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Master's Dissertation
DOI
Document
Author
Full name
Elizbeth Chipa Bedia
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2016
Supervisor
Committee
Gallo, Alexsandro Giacomo Grimbert (President)
Coletti, Cristian Favio
Leonardi, Florencia Graciela
Title in Portuguese
Conectividade do grafo aleatório de Erdös-Rényi, e de uma variante com conexões locais
Keywords in Portuguese
Conectividade
Grafos aleatórios
Probabilidade
Transição de fase
Abstract in Portuguese
Dizemos que um grafo e conectado se existe um caminho de arestas entre quaisquer par de vértices. O grafo aleatório de Erdös-Rényi com n vértices e obtido conectando cada par de vértice com probabilidade pn ∈ (0, 1), independentemente dos outros. Neste trabalho, estudamos em detalhe o limiar da conectividade na probabilidade de conexão pn para grafos aleatórios Erdös-Rényi quando o número de vértices n diverge. Para este estudo, revisamos algumas ferramentas probabilísticas básicas (convergência de variáveis aleatórias e Métodos do primeiro e segundo momento), que também irão auxiliar ao melhor entendimento de resultados mais complexos. Além disto, aplicamos os conceitos anteriores para um modelo com uma topologia simples, mais especificamente estudamos o comportamento assintótico da probabilidade de não existência de vértices isolados, e discutimos a conectividade ou não do grafo. Por m mostramos a convergência em distrubuição do número de vértices isolados para uma Distribuição Poisson do modelo estudado.
Title in English
Connectivity for the Erdös-Rényi random graph, and a variant with local connections
Keywords in English
Connectivity
Phase transition
Probability
Random graphs
Abstract in English
We say that a graph is connected if there is a path edges between any pair of vertices. Random graph Erdös-Rényi with n vertices is obtained by connecting each pair of vertex with probability pn ∈ (0, 1) independently of the others. In this work, we studied in detail the connectivity threshold in the connection probability pn for random graphs Erdös-Rényi when the number of vertices n diverges. For this study, we review some basic probabilistic tools (convergence of random variables and methods of the first and second moment), which will lead to a better understanding of more complex results. In addition, we apply the above concepts for a model with a simple topology, specifically studied the asymptotic behavior of the probability of non-existence of isolated vertices, and we discussed the connectivity or not of the graph. Finally we show the convergence in distribution of the number of isolated vertices for a Poisson distribution of the studied model.
 
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Publishing Date
2019-08-12
 
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