Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

Constellations and the unsupervised learning of graphs

In this paper, we propose a novel method for the unsupervised clustering of graphs in the context of the constellation approach to object recognition. Such method is an EM central clustering algorithm which builds prototypical graphs on the basis of fast matching with graph transformations. Our experiments, both with random graphs and in realistic situations (visual localization), show that our prototypes improve the set median graphs and also the prototypes derived from our previous incremental method. We also discuss how the method scales with a growing number of images.

Random graph coloring:Statistical physics approach

The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte Carlo simulations. We critically discuss the merits and shortcomings of the various methods, and interpret the results obtained. We present an exact analytical expression for the two-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges. ©2002 The American Physical Society.

Properties of sparse random matrices over finite fields

Typical properties of sparse random matrices over finite (Galois) fields are studied, in the limit of large matrices, using techniques from the physics of disordered systems. For the case of a finite field GF(q) with prime order q, we present results for the average kernel dimension, average dimension of the eigenvector spaces and the distribution of the eigenvalues. The number of matrices for a given distribution of entries is also calculated for the general case. The significance of these results to error-correcting codes and random graphs is also discussed.

A linear optimization technique for graph pebbling

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is NP 2 -complete. In this paper we develop a tool, called theWeight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply theWeight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.

Error and attack tolerance of layered complex networks.

Many complex systems may be described by not one but a number of complex networks mapped on each other in a multi-layer structure. Because of the interactions and dependencies between these layers, the state of a single layer does not necessarily reflect well the state of the entire system. In this paper we study the robustness of five examples of two-layer complex systems: three real-life data sets in the fields of communication (the Internet), transportation (the European railway system), and biology (the human brain), and two models based on random graphs. In order to cover the whole range of features specific to these systems, we focus on two extreme policies of system's response to failures, no rerouting and full rerouting. Our main finding is that multi-layer systems are much more vulnerable to errors and intentional attacks than they appear from a single layer perspective.

Evolutionary games on complex networks : the emergence and maintenance of cooperation

Abstract The object of game theory lies in the analysis of situations where different social actors have conflicting requirements and where their individual decisions will all influence the global outcome. In this framework, several games have been invented to capture the essence of various dilemmas encountered in many common important socio-economic situations. Even though these games often succeed in helping us understand human or animal behavior in interactive settings, some experiments have shown that people tend to cooperate with each other in situations for which classical game theory strongly recommends them to do the exact opposite. Several mechanisms have been invoked to try to explain the emergence of this unexpected cooperative attitude. Among them, repeated interaction, reputation, and belonging to a recognizable group have often been mentioned. However, the work of Nowak and May (1992) showed that the simple fact of arranging the players according to a spatial structure and only allowing them to interact with their immediate neighbors is sufficient to sustain a certain amount of cooperation even when the game is played anonymously and without repetition. Nowak and May's study and much of the following work was based on regular structures such as two-dimensional grids. Axelrod et al. (2002) showed that by randomizing the choice of neighbors, i.e. by actually giving up a strictly local geographical structure, cooperation can still emerge, provided that the interaction patterns remain stable in time. This is a first step towards a social network structure. However, following pioneering work by sociologists in the sixties such as that of Milgram (1967), in the last few years it has become apparent that many social and biological interaction networks, and even some technological networks, have particular, and partly unexpected, properties that set them apart from regular or random graphs. Among other things, they usually display broad degree distributions, and show small-world topological structure. Roughly speaking, a small-world graph is a network where any individual is relatively close, in terms of social ties, to any other individual, a property also found in random graphs but not in regular lattices. However, in contrast with random graphs, small-world networks also have a certain amount of local structure, as measured, for instance, by a quantity called the clustering coefficient. In the same vein, many real conflicting situations in economy and sociology are not well described neither by a fixed geographical position of the individuals in a regular lattice, nor by a random graph. Furthermore, it is a known fact that network structure can highly influence dynamical phenomena such as the way diseases spread across a population and ideas or information get transmitted. Therefore, in the last decade, research attention has naturally shifted from random and regular graphs towards better models of social interaction structures. The primary goal of this work is to discover whether or not the underlying graph structure of real social networks could give explanations as to why one finds higher levels of cooperation in populations of human beings or animals than what is prescribed by classical game theory. To meet this objective, I start by thoroughly studying a real scientific coauthorship network and showing how it differs from biological or technological networks using divers statistical measurements. Furthermore, I extract and describe its community structure taking into account the intensity of a collaboration. Finally, I investigate the temporal evolution of the network, from its inception to its state at the time of the study in 2006, suggesting also an effective view of it as opposed to a historical one. Thereafter, I combine evolutionary game theory with several network models along with the studied coauthorship network in order to highlight which specific network properties foster cooperation and shed some light on the various mechanisms responsible for the maintenance of this same cooperation. I point out the fact that, to resist defection, cooperators take advantage, whenever possible, of the degree-heterogeneity of social networks and their underlying community structure. Finally, I show that cooperation level and stability depend not only on the game played, but also on the evolutionary dynamic rules used and the individual payoff calculations. Synopsis Le but de la théorie des jeux réside dans l'analyse de situations dans lesquelles différents acteurs sociaux, avec des objectifs souvent conflictuels, doivent individuellement prendre des décisions qui influenceront toutes le résultat global. Dans ce cadre, plusieurs jeux ont été inventés afin de saisir l'essence de divers dilemmes rencontrés dans d'importantes situations socio-économiques. Bien que ces jeux nous permettent souvent de comprendre le comportement d'êtres humains ou d'animaux en interactions, des expériences ont montré que les individus ont parfois tendance à coopérer dans des situations pour lesquelles la théorie classique des jeux prescrit de faire le contraire. Plusieurs mécanismes ont été invoqués pour tenter d'expliquer l'émergence de ce comportement coopératif inattendu. Parmi ceux-ci, la répétition des interactions, la réputation ou encore l'appartenance à des groupes reconnaissables ont souvent été mentionnés. Toutefois, les travaux de Nowak et May (1992) ont montré que le simple fait de disposer les joueurs selon une structure spatiale en leur permettant d'interagir uniquement avec leurs voisins directs est suffisant pour maintenir un certain niveau de coopération même si le jeu est joué de manière anonyme et sans répétitions. L'étude de Nowak et May, ainsi qu'un nombre substantiel de travaux qui ont suivi, étaient basés sur des structures régulières telles que des grilles à deux dimensions. Axelrod et al. (2002) ont montré qu'en randomisant le choix des voisins, i.e. en abandonnant une localisation géographique stricte, la coopération peut malgré tout émerger, pour autant que les schémas d'interactions restent stables au cours du temps. Ceci est un premier pas en direction d'une structure de réseau social. Toutefois, suite aux travaux précurseurs de sociologues des années soixante, tels que ceux de Milgram (1967), il est devenu clair ces dernières années qu'une grande partie des réseaux d'interactions sociaux et biologiques, et même quelques réseaux technologiques, possèdent des propriétés particulières, et partiellement inattendues, qui les distinguent de graphes réguliers ou aléatoires. Entre autres, ils affichent en général une distribution du degré relativement large ainsi qu'une structure de "petit-monde". Grossièrement parlant, un graphe "petit-monde" est un réseau où tout individu se trouve relativement près de tout autre individu en termes de distance sociale, une propriété également présente dans les graphes aléatoires mais absente des grilles régulières. Par contre, les réseaux "petit-monde" ont, contrairement aux graphes aléatoires, une certaine structure de localité, mesurée par exemple par une quantité appelée le "coefficient de clustering". Dans le même esprit, plusieurs situations réelles de conflit en économie et sociologie ne sont pas bien décrites ni par des positions géographiquement fixes des individus en grilles régulières, ni par des graphes aléatoires. De plus, il est bien connu que la structure même d'un réseau peut passablement influencer des phénomènes dynamiques tels que la manière qu'a une maladie de se répandre à travers une population, ou encore la façon dont des idées ou une information s'y propagent. Ainsi, durant cette dernière décennie, l'attention de la recherche s'est tout naturellement déplacée des graphes aléatoires et réguliers vers de meilleurs modèles de structure d'interactions sociales. L'objectif principal de ce travail est de découvrir si la structure sous-jacente de graphe de vrais réseaux sociaux peut fournir des explications quant aux raisons pour lesquelles on trouve, chez certains groupes d'êtres humains ou d'animaux, des niveaux de coopération supérieurs à ce qui est prescrit par la théorie classique des jeux. Dans l'optique d'atteindre ce but, je commence par étudier un véritable réseau de collaborations scientifiques et, en utilisant diverses mesures statistiques, je mets en évidence la manière dont il diffère de réseaux biologiques ou technologiques. De plus, j'extrais et je décris sa structure de communautés en tenant compte de l'intensité d'une collaboration. Finalement, j'examine l'évolution temporelle du réseau depuis son origine jusqu'à son état en 2006, date à laquelle l'étude a été effectuée, en suggérant également une vue effective du réseau par opposition à une vue historique. Par la suite, je combine la théorie évolutionnaire des jeux avec des réseaux comprenant plusieurs modèles et le réseau de collaboration susmentionné, afin de déterminer les propriétés structurelles utiles à la promotion de la coopération et les mécanismes responsables du maintien de celle-ci. Je mets en évidence le fait que, pour ne pas succomber à la défection, les coopérateurs exploitent dans la mesure du possible l'hétérogénéité des réseaux sociaux en termes de degré ainsi que la structure de communautés sous-jacente de ces mêmes réseaux. Finalement, je montre que le niveau de coopération et sa stabilité dépendent non seulement du jeu joué, mais aussi des règles de la dynamique évolutionnaire utilisées et du calcul du bénéfice d'un individu.

Deciphering the global organization of clustering in real complex networks

We uncover the global organization of clustering in real complex networks. To this end, we ask whether triangles in real networks organize as in maximally random graphs with given degree and clustering distributions, or as in maximally ordered graph models where triangles are forced into modules. The answer comes by way of exploring m-core landscapes, where the m-core is defined, akin to the k-core, as the maximal subgraph with edges participating in at least m triangles. This property defines a set of nested subgraphs that, contrarily to k-cores, is able to distinguish between hierarchical and modular architectures. We find that the clustering organization in real networks is neither completely random nor ordered although, surprisingly, it is more random than modular. This supports the idea that the structure of real networks may in fact be the outcome of self-organized processes based on local optimization rules, in contrast to global optimization principles.

Periodic Reordering

For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.

Small-world effects in lattice stochastic diffusion search

Stochastic Diffusion Search is an efficient probabilistic bestfit search technique, capable of transformation invariant pattern matching. Although inherently parallel in operation it is difficult to implement efficiently in hardware as it requires full inter-agent connectivity. This paper describes a lattice implementation, which, while qualitatively retaining the properties of the original algorithm, restricts connectivity, enabling simpler implementation on parallel hardware. Diffusion times are examined for different network topologies, ranging from ordered lattices, over small-world networks to random graphs.

Bistability through triadic closure

We propose and analyse a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, on-line social behaviour and information processing in neuroscience. We model the evolving network as a discrete time Markov chain, and study a very general framework where, conditioned on the current state, edges appear or disappear independently at the next timestep. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cell phone data, and find support for a stratified model, where individuals are assigned to one of two distinct groups having different within-group and across-group dynamics.

A pattern recognition approach to complex networks

Complex networks exist in many areas of science such as biology, neuroscience, engineering, and sociology. The growing development of this area has led to the introduction of several topological and dynamical measurements, which describe and quantify the structure of networks. Such characterization is essential not only for the modeling of real systems but also for the study of dynamic processes that may take place in them. However, it is not easy to use several measurements for the analysis of complex networks, due to the correlation between them and the difficulty of their visualization. To overcome these limitations, we propose an effective and comprehensive approach for the analysis of complex networks, which allows the visualization of several measurements in a few projections that contain the largest data variance and the classification of networks into three levels of detail, vertices, communities, and the global topology. We also demonstrate the efficiency and the universality of the proposed methods in a series of real-world networks in the three levels.

Conceitos e técnicas da mecânica estatística e termodinâmica aplicados ao estudo dos grafos aleatórios

This dissertation briefly presents the random graphs and the main quantities calculated from them. At the same time, basic thermodynamics quantities such as energy and temperature are associated with some of their characteristics. Approaches commonly used in Statistical Mechanics are employed and rules that describe a time evolution for the graphs are proposed in order to study their ergodicity and a possible thermal equilibrium between them

Yang-Lee zeros of the two- and three-state Potts model defined on π3 Feynman diagrams

We present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q=2 state (Ising) and the q=3 state Potts model defined on phi(3) Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model, an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the q=3 state Potts model, our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

Predicting epidemic outbreak from individual features of the spreaders

Knowing which individuals can be more efficient in spreading a pathogen throughout a determinate environment is a fundamental question in disease control. Indeed, over recent years the spread of epidemic diseases and its relationship with the topology of the involved system have been a recurrent topic in complex network theory, taking into account both network models and real-world data. In this paper we explore possible correlations between the heterogeneous spread of an epidemic disease governed by the susceptible-infected-recovered (SIR) model, and several attributes of the originating vertices, considering Erdos-Renyi (ER), Barabasi-Albert (BA) and random geometric graphs (RGG), as well as a real case study, the US air transportation network, which comprises the 500 busiest airports in the US along with inter-connections. Initially, the heterogeneity of the spreading is achieved by considering the RGG networks, in which we analytically derive an expression for the distribution of the spreading rates among the established contacts, by assuming that such rates decay exponentially with the distance that separates the individuals. Such a distribution is also considered for the ER and BA models, where we observe topological effects on the correlations. In the case of the airport network, the spreading rates are empirically defined, assumed to be directly proportional to the seat availability. Among both the theoretical and real networks considered, we observe a high correlation between the total epidemic prevalence and the degree, as well as the strength and the accessibility of the epidemic sources. For attributes such as the betweenness centrality and the k-shell index, however, the correlation depends on the topology considered.