Neste estudo são apresentados os resultados do trabalho sobre simulações de redes de conexões complexas. Foram simuladas redes regulares, intermediárias e aleatórias com o número de nós e de conexões variando entre 103 e 5x103 e entre 2x104 e 105, respectivamente, e com probabilidade variando de 0 a 1 com passo de 0.1, com o enfoque na Teoria Espectral. Utilizando a linguagem C e o software Matlab, as redes são representadas pela sua matriz adjacência, com o objetivo de observar-se o comportamento de seus autovalores através de histogramas. A finalidade é a caracterização de redes complexas. Observa-se que a distribuição dos autovalores segue a lei semicircular de Wigner. / This study presents the results of the work about simulations of networks of complex connections. They were simulate regular networks, middlemen and aleatory with the number of nodes and of connections varying between 103 and 5x104 and between 2x104 and 105, respectively, and with probability varying from 0 to 1 with step of 0.1, with the focus in the Spectral Theory. Using the language C and the software Matlab, the networks are represented by its adjacency matrix, with the objective of observing the behavior of its eigenvalues through histograms. The purpose is the characterization of complex networks. Its observed that the eigenvalues distribution follows the Wigners semicircular law.

Neste estudo são apresentados os resultados do trabalho sobre simulações de redes de conexões complexas. Foram simuladas redes regulares, intermediárias e aleatórias com o número de nós e de conexões variando entre 103 e 5x103 e entre 2x104 e 105, respectivamente, e com probabilidade variando de 0 a 1 com passo de 0.1, com o enfoque na Teoria Espectral. Utilizando a linguagem C e o software Matlab, as redes são representadas pela sua matriz adjacência, com o objetivo de observar-se o comportamento de seus autovalores através de histogramas. A finalidade é a caracterização de redes complexas. Observa-se que a distribuição dos autovalores segue a lei semicircular de Wigner. / This study presents the results of the work about simulations of networks of complex connections. They were simulate regular networks, middlemen and aleatory with the number of nodes and of connections varying between 103 and 5x104 and between 2x104 and 105, respectively, and with probability varying from 0 to 1 with step of 0.1, with the focus in the Spectral Theory. Using the language C and the software Matlab, the networks are represented by its adjacency matrix, with the objective of observing the behavior of its eigenvalues through histograms. The purpose is the characterization of complex networks. Its observed that the eigenvalues distribution follows the Wigners semicircular law.

Made available in DSpace on 2016-03-15T19:37:45Z (GMT). No. of bitstreams: 1 Wander Lairson Costa.pdf: 1957133 bytes, checksum: 6819580d97bb5eaca5ea04352fcda0b8 (MD5) Previous issue date: 2013-03-15 / Universidade Presbiteriana Mackenzie / Cellular automata are totally discrete systems that act locally in a simple and deterministic way, but whose resulting global behavior can be extremely complex. The set of possible global configurations in one finite time step for a CA can be described by a regular language, which in turn can be represented by a finite automaton, more precisely the so-called process graph, in which all states are initial and final. Here, we study the temporal evolution complexity of the elementary cellular automata (i.e., one-dimensional, binary, with radius 1), and related previous works are revisited and discussed, indicating problems and their consequences. We also start up a novel approach for the problem, substituting the process graph based representation that describes the configuration at each time step by adjacency matrices derived from them. In fact, we extend the classical adjacency matrix notation, as they cannot fully represent process graphs. With this new notation, we show that it is possible to obtain the algorithm to generate a process graph for an arbitrary finite time step for each of the rules at study. In conclusion, although advancing the limit graph problem, it still remains open, and we provide suggestions for further research. / Autômatos celulares são sistemas totalmente discretos que agem localmente de forma simples e determinística, mas cujo comportamento global resultante pode ser extremamente complexo. O conjunto de possíveis configurações globais em um passo de tempo t finito para um autômato celular pode ser descrito por uma linguagem regular, a qual por sua vez pode ser representada por meio de um autômato finito, mais precisamente, pelo chamado grafo de processo, em que todos os estados são iniciais e finais. Estuda-se aqui a complexidade da evolução temporal dos autômatos celulares elementares (i.e., unidimensionais, binários, de raio 1), e trabalhos anteriores são revisitados e discutidos, no quais apontam-se problemas e suas consequências. Também inicia-se uma nova abordagem para o problema, substituindo a representação dos grafos de processo que descrevem a configuração a cada passo de tempo por matrizes de adjacência deles derivadas. De fato, estende-se a notação clássica de matriz de adjacência, já que ela se mostra insuficiente para descrever completamente os grafos de processo em questão. Com essa nova notação, mostra-se que é possível obter o algoritmo que gere o grafo de processo de tempo t para cada uma das regras estudadas. Conclui-se que, embora houve avanços para o problema do grafo limite, este ainda permanece aberto, e sugestões para continuação da pesquisa são dadas.

La vie privée est une préoccupation des utilisateurs des réseaux sociaux. Les réseaux sociaux sont une source de données précieuses pour des analyses scientifiques ou commerciales. Cette thèse aborde trois problèmes de confidentialité des réseaux sociaux: l'anonymisation de graphes sociaux, la détection de communautés privées et l'échange de liens privés. Nous abordons le problème d'anonymisation de graphes via la sémantique de l'incertitude et l'intimité différentielle. Pour la première, nous proposons un modèle général appelé Uncertain Adjacency Matrix (UAM) qui préserve dans le graphe anonymisé les degrés des nœuds du graphe non-anonymisé. Nous analysons deux schémas proposés récemment et montrons leur adaptation dans notre modèle. Nous aussi présentons notre approche dite MaxVar. Pour la technique d'intimité différentielle, le problème devient difficile en raison de l'énorme espace des graphes anonymisés possibles. Un grand nombre de systèmes existants ne permettent pas de relâcher le budget contrôlant la vie privée, ni de déterminer sa borne supérieure. Dans notre approche nous pouvons calculer cette borne. Nous introduisons le nouveau schéma Top-m-Filter de complexité linéaire et améliorons la technique récente EdgeFlip. L'évaluation de ces algorithmes sur une large gamme de graphes donne un panorama de l'état de l'art. Nous présentons le problème original de la détection de la communauté dans le cadre de l'intimité différentielle. Nous analysons les défis majeurs du problème et nous proposons quelques approches pour les aborder sous deux angles: par perturbation d'entrée (schéma LouvainDP) et par perturbation d'algorithme (schéma ModDivisive) / Privacy is a serious concern of users in daily usage of social networks. Social networks are a valuable data source for large-scale studies on social organization and evolution and are usually published in anonymized forms. This thesis addresses three privacy problems of social networks: graph anonymization, private community detection and private link exchange. First, we tackle the problem of graph anonymization via uncertainty semantics and differential privacy. As for uncertainty semantics, we propose a general obfuscation model called Uncertain Adjacency Matrix (UAM) that keep expected node degrees equal to those in the unanonymized graph. We analyze two recently proposed schemes and show their fitting into the model. We also present our scheme Maximum Variance (MaxVar) to fill the gap between them. Using differential privacy, the problem is very challenging because of the huge output space of noisy graphs. A large body of existing schemes on differentially private release of graphs are not consistent with increasing privacy budgets as well as do not clarify the upper bounds of privacy budgets. In this thesis, such a bound is provided. We introduce the new linear scheme Top-m-Filter (TmF) and improve the existing technique EdgeFlip. Thorough comparative evaluation on a wide range of graphs provides a panorama of the state-of-the-art's performance as well as validates our proposed schemes. Second, we present the problem of community detection under differential privacy. We analyze the major challenges behind the problem and propose several schemes to tackle them from two perspectives: input perturbation (LouvainDP) and algorithm perturbation (ModDivisive)

圖的譜半徑在數學方面以及其他領域有非常多的應用。在這篇論文裡，我們整理有關連通圖的拉普拉斯與無符號拉普拉斯譜半徑的論文。本文一開始探討一些圖的譜理論，並找出這些界限的關係。然後，我們將討論更精確的圖之拉普拉斯與無符號拉普拉斯譜半徑。最後，我們給一個例子，並使用前面所探討過的性質分析之。 / The spectral radius of a graph has been applied in mathenatics and in diverse disciplines.In this thesis, we survey some papers about the Laplacian spectral radius and the signless Laplacian spectral radius of a connected graph. Initially, we discuss some properties about the spectral graphs and find the relations between these bounds. Then, we discuss the upper bounds and lower bounds of the Laplacian and signless Laplacian spectral radius of a graph. In the end, we give an example and analyze it.

This Master Thesis deals with the drawing algorithms of graphs known from the mathematical theory. These algorithms deals with an appropriate distribution of the graph vertices in order to obtain the most clear and readable graphs for human readers. The main objective of this work was also to implement the drawing algorithm in the application that would allow to edit the graph. This work deals also with graphs representation in computers.