epidemics and vaccination on weighted graphs
Deijfen, Maria
2011-01-01
A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge $(u,v)$ in the graph is equipped with two weights $W_{(u,v)}$ and $W_{(v,u)}$ that represent the (subjective) strength of the connection and determine the probability that $u$ infects $v$ in case $u$ is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d.\\ weights and for weights that are functions of the degrees. For i.i.d.\\ weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.
How mutation affects evolutionary games on graphs.
Allen, Benjamin; Traulsen, Arne; Tarnita, Corina E; Nowak, Martin A
2012-04-21
Evolutionary dynamics are affected by population structure, mutation rates and update rules. Spatial or network structure facilitates the clustering of strategies, which represents a mechanism for the evolution of cooperation. Mutation dilutes this effect. Here we analyze how mutation influences evolutionary clustering on graphs. We introduce new mathematical methods to evolutionary game theory, specifically the analysis of coalescing random walks via generating functions. These techniques allow us to derive exact identity-by-descent (IBD) probabilities, which characterize spatial assortment on lattices and Cayley trees. From these IBD probabilities we obtain exact conditions for the evolution of cooperation and other game strategies, showing the dual effects of graph topology and mutation rate. High mutation rates diminish the clustering of cooperators, hindering their evolutionary success. Our model can represent either genetic evolution with mutation, or social imitation processes with random strategy exploration.
Graph theoretical model of a sensorimotor connectome in zebrafish.
Michael Stobb
Full Text Available Mapping the detailed connectivity patterns (connectomes of neural circuits is a central goal of neuroscience. The best quantitative approach to analyzing connectome data is still unclear but graph theory has been used with success. We present a graph theoretical model of the posterior lateral line sensorimotor pathway in zebrafish. The model includes 2,616 neurons and 167,114 synaptic connections. Model neurons represent known cell types in zebrafish larvae, and connections were set stochastically following rules based on biological literature. Thus, our model is a uniquely detailed computational representation of a vertebrate connectome. The connectome has low overall connection density, with 2.45% of all possible connections, a value within the physiological range. We used graph theoretical tools to compare the zebrafish connectome graph to small-world, random and structured random graphs of the same size. For each type of graph, 100 randomly generated instantiations were considered. Degree distribution (the number of connections per neuron varied more in the zebrafish graph than in same size graphs with less biological detail. There was high local clustering and a short average path length between nodes, implying a small-world structure similar to other neural connectomes and complex networks. The graph was found not to be scale-free, in agreement with some other neural connectomes. An experimental lesion was performed that targeted three model brain neurons, including the Mauthner neuron, known to control fast escape turns. The lesion decreased the number of short paths between sensory and motor neurons analogous to the behavioral effects of the same lesion in zebrafish. This model is expandable and can be used to organize and interpret a growing database of information on the zebrafish connectome.
Graph theoretical model of a sensorimotor connectome in zebrafish.
Stobb, Michael; Peterson, Joshua M; Mazzag, Borbala; Gahtan, Ethan
2012-01-01
Mapping the detailed connectivity patterns (connectomes) of neural circuits is a central goal of neuroscience. The best quantitative approach to analyzing connectome data is still unclear but graph theory has been used with success. We present a graph theoretical model of the posterior lateral line sensorimotor pathway in zebrafish. The model includes 2,616 neurons and 167,114 synaptic connections. Model neurons represent known cell types in zebrafish larvae, and connections were set stochastically following rules based on biological literature. Thus, our model is a uniquely detailed computational representation of a vertebrate connectome. The connectome has low overall connection density, with 2.45% of all possible connections, a value within the physiological range. We used graph theoretical tools to compare the zebrafish connectome graph to small-world, random and structured random graphs of the same size. For each type of graph, 100 randomly generated instantiations were considered. Degree distribution (the number of connections per neuron) varied more in the zebrafish graph than in same size graphs with less biological detail. There was high local clustering and a short average path length between nodes, implying a small-world structure similar to other neural connectomes and complex networks. The graph was found not to be scale-free, in agreement with some other neural connectomes. An experimental lesion was performed that targeted three model brain neurons, including the Mauthner neuron, known to control fast escape turns. The lesion decreased the number of short paths between sensory and motor neurons analogous to the behavioral effects of the same lesion in zebrafish. This model is expandable and can be used to organize and interpret a growing database of information on the zebrafish connectome.
Wenjun Xiao
2002-01-01
Wu, Lakshmivarahan and Dhall[5] recently described a deterministic, distributed routing scheme for some special classes of metacyclic graphs. However they have no proof of correctness that the scheme is a shortest path routing algorithm. In the note we give a suboptimal, deterministic routing algorithm.
Nemirovsky, Ricardo; Tierney, Cornelia; Wright, Tracy
1998-01-01
Analyzed two children's use of a computer-based motion detector to make sense of symbolic expressions (Cartesian graphs). Found three themes: (1) tool perspectives, efforts to understand graphical responses to body motion; (2) fusion, emergent ways of talking and behaving that merge symbols and referents; and (3) graphical spaces, when changing…
Pitts Bannister, Vanessa R.; Jamar, Idorenyin; Mutegi, Jomo W.
2007-01-01
In this article, the learning progress of one fifth-grade student is examined with regard to the development of her graph interpretation skills as she participated in the Junior Science Institute (JSI), a two-week, science intensive summer camp in which participants engaged in microbiology research and application. By showcasing the student's…
S.M. Heditniemi (Sandra); R.C. Laskar (R.C.); H.M. Mulder (Martyn)
2012-01-01
textabstractLet $G = (V,E)$ be a graph. A partition $\\pi = \\{V_1, V_2, \\ldots, V_k \\}$ of the vertices $V$ of $G$ into $k$ {\\it color classes} $V_i$, with $1 \\leq i \\leq k$, is called a {\\it quorum coloring} if for every vertex $v \\in V$, at least half of the vertices in the closed neighborhood
Neural networks and graph theory
许进; 保铮
2002-01-01
The relationships between artificial neural networks and graph theory are considered in detail. The applications of artificial neural networks to many difficult problems of graph theory, especially NP-complete problems, and the applications of graph theory to artificial neural networks are discussed. For example graph theory is used to study the pattern classification problem on the discrete type feedforward neural networks, and the stability analysis of feedback artificial neural networks etc.
Temporal Representation in Semantic Graphs
Levandoski, J J; Abdulla, G M
2007-08-07
A wide range of knowledge discovery and analysis applications, ranging from business to biological, make use of semantic graphs when modeling relationships and concepts. Most of the semantic graphs used in these applications are assumed to be static pieces of information, meaning temporal evolution of concepts and relationships are not taken into account. Guided by the need for more advanced semantic graph queries involving temporal concepts, this paper surveys the existing work involving temporal representations in semantic graphs.
李浩; 刘群
1989-01-01
Because of the widespread applications of tree and treee graph in computer science,we are interested in studying the reee graph.M.Farber,B.Richter and H.Shang in [1] showed that the graph τ2(G)is 2-edge-connected as |V(G)）≥3，at the same time,we will show the best lower bounds about vertex number and minimum degree of graph τ2(G）.
Cascades on clique-based graphs
Hackett, Adam
2013-01-01
We present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of highly-clustered random graphs introduced in [J. P. Gleeson, Phys. Rev. E 80, 036107 (2009)]. A condition for the existence of global cascades is also derived. Applications of this approach include analyses of percolation, and Watts's model. We show how our techniques can be used to study the effects of in-group bias in cascades on social networks.
Winlaw, Manda [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); De Sterck, Hans [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Sanders, Geoffrey [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2015-10-26
In very simple terms a network can be de ned as a collection of points joined together by lines. Thus, networks can be used to represent connections between entities in a wide variety of elds including engi- neering, science, medicine, and sociology. Many large real-world networks share a surprising number of properties, leading to a strong interest in model development research and techniques for building synthetic networks have been developed, that capture these similarities and replicate real-world graphs. Modeling these real-world networks serves two purposes. First, building models that mimic the patterns and prop- erties of real networks helps to understand the implications of these patterns and helps determine which patterns are important. If we develop a generative process to synthesize real networks we can also examine which growth processes are plausible and which are not. Secondly, high-quality, large-scale network data is often not available, because of economic, legal, technological, or other obstacles [7]. Thus, there are many instances where the systems of interest cannot be represented by a single exemplar network. As one example, consider the eld of cybersecurity, where systems require testing across diverse threat scenarios and validation across diverse network structures. In these cases, where there is no single exemplar network, the systems must instead be modeled as a collection of networks in which the variation among them may be just as important as their common features. By developing processes to build synthetic models, so-called graph generators, we can build synthetic networks that capture both the essential features of a system and realistic variability. Then we can use such synthetic graphs to perform tasks such as simulations, analysis, and decision making. We can also use synthetic graphs to performance test graph analysis algorithms, including clustering algorithms and anomaly detection algorithms.
Skurnick, Ronald; Davi, Charles; Skurnick, Mia
2005-01-01
Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian…
Mining and Indexing Graph Databases
Yuan, Dayu
2013-01-01
Graphs are widely used to model structures and relationships of objects in various scientific and commercial fields. Chemical molecules, proteins, malware system-call dependencies and three-dimensional mechanical parts are all modeled as graphs. In this dissertation, we propose to mine and index those graph data to enable fast and scalable search.…
Text analysis for knowledge graphs
Popping, Roel
2007-01-01
The concept of knowledge graphs is introduced as a method to represent the state of the art in a specific scientific discipline. Next the text analysis part in the construction of such graphs is considered. Here the 'translation' from text to graph takes place. The method that is used here is compar
Hopkins, Brian
2004-01-01
The interconnected world of actors and movies is a familiar, rich example for graph theory. This paper gives the history of the "Kevin Bacon Game" and makes extensive use of a Web site to analyze the underlying graph. The main content is the classroom development of the weighted average to determine the best choice of "center" for the graph. The…
Mining and Indexing Graph Databases
Yuan, Dayu
2013-01-01
Graphs are widely used to model structures and relationships of objects in various scientific and commercial fields. Chemical molecules, proteins, malware system-call dependencies and three-dimensional mechanical parts are all modeled as graphs. In this dissertation, we propose to mine and index those graph data to enable fast and scalable search.…
Submanifolds Weakly Associated with Graphs
A Carriazo; L M Fernández; A Rodríguez-Hidalgo
2009-06-01
We establish an interesting link between differential geometry and graph theory by defining submanifolds weakly associated with graphs. We prove that, in a local sense, every submanifold satisfies such an association, and other general results. Finally, we study submanifolds associated with graphs either in low dimensions or belonging to some special families.
On the mixing time of geographical threshold graphs
Bradonjic, Milan [Los Alamos National Laboratory
2009-01-01
In this paper, we study the mixing time of 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). We specifically study the mixing times of random walks on 2-dimensional GTGs near the connectivity threshold. We provide a set of criteria on the distribution of vertex weights that guarantees that the mixing time is {Theta}(n log n).
Investigation of Zero Knowledge Proof Approaches Based on Graph Theory
2011-02-01
instances (architectural floor plans ) making it difficult to evaluate in terms of the more general instances seen in the graph databases...Intelligent Control and Automation: 7902-7906, June 2008. Notes: Introduces a new ACO (ant colony optimization) algorithm and tests it against...random graphs G(n, p). Presents polynomial-time optimal algorithms for specific ranges of values for p. 74. I. Devarenne, A. Caminada, H. Mabed
Harary, Frank
2015-01-01
Presented in 1962-63 by experts at University College, London, these lectures offer a variety of perspectives on graph theory. Although the opening chapters form a coherent body of graph theoretic concepts, this volume is not a text on the subject but rather an introduction to the extensive literature of graph theory. The seminar's topics are geared toward advanced undergraduate students of mathematics.Lectures by this volume's editor, Frank Harary, include ""Some Theorems and Concepts of Graph Theory,"" ""Topological Concepts in Graph Theory,"" ""Graphical Reconstruction,"" and other introduc
Dynamic Representations of Sparse Graphs
Brodal, Gerth Stølting; Fagerberg, Rolf
1999-01-01
We present a linear space data structure for maintaining graphs with bounded arboricity—a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth—under edge insertions, edge deletions, and adjacency queries. The data structure supports adjacency queries in worst...... case O(c) time, and edge insertions and edge deletions in amortized O(1) and O(c+log n) time, respectively, where n is the number of nodes in the graph, and c is the bound on the arboricity....
Managing and Mining Graph Data
Aggarwal, Charu C
2010-01-01
Managing and Mining Graph Data is a comprehensive survey book in graph management and mining. It contains extensive surveys on a variety of important graph topics such as graph languages, indexing, clustering, data generation, pattern mining, classification, keyword search, pattern matching, and privacy. It also studies a number of domain-specific scenarios such as stream mining, web graphs, social networks, chemical and biological data. The chapters are written by well known researchers in the field, and provide a broad perspective of the area. This is the first comprehensive survey book in t
Resolvability in Circulant Graphs
Muhammad SALMAN; Imran JAVAID; Muhammad Anwar CHAUDHRY
2012-01-01
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u,v ∈ V(G) there is a vertex w ∈ W such that d(u,w) ≠ d(v,w).A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G,denoted by dim(G).For a vertex u of G and a subset S of V(G),the distance between u and S is the number mins∈s d(u,s).A k-partition H ={S1,S2,...,Sk} of V(G) is called a resolving partition if for every two distinct vertices u,v ∈ V(G) there is a set Si in Π such that d(u,Si) ≠ d(v,Si).The minimum k for which there is a resolving k-partition of V(G) is called the partition dimension of G,denoted by pd(G).The circulant graph is a graph with vertex set Zn,an additive group ofintegers modulo n,and two vertices labeled i and j adjacent if and only if i - j (mod n) ∈ C,where C C Zn has the property that C =-C and 0(∈) C.The circulant graph is denoted by Xn,△ where A =|C|.In this paper,we study the metric dimension of a family of circulant graphs Xn,3 with connection set C ={1,-n/2,n - 1} and prove that dim(Xn,3) is independent of choice of n by showing that 3 for all n =0 (mod 4),dim(X,n,3) ={ 4 for all n =2 (mod 4).We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C ={±1,±2} and prove that pd(Xn,4) is independent of choice of n and show that pd(X5,4) =5 and 3 forall odd n≥9,pd(Xn,4) ={ 4 for all even n ≥ 6 and n =7.
Horizontal visibility graphs from integer sequences
Lacasa, Lucas
2016-09-01
The horizontal visibility graph (HVG) is a graph-theoretical representation of a time series and builds a bridge between dynamical systems and graph theory. In recent years this representation has been used to describe and theoretically compare different types of dynamics and has been applied to characterize empirical signals, by extracting topological features from the associated HVGs which have shown to be informative on the class of dynamics. Among some other measures, it has been shown that the degree distribution of these graphs is a very informative feature that encapsulates nontrivial information of the series's generative dynamics. In particular, the HVG associated to a bi-infinite real-valued series of independent and identically distributed random variables is a universal exponential law P(k)=(1/3){(2/3)}k-2, independent of the series marginal distribution. Most of the current applications have however only addressed real-valued time series, as no exact results are known for the topological properties of HVGs associated to integer-valued series. In this paper we explore this latter situation and address univariate time series where each variable can only take a finite number n of consecutive integer values. We are able to construct an explicit formula for the parametric degree distribution {P}n(k), which we prove to converge to the continuous case for large n and deviates otherwise. A few applications are then considered.
Conditional coloring of some parameterized graphs
Reddy, P Venkata Subba
2010-01-01
For integers k>0 and r>0, a conditional (k,r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r,d(v)} different colors. The smallest integer k for which a graph G has a conditional (k,r)-coloring is called the rth order conditional chromatic number, denoted by $\\chi_r(G)$. For different values of r we obtain $\\chi_r(G)$ of certain parameterized graphs viz., Windmill graph, line graph of Windmill graph, middle graph of Friendship graph, middle graph of a cycle, line graph of Friendship graph, middle graph of complete k-partite graph and middle graph of a bipartite graph.
Hendrix, William; Jenkins, John; Padmanabhan, Kanchana; Chakraborty, Arpan
2014-01-01
Practical Graph Mining with R presents a "do-it-yourself" approach to extracting interesting patterns from graph data. It covers many basic and advanced techniques for the identification of anomalous or frequently recurring patterns in a graph, the discovery of groups or clusters of nodes that share common patterns of attributes and relationships, the extraction of patterns that distinguish one category of graphs from another, and the use of those patterns to predict the category of new graphs. Hands-On Application of Graph Data Mining Each chapter in the book focuses on a graph mining task, such as link analysis, cluster analysis, and classification. Through applications using real data sets, the book demonstrates how computational techniques can help solve real-world problems. The applications covered include network intrusion detection, tumor cell diagnostics, face recognition, predictive toxicology, mining metabolic and protein-protein interaction networks, and community detection in social networks. De...
Hierarchy of Modular Graph Identities
D'Hoker, Eric
2016-01-01
The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues are calculated analy...
Valiant Transform of Forney Graphs
Al-Bashabsheh, Ali
2010-01-01
The introduction of Forney graphs, or normal graphs, and the duality result therein [1] is a landmark in the theory of codes on graphs and in graph-based iterative decoding. A generic modeling framework for codes and systems, Forney graphs have since found various applications. It is unfortunate however that the development of the theory and application of Forney graphs to date has been restricted to the context of linear (and group) codes and systems, and the primary tool of Forney graphs is the duality result introduced in [1]. In a rather distant area of computer science, Valiant has recently presented a powerful family of new algorithms, which he calls holographic algorithms [2]. Using holographic algorithms, Valiant provides polynomial-time solutions to families of problems previously unknown to be tractable. At the heart of Valiant's holographic algorithms is the notion of "holographic reduction", which is the engine used in holographic algorithms to reduce from one problem to another. Recognizing the c...
Bond percolation on isoradial graphs
Grimmett, Geoffrey
2012-01-01
In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.
Projective modules and complete intersections
Mandal, Satya
1997-01-01
In these notes on "Projective Modules and Complete Intersections" an account on the recent developments in research on this subject is presented. The author's preference for the technique of Patching isotopic isomorphisms due to Quillen, formalized by Plumsted, over the techniques of elementary matrices is evident here. The treatment of Basic Element theory here incorporates Plumstead's idea of the "generalized dimension functions". These notes are highly selfcontained and should be accessible to any graduate student in commutative algebra or algebraic geometry. They include fully self-contained presentations of the theorems of Ferrand-Szpiro, Cowsik-Nori and the techniques of Lindel.
Normal Order: Combinatorial Graphs
Solomon, A I; Blasiak, P; Horzela, A; Penson, K A; Solomon, Allan I.; Duchamp, Gerard; Blasiak, Pawel; Horzela, Andrzej; Penson, Karol A.
2004-01-01
A conventional context for supersymmetric problems arises when we consider systems containing both boson and fermion operators. In this note we consider the normal ordering problem for a string of such operators. In the general case, upon which we touch briefly, this problem leads to combinatorial numbers, the so-called Rook numbers. Since we assume that the two species, bosons and fermions, commute, we subsequently restrict ourselves to consideration of a single species, single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, specifically Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. In this note we concentrate on the combinatorial graph approach, showing how some important classical results of graph theory lead to transparent representations of the combinatorial numbers associated with the boson normal ordering problem.
2010-12-02
evaluating the function ΘP (A) for any fixed A,P is equivalent to solving the so-called Quadratic Assignment Problem ( QAP ), and thus we can employ various...tractable linear programming, spectral, and SDP relaxations of QAP [40, 11, 33]. In particular we discuss recent work [14] on exploiting group...symmetry in SDP relaxations of QAP , which is useful for approximately computing elementary convex graph invariants in many interesting cases. Finally in
Syed, M. Qasim; Lovatt, Ian
2014-01-01
This paper is an addition to the series of papers on the exponential function begun by Albert Bartlett. In particular, we ask how the graph of the exponential function y = e[superscript -t/t] would appear if y were plotted versus ln t rather than the normal practice of plotting ln y versus t. In answering this question, we find a new way to…
Zhou, Feng; de la Torre, Fernando
2015-11-19
Graph matching (GM) is a fundamental problem in computer science, and it plays a central role to solve correspondence problems in computer vision. GM problems that incorporate pairwise constraints can be formulated as a quadratic assignment problem (QAP). Although widely used, solving the correspondence problem through GM has two main limitations: (1) the QAP is NP-hard and difficult to approximate; (2) GM algorithms do not incorporate geometric constraints between nodes that are natural in computer vision problems. To address aforementioned problems, this paper proposes factorized graph matching (FGM). FGM factorizes the large pairwise affinity matrix into smaller matrices that encode the local structure of each graph and the pairwise affinity between edges. Four are the benefits that follow from this factorization: (1) There is no need to compute the costly (in space and time) pairwise affinity matrix; (2) The factorization allows the use of a path-following optimization algorithm, that leads to improved optimization strategies and matching performance; (3) Given the factorization, it becomes straight-forward to incorporate geometric transformations (rigid and non-rigid) to the GM problem. (4) Using a matrix formulation for the GM problem and the factorization, it is easy to reveal commonalities and differences between different GM methods. The factorization also provides a clean connection with other matching algorithms such as iterative closest point; Experimental results on synthetic and real databases illustrate how FGM outperforms state-of-the-art algorithms for GM. The code is available at http://humansensing.cs.cmu.edu/fgm.
Lorscheid, Oliver
2010-01-01
Let $X$ be a curve over $\\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the g...
Kinetic Stable Delaunay Graphs
Agarwal, Pankaj K; Guibas, Leonidas J; Kaplan, Haim; Koltun, Vladlen; Rubin, Natan; Sharir, Micha
2011-01-01
We consider the problem of maintaining the Euclidean Delaunay triangulation $\\DT$ of a set $P$ of $n$ moving points in the plane, along algebraic trajectories of constant description complexity. Since the best known upper bound on the number of topological changes in the full $\\DT$ is nearly cubic, we seek to maintain a suitable portion of it that is less volatile yet retains many useful properties. We introduce the notion of a stable Delaunay graph, which is a dynamic subgraph of the Delaunay triangulation. The stable Delaunay graph (a) is easy to define, (b) experiences only a nearly quadratic number of discrete changes, (c) is robust under small changes of the norm, and (d) possesses certain useful properties. The stable Delaunay graph ($\\SDG$ in short) is defined in terms of a parameter $\\alpha>0$, and consists of Delaunay edges $pq$ for which the angles at which $p$ and $q$ see their Voronoi edge $e_{pq}$ are at least $\\alpha$. We show that (i) $\\SDG$ always contains at least roughly one third of the Del...
The phylogeny graphs of doubly partial orders
Park, Boram
2011-01-01
The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three as an induced subgraph, respectively. Phylogeny graphs are variant of competition graphs. The phylogeny graph $P(D)$ of a digraph $D$ is the (simple undirected) graph defined by $V(P(D)):=V(D)$ and $E(P(D)):=\\{xy \\mid N^+_D(x) \\cap N^+_D(y) \
Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures
Tyler Reese
2016-02-01
Full Text Available We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell’s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are “dual” Kirchhoff graphs. Given an example in which Maxwell’s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs.
Tropical intersection theory on R^n
Flossmann, Simon
2014-01-01
In these notes we survey the tropical intersection theory on R^n by deriving the properties for tropical cycles from the corresponding properties in Chow cohomology. For this we review the stable intersection product introduced by Mikhalkin and the push forward of tropical cycles defined by Allermann and Rau. Furthermore we define a pull back for tropical cycles based on the pull back of Minkowski weights. This pull back commutes with the tropical intersection product and satisfies the projec...
Tropical intersection theory on R^n
Flossmann, Simon
2014-01-01
In these notes we survey the tropical intersection theory on R^n by deriving the properties for tropical cycles from the corresponding properties in Chow cohomology. For this we review the stable intersection product introduced by Mikhalkin and the push forward of tropical cycles defined by Allermann and Rau. Furthermore we define a pull back for tropical cycles based on the pull back of Minkowski weights. This pull back commutes with the tropical intersection product and satisfies the projec...
2008-01-01
In this paper the author first introduce a new concept of Lp-dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of Lp- intersection body to Lp-mixed intersection body. Inequalities for Lp-dual mixed volumes of Lp-mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.
SATISFIABILITY METHODS FOR COLOURING GRAPHS
Munmun Dey
2013-02-01
Full Text Available The graph colouring problem can be solved using methods based on Satisfiability (SAT. An instance of the problem is defined by a Boolean expression written using Boolean variables and the logical connectives AND, OR and NOT. It has to be determined whether there is an assignment of TRUE and FALSE values to the variables that makes the entire expression true.A SAT problem is syntactically and semantically quite simple. Many Constraint Satisfaction Problems (CSPsin AI and OR can be formulated in SAT. These make use of two kinds of searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood search. Such methodshave been applied very successfully to find good solutions to hard decision problems
DESIGNING AN EFFECTIVE INTERSECTION USING CAD ENVIRONMENT
CRISAN George-Horea
2017-05-01
Full Text Available Ensuring the safety and streamline in road traffic are very important aims, with regard to the nowadays people mobility level. Road infrastructure is an essential element that can meet these requirements. Thus, it is proposed to develop an effective model of intersection by using CAD software tools. This type of intersection can be successfully used on almost any category of roads, increasing road traffic safety, reducing passing times through the intersection and in the same time, reducing conflict points and increase the intersection capacity.
POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS
无
2000-01-01
A vanishing theorem is proved for -adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibility of the poles or equivalently an improvement of the polar part of the AxKatz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.
Graphs cospectral with a friendship graph or its complement
Alireza Abdollahi
2013-12-01
Full Text Available Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length $4$ or $5$, then $Gcong F_n$. Moreover if $G$ is connected and planar then $Gcong F_n$.All but one of connected components of $G$ are isomorphic to $K_2$.The complement $overline{F_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $overline{F_n}$ is cospectral with a graph $H$, then $Hcong overline{F_n}$.
Evaluation of Graph Pattern Matching Workloads in Graph Analysis Systems
Hong, Seokyong [North Carolina State University (NCSU), Raleigh; Lee, Sangkeun (Matt) [ORNL; Lim, Seung-Hwan [ORNL; Sukumar, Sreenivas Rangan [ORNL; Vatsavai, Raju [North Carolina State University (NCSU), Raleigh
2016-01-01
Graph analysis has emerged as a powerful method for data scientists to represent, integrate, query, and explore heterogeneous data sources. As a result, graph data management and mining became a popular area of research, and led to the development of plethora of systems in recent years. Unfortunately, the number of emerging graph analysis systems and the wide range of applications, coupled with a lack of apples-to-apples comparisons, make it difficult to understand the trade-offs between different systems and the graph operations for which they are designed. A fair comparison of these systems is a challenging task for the following reasons: multiple data models, non-standardized serialization formats, various query interfaces to users, and diverse environments they operate in. To address these key challenges, in this paper we present a new benchmark suite by extending the Lehigh University Benchmark (LUBM) to cover the most common capabilities of various graph analysis systems. We provide the design process of the benchmark, which generalizes the workflow for data scientists to conduct the desired graph analysis on different graph analysis systems. Equipped with this extended benchmark suite, we present performance comparison for nine subgraph pattern retrieval operations over six graph analysis systems, namely NetworkX, Neo4j, Jena, Titan, GraphX, and uRiKA. Through the proposed benchmark suite, this study reveals both quantitative and qualitative findings in (1) implications in loading data into each system; (2) challenges in describing graph patterns for each query interface; and (3) different sensitivity of each system to query selectivity. We envision that this study will pave the road for: (i) data scientists to select the suitable graph analysis systems, and (ii) data management system designers to advance graph analysis systems.
An Algebraic Representation of Graphs and Applications to Graph Enumeration
Ângela Mestre
2013-01-01
Full Text Available We give a recursion formula to generate all the equivalence classes of connected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We use an algebraic graph representation to apply the result to the enumeration of connected graphs, all of whose biconnected components have the same number of vertices and edges. The proof uses Abel’s binomial theorem and generalizes Dziobek’s induction proof of Cayley’s formula.
Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs
Bensmail, Julien; Renault, Gabriel
2016-01-01
An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E1∪E2∪⋯∪Ek of the edge set E(G) such that each Ei induces a locally irregular graph. It was recently co...
Spectral Radius of Hamiltonian Planar Graphs and Outerplanar Graphs
周建; 林翠琴; 胡冠章
2001-01-01
The spectral radius is an important parameter of a graph related to networks. A method forestimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of aHamiltonian planar graph of order n ≥ 4 is less than or equal toand the spectral radius of theouterplanar graph of order n ≥ 6 is less than or equal to, which are improvements overprevious results. A direction for further study is then suggested.``
The traveling salesman problem on cubic and subcubic graphs
Boyd, Sylvia; van der Ster, Suzanne; Stougie, Leen
2011-01-01
We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on $n$ vertices a tour of length 4n/3-2 exists, which also implies the 4/3 conjecture, as an upper bound, for this class of graph-TSP. Recently, M\\"omke and Svensson presented a randomized algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3 conjectur...
Low-algorithmic-complexity entropy-deceiving graphs
Zenil, Hector; Kiani, Narsis A.; Tegnér, Jesper
2017-07-01
In estimating the complexity of objects, in particular, of graphs, it is common practice to rely on graph- and information-theoretic measures. Here, using integer sequences with properties such as Borel normality, we explain how these measures are not independent of the way in which an object, such as a graph, can be described or observed. From observations that can reconstruct the same graph and are therefore essentially translations of the same description, we see that when applying a computable measure such as the Shannon entropy, not only is it necessary to preselect a feature of interest where there is one, and to make an arbitrary selection where there is not, but also more general properties, such as the causal likelihood of a graph as a measure (opposed to randomness), can be largely misrepresented by computable measures such as the entropy and entropy rate. We introduce recursive and nonrecursive (uncomputable) graphs and graph constructions based on these integer sequences, whose different lossless descriptions have disparate entropy values, thereby enabling the study and exploration of a measure's range of applications and demonstrating the weaknesses of computable measures of complexity.
Studying the corona product of graphs under some graph invariants
M. Tavakoli
2014-09-01
Full Text Available The corona product $Gcirc H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V(G|$ copies of $H$; and by joining each vertex of the $i$-th copy of $H$ to the $i$-th vertex of $G$, where $1 leq i leq |V(G|$. In this paper, exact formulas for the eccentric distance sum and the edge revised Szeged indices of the corona product of graphs are presented. We also study the conditions under which the corona product of graphs produces a median graph.
Graph Coarsening for Path Finding in Cybersecurity Graphs
Hogan, Emilie A.; Johnson, John R.; Halappanavar, Mahantesh
2013-01-01
n the pass-the-hash attack, hackers repeatedly steal password hashes and move through a computer network with the goal of reaching a computer with high level administrative privileges. In this paper we apply graph coarsening in network graphs for the purpose of detecting hackers using this attack or assessing the risk level of the network's current state. We repeatedly take graph minors, which preserve the existence of paths in the graph, and take powers of the adjacency matrix to count the paths. This allows us to detect the existence of paths as well as find paths that have high risk of being used by adversaries.
Eilers, Søren; Sørensen, Adam P W
2011-01-01
We provide a complete invariant for graph C*-algebras which are amplified in the sense that whenever there is an edge between two vertices, there are infinitely many. The invariant used is the standard primitive ideal space adorned with a map into {-1,0,1,2,...}, and we prove that the classification result is strong in the sense that isomorphisms at the level of the invariant always lift. We extend the classification result to cover more graphs, and give a range result for the invariant (in the vein of Effros-Handelman-Shen) which is further used to prove that extensions of graph C*-algebras associated to amplified graphs are again graph C*-algebras of amplified graphs.
Dowding, Dawn; Merrill, Jacqueline A; Onorato, Nicole; Barrón, Yolanda; Rosati, Robert J; Russell, David
2017-04-27
To explore home care nurses' numeracy and graph literacy and their relationship to comprehension of visualized data. A multifactorial experimental design using online survey software. Nurses were recruited from 2 Medicare-certified home health agencies. Numeracy and graph literacy were measured using validated scales. Nurses were randomized to 1 of 4 experimental conditions. Each condition displayed data for 1 of 4 quality indicators, in 1 of 4 different visualized formats (bar graph, line graph, spider graph, table). A mixed linear model measured the impact of numeracy, graph literacy, and display format on data understanding. In all, 195 nurses took part in the study. They were slightly more numerate and graph literate than the general population. Overall, nurses understood information presented in bar graphs most easily (88% correct), followed by tables (81% correct), line graphs (77% correct), and spider graphs (41% correct). Individuals with low numeracy and low graph literacy had poorer comprehension of information displayed across all formats. High graph literacy appeared to enhance comprehension of data regardless of numeracy capabilities. Clinical dashboards are increasingly used to provide information to clinicians in visualized format, under the assumption that visual display reduces cognitive workload. Results of this study suggest that nurses' comprehension of visualized information is influenced by their numeracy, graph literacy, and the display format of the data. Individual differences in numeracy and graph literacy skills need to be taken into account when designing dashboard technology.
Dettlaff, Magda; Yero, Ismael G
2012-01-01
The bondage number $b(G)$ of a nonempty graph $G$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some Cartesian product, strong product or direct product of two paths.
Dettlaff, Magda; Lemanska, Magdalena; Yero, Ismael G.
2012-01-01
The bondage number $b(G)$ of a nonempty graph $G$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.
Opposed turns at signalized intersections
Akcelik, R.
1989-06-01
The 1985 Highway Capacity Manual (HCM) brought the U.S. and Australian methodologies for signalized intersections closer together. An important element in this methodology is the techniques used for the estimation of opposed (permissive) turn saturation flows. Although the basic modeling philosophies of the HCM and Australian methods are similar, there are significant differences in the procedures used and therefore in the results from the two methods. In particular, the latest methodology employed in the SIDRA software has eliminated the use of opposed turn adjustment factors for lane groups and adopted an explicit and direct method of modeling individual lanes. The purpose of this paper is to bring these new methods to the attention of the U.S. researchers since it is understood that efforts are being made to improve the 1985 HCM method.
Tracks, intersections and dead ends
Siim, Birte; Skjeie, Hege
2008-01-01
of organisations of civil society in political power. The second part explores the framing of the hijab as a political issue of "intersections" of gender equality versus religious belongings. The third part investigates what we see as a "dead end" in policy making to prevent violations of women's rights......The article discusses multicultural challenges to state feminism in Denmark and Norway focusing both on similarities and differences in the two countries policy responses. In spite of important differences, we point towards similar problems and dilemmas in the public responses to multiculturalism......; that is the general, age based, restrictions on family unification as a means to combat forced marriages. Finally, in the conclusion, we emphasise the importance of participatory women-friendly politics that include all who are affected by political decisions. ...
Graphs Theory and Applications
Fournier, Jean-Claude
2008-01-01
This book provides a pedagogical and comprehensive introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings and the timetabling problem, matchings and the optimal assignment problem, and Hamiltonian cycles and the traveling salesman problem, to name but a few. Exercises at various levels are given at the end of each chapter, and a final chapter presents a few general problems with hints for solutions, thus providing the reader with the opportunity to test and refine their knowledge on the
Burleigh, Scott C.
2011-01-01
Contact Graph Routing (CGR) is a dynamic routing system that computes routes through a time-varying topology of scheduled communication contacts in a network based on the DTN (Delay-Tolerant Networking) architecture. It is designed to enable dynamic selection of data transmission routes in a space network based on DTN. This dynamic responsiveness in route computation should be significantly more effective and less expensive than static routing, increasing total data return while at the same time reducing mission operations cost and risk. The basic strategy of CGR is to take advantage of the fact that, since flight mission communication operations are planned in detail, the communication routes between any pair of bundle agents in a population of nodes that have all been informed of one another's plans can be inferred from those plans rather than discovered via dialogue (which is impractical over long one-way-light-time space links). Messages that convey this planning information are used to construct contact graphs (time-varying models of network connectivity) from which CGR automatically computes efficient routes for bundles. Automatic route selection increases the flexibility and resilience of the space network, simplifying cross-support and reducing mission management costs. Note that there are no routing tables in Contact Graph Routing. The best route for a bundle destined for a given node may routinely be different from the best route for a different bundle destined for the same node, depending on bundle priority, bundle expiration time, and changes in the current lengths of transmission queues for neighboring nodes; routes must be computed individually for each bundle, from the Bundle Protocol agent's current network connectivity model for the bundle s destination node (the contact graph). Clearly this places a premium on optimizing the implementation of the route computation algorithm. The scalability of CGR to very large networks remains a research topic
Yap, Hian-Poh
1996-01-01
This book provides an up-to-date and rapid introduction to an important and currently active topic in graph theory. The author leads the reader to the forefront of research in this area. Complete and easily readable proofs of all the main theorems, together with numerous examples, exercises and open problems are given. The book is suitable for use as a textbook or as seminar material for advanced undergraduate and graduate students. The references are comprehensive and so it will also be useful for researchers as a handbook.
Kucharik, Marcel; Hofacker, Ivo; Stadler, Peter
2014-01-01
Motivation RNA folding is a complicated kinetic process. The minimum free energy structure provides only a static view of the most stable conformational state of the system. It is insufficient to give detailed insights into the dynamic behavior of RNAs. A sufficiently sophisticated analysis...... of the folding free energy landscape, however, can provide the relevant information. Results We introduce the basin hopping graph (BHG) as a novel coarse-grained model of folding landscapes. Each vertex of the BHG is a local minimum, which represents the corresponding basin in the landscape. Its edges connect...
Zeps, Dainis
2009-01-01
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces. Further, we define multiplication of these objects, that coincides with the multiplication of permutations. We consider closed under multiplication classes of combinatorial maps that consist of closed classes of combinatorial maps with fixed edges where each such class is defined by a knot. One class among them is special, containing selfconjugate maps.
Learning Probabilistic Decision Graphs
Jaeger, Manfred; Dalgaard, Jens; Silander, Tomi
2004-01-01
Probabilistic decision graphs (PDGs) are a representation language for probability distributions based on binary decision diagrams. PDGs can encode (context-specific) independence relations that cannot be captured in a Bayesian network structure, and can sometimes provide computationally more...... efficient representations than Bayesian networks. In this paper we present an algorithm for learning PDGs from data. First experiments show that the algorithm is capable of learning optimal PDG representations in some cases, and that the computational efficiency of PDG models learned from real-life data...
Endomorphisms of graph algebras
Conti, Roberto; Hong, Jeong Hee; Szymanski, Wojciech
2012-01-01
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D...... that the restriction to the diagonal MASA of an automorphism which globally preserves both D_E and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C...
Partitions of generalized split graphs
Shklarsky, Oren
2012-01-01
We discuss matrix partition problems for graphs that admit a partition into k independent sets and ` cliques. We show that when k + ` 6 2, any matrix M has finitely many (k; `) minimal obstructions and hence all of these problems are polynomial time solvable. We provide upper bounds for the size of any (k; `) minimal obstruction when k = ` = 1 (split graphs), when k = 2; ` = 0 (bipartite graphs), and when k = 0; ` = 2 (co-bipartite graphs). When k = ` = 1, we construct an exponential size spl...
Nested Dynamic Condition Response Graphs
Hildebrandt, Thomas; Mukkamala, Raghava Rao; Slaats, Tijs
2012-01-01
We present an extension of the recently introduced declarative process model Dynamic Condition Response Graphs ( DCR Graphs) to allow nested subgraphs and a new milestone relation between events. The extension was developed during a case study carried out jointly with our industrial partner...... Exformatics, a danish provider of case and workflow management systems. We formalize the semantics by giving first a map from Nested to (flat) DCR Graphs with milestones, and then extending the previously given mapping from DCR Graphs to Buchi-automata to include the milestone relation....
Edge Ideals of Weighted Graphs
Paulsen, Chelsey
2012-01-01
We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles and trees are unmixed and which ones are Cohen-Macaulay, and we prove that all weighted complete graphs are Cohen-Macaulay.
Intuitionistic Fuzzy Graphs with Categorical Properties
Hossein Rashmanlou
2015-09-01
Full Text Available The main purpose of this paper is to show the rationality of some operations, defined or to be defined, on intuitionistic fuzzy graphs. Firstly, three kinds of new product operations (called direct product, lexicographic product, and strong product are defined in intuitionistic fuzzy graphs, and some important notions on intuitionistic fuzzy graphs are demonstrated by characterizing these notions and their level counterparts graphs such as intuitionistic fuzzy complete graph, cartesian product of intuitionistic fuzzy graphs, composition of intuitionistic fuzzy graphs, union of intuitionistic fuzzy graphs, and join of intuitionistic fuzzy graphs. As a result, a kind of representations of intuitionistic fuzzy graphs and intuitionistic fuzzy complete graphs are given. Next, categorical goodness of intuitionistic fuzzy graphs is illustrated by proving that the category of intuitionistic fuzzy graphs and homomorphisms between them is isomorphic-closed, complete, and co-complete.
Framing Gender Intersections in the European Union
Lombardo, Emanuela; Agustin, Lise Rolandsen
2012-01-01
This article explores the extent to which the emergence of an antidiscrimination policy in the European Union (EU) implies a shift in EU gender equality policies towards an intersectional approach. The frame analysis of EU gender equality policy documents shows that intersectional dimensions...
Computations in intersection rings of flag bundles
Grayson, Daniel R; Stillman, Michael E
2012-01-01
Intersection rings of flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern classes. In this paper we describe the Groebner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.
Framing Gender Intersections in the European Union
Lombardo, Emanuela; Agustin, Lise Rolandsen
2012-01-01
This article explores the extent to which the emergence of an antidiscrimination policy in the European Union (EU) implies a shift in EU gender equality policies towards an intersectional approach. The frame analysis of EU gender equality policy documents shows that intersectional dimensions...
Graphical literacy: the quality of graphs in a large-circulation journal.
Cooper, Richelle J; Schriger, David L; Close, Reb J H
2002-09-01
We sought to characterize the quantity and quality of graphs in the Journal of American Medicine (JAMA), contrasting articles published in 1999 with those published in 2000 after the addition of a dedicated tables and figures editor. We also sought to compare the quality of graphs in JAMA with the quality of graphs in Annals of Emergency Medicine. Two reviewers independently assessed all graphs in original research articles from 12 randomly chosen issues of JAMA, 6 each from 1999 and 2000, using a standardized abstraction form. We noted graph type, clarity, and completeness and identified internal discrepancies. We examined the graphs and articles to observe discrepancies with text, to observe efficiency of graph presentation, and to determine whether the graph portrayed unaggregated data. Results were compared with results from a previously published study of graphs from 18 consecutive issues of Annals of Emergency Medicine beginning in January 1998. The 12 JAMA issues contained 56 research articles, with 64 graphs in the 37 articles that had graphs (28 in 27 1999 articles, 36 in 29 2000 articles). Simple bar or point charts (63%) predominated. We rarely encountered internal errors (8%), contradictions with text (3%), numeric distortion (6%), lack of visual clarity (5%), nonstandard graphing conventions (11%), or extraneous decoration (0%). Graphs generally defined all symbols (98%), but 31% were not self-explanatory; that is, despite knowing the study's design and reading the figure's legend, we could not unambiguously interpret the graph. Fourteen percent contained redundancies. Graphs infrequently portrayed by-subject data (9%) or advanced features (15%) such as pairing, symbolic dimensionality, or small multiples. Forty-eight percent (21/44) of graphs did not illustrate the underlying distribution, 48% (26/54) did not depict important covariates, and 67% (14/21) did not portray pairing inherent in the data. There were no differences between 1999 and 2000
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.
Double-Critical Graphs and Complete Minors
Kawarabayashi, Ken-ichi; Pedersen, Anders Sune; Toft, Bjarne
2010-01-01
A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case...
Tutte Polynomial of Multi-Bridge Graphs
Julian A. Allagan
2013-10-01
Full Text Available In this paper, using a well-known recursion for computing the Tutte polynomial of any graph, we found explicit formulae for the Tutte polynomials of any multi-bridge graph and some $2-$tree graphs. Further, several recursive formulae for other graphs such as the fan and the wheel graphs are also discussed.
A Modal-Logic Based Graph Abstraction
Bauer, J.; Boneva, I.B.; Kurban, M.E.; Rensink, A.; Ehrig, H.; Heckel, R.; Rozenberg, G.; Taentzer, G.
2008-01-01
Infinite or very large state spaces often prohibit the successful verification of graph transformation systems. Abstract graph transformation is an approach that tackles this problem by abstracting graphs to abstract graphs of bounded size and by lifting application of productions to abstract graphs
Wang, Suijie
2010-01-01
In this paper, we give a Laplacian characterization of the product of the complete graphs $K_m$ with trees, unicyclic graphs, and bicyclic graphs. More precisely, let $G$ be a connected graph with at most two independent cycles. If $G$ is neither $C_{6}$ nor $\\Theta_{3,2,5}$ and determined by its Laplacain spectrum, then the product $G\\times K_{m}$ is also a graph determined by its Laplacian spectrum. In addition, we find the cosepctral graphs of $C_{6}\\times K_{m}$ and $\\Theta_{3,2,5}\\times K_{m}$, where the case $m=1$ is shown in Figure \\ref{F1} and \\ref{F2}.
An intersection algorithm based on transformation
CHEN Xiao-xia; YONG Jun-hai; CHEN Yu-jian
2006-01-01
How to obtain intersection of curves and surfaces is a fundamental problem in many areas such as computer graphics,CAD/CAM,computer animation,and robotics.Especially,how to deal with singular cases,such as tangency or superposition,is a key problem in obtaining intersection results.A method for solving the intersection problem based on the coordinate transformation is presented.With the Lagrange multiplier method,the minimum distance between the center of a circle and a quadric surface is given as well.Experience shows that the coordinate transformation could significantly simplify the method for calculating intersection to the tangency condition.It can improve the stability of the intersection of given curves and surfaces in singularity cases.The new algorithm is applied in a three dimensional CAD software (GEMS),produced by Tsinghua University.
Detecting alternative graph clusterings.
Mandala, Supreet; Kumara, Soundar; Yao, Tao
2012-07-01
The problem of graph clustering or community detection has enjoyed a lot of attention in complex networks literature. A quality function, modularity, quantifies the strength of clustering and on maximization yields sensible partitions. However, in most real world networks, there are an exponentially large number of near-optimal partitions with some being very different from each other. Therefore, picking an optimal clustering among the alternatives does not provide complete information about network topology. To tackle this problem, we propose a graph perturbation scheme which can be used to identify an ensemble of near-optimal and diverse clusterings. We establish analytical properties of modularity function under the perturbation which ensures diversity. Our approach is algorithm independent and therefore can leverage any of the existing modularity maximizing algorithms. We numerically show that our methodology can systematically identify very different partitions on several existing data sets. The knowledge of diverse partitions sheds more light into the topological organization and helps gain a more complete understanding of the underlying complex network.