Recentemente, a teoria de grafos aleatórios vem sendo aplicada para modelar interações neurais do cérebro. Enquanto as propriedades dos grafos aleatórios vem sendo vastamente estudadas na literatura, o desenvolvimento de métodos de inferência estatística para essa classe de objetos tem recebido menos atenção. Nesse trabalho propomos um teste de hipóteses não paramétrico para testar se duas amostras de grafos aleatórios provém da mesma distribuição de probabilidade. Nós provamos como computar de maneira eficiente a estatística do teste e estudamos o desempenho do teste em dados simulados de grafos. A principal motivação deste trabalho é a aplicação do teste proposto em dados de eletroencefalograma. / The theory of random graphs has been successfully applied in recent years to model neural interactions in the brain. While the probabilistic properties of random graphs has been extensively studied in the literature, the development of statistical inference methods for this class of objects has received less attention. In this work we propose a non parametric test of hypotheses to decide if two samples of random graphs are originated from the same probability distribution. We show how to compute efficiently the test statistic and we study the performance of the test on simulated data. The main motivation of this work is to apply this test to analyze neural networks constructed from electroencephalographic data.

Estudamos o comportamento da familia aresta-triangulo de grafos aleatorios exponenciais (ERG) usando metodos de Monte Carlo baseados em Cadeias de Markov. Comparamos contagens de subgrafos e correlacoes entre arestas de ergs as de Grafos Aleatorios Binomiais (BRG, tambem chamados de Erdos-Renyi). E um resultado teorico conhecido que para algumas parametrizacoes os limites das contagens de subgrafos de ERGs convergem para os de BRGs, assintoticamente no numero de vertices [BBS11, CD11]. Observamos esse fenomeno em grafos com poucos (20) vertices em nossas simulacoes. / We study the behavior of the edge-triangle family of exponential random graphs (ERG) using the Markov Chain Monte Carlo method. We compare ERG subgraph counts and edge correlations to those of the classic Binomial Random Graph (BRG, also called Erdos-Renyi model). It is a known theoretical result that for some parameterizations the limit ERG subgraph counts converge to those of BRGs, as the number of vertices grows [BBS11, CD11]. We observe this phenomenon on graphs with few (20) vertices in our simulations.

A connected dominating set D is a set of vertices of a graph G=(V,E) such that every vertex in V-D is adjacent to at least one vertex in D and the subgraph induced by the set D is connected. The connected domination number is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D.

Interactions are what gives us the knowledge of the world around us. Interactions on all levels may fundamentally be seen as an exchange of information and a possible response of the same. Whether it is an electron in an electrical field or a handsome dude in a bar responding to a flirtation---interactions make things happen. In this sense we can see that objects without the capability of interacting with each other also are invisible to each other. Chains of pairwise interacting entities can serve as mediators of indirect interactions between objects. Nonetheless, in the limit of no interactions, we get into a philosophical debate whether we actually may consider anything to exist since it can not be detected in any way. Interactions between matter tend to be organized and show a hierarchical structure in which smaller sub-systems can be seen as parts of a bigger system, which in turn might be a smaller part of an even bigger system. This is reflected by the fact that we have sciences that successfully study specific interactions between objects or matter---physics, chemistry, biology, ecology, sociology,... What happens in a situation where all length scales are important? How does the structure of the underlying network of interactions affect the dynamical properties of a system? What network structures do we find and how are they created? This thesis is a physicist's view of collective dynamics, from superconductors to social systems and navigation in city street networks.

Spectral graph theory widely increases the interests in not only discovering new properties of well known graphs but also proving the well known properties for the new type of graphs. In fact all spectral properties of proverbial graphs are not acknowledged to us and in other hand due to the structure of nature, new classes of graphs are required to explain the phenomena around us and the spectral properties of these graphs can tell us more about the structure of them. These both themes are the body of our work here. We introduce here three models of random graphs and show that the eigenvalue counting function of Laplacians on these graphs has exponential decay bound. Since our methods heavily depend on the first nonzero eigenvalue of Laplacian, we study also this eigenvalue for the graph in both random and nonrandom cases.

Thesis (M.S.)--University of North Carolina at Wilmington, 2005. / Includes bibliographical references (leaves: 60-61)

A caracterização de redes e o estudo de sistemas, ambos utilizando grafos, é algo muito usado por várias áreas científicas. Uma das linhas deste estudo é denominada de grafos aleatórios, que por sua vez auxilia na criação de modelos para análise de redes reais. Consideramos um modelo de grafo aleatório não homogêneo criado por Kang, Pachón e Rodríguez (2016), cuja construção é feita a partir da realização do grafo binomial G(n; p). Para este modelo, estudamos argumentos e métodos usados para encontrar resultados sobre o limiar de conectividade, importante propriedade relacionada a existência assintótica de vértices e componentes isolados. Em seguida, constatamos algumas características positivas e negativas a respeito da utilização do grafo para modelar redes reais complexas, onde usamos de simulações computacionais e medidas topológicas. / The characterization of networks and the study of systems, both using graphs, is very used by several scientific areas. One of the lines of this study is called random graphs, which in turn assists in creating models for the analysis of real networks. We consider an inhomogeneous random graph model created by Kang, Pachón e Rodríguez (2016), where its construction is made from the realization of the binomial graph G(n; p). For this model, we studied the arguments and methods used to find results on the connectivity threshold, important property related to asymptotic existence of vertices and isolated components. Then we found some positive and negative characteristics about the use of the graph to model complex real networks, using computer simulations and topological measures.

Estudamos o comportamento da familia aresta-triangulo de grafos aleatorios exponenciais (ERG) usando metodos de Monte Carlo baseados em Cadeias de Markov. Comparamos contagens de subgrafos e correlacoes entre arestas de ergs as de Grafos Aleatorios Binomiais (BRG, tambem chamados de Erdos-Renyi). E um resultado teorico conhecido que para algumas parametrizacoes os limites das contagens de subgrafos de ERGs convergem para os de BRGs, assintoticamente no numero de vertices [BBS11, CD11]. Observamos esse fenomeno em grafos com poucos (20) vertices em nossas simulacoes. / We study the behavior of the edge-triangle family of exponential random graphs (ERG) using the Markov Chain Monte Carlo method. We compare ERG subgraph counts and edge correlations to those of the classic Binomial Random Graph (BRG, also called Erdos-Renyi model). It is a known theoretical result that for some parameterizations the limit ERG subgraph counts converge to those of BRGs, as the number of vertices grows [BBS11, CD11]. We observe this phenomenon on graphs with few (20) vertices in our simulations.

La richesse de la programmation par contraintes repose sur la très large variété des algorithmes qu’elle utilise en puisant dans les grands domaines de l’Intelligence Artificielle, de la Programmation Logique et de la Recherche Opérationnelle. Cependant, cette richesse, qui offre aux spécialistes une palette quasi-illimitée de configurations possibles pour attaquer des problèmes combinatoires, devient une frein à la diffusion plus large du paradigme, car les outils actuels sont très loin d’une boîte noire, et leur utilisation suppose une bonne connaissance du domaine, notamment en ce qui concerne leur paramétrage. Dans cette thèse, nous proposons d’analyser le comportement des contraintes de cardinalité avec des modèles probabilistes et des outils de dénombrement, pour paramétrer automatiquement les solveurs de contraintes : heuristiques de choix de variables et de choix de valeurs et stratégies de recherche. / The main asset of constraint programming is its wide variety of algorithms that comes from the major areas of artificial intelligence, logic programming and operational research. It offers specialists a limitless range of possible configurations to tackle combinatorial problems, but it becomes an obstacle to the wider diffusion of the paradigm. The current tools are very far from being used as a black-box tool, and it assumes a good knowledge of the field, in particular regarding the parametrization of solvers.In this thesis, we propose to analyze the behavior of cardinality constraints with probabilistic models and counting tools, to automatically parameterize constraint solvers: heuristics of choice of variables and choice of values and search strategies.

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