Análise topológica de redes q-exponenciais aleatórias

Abstract

This thesis explores the relevance and fundamental properties of networks with q-exponential degree distribution, a theme that pervades a variety of domains, including plasma physics, econometrics, biophysics, and more. The congruence between experiments, numerical analyses, and theoretical models with the q-exponential distribution is notably observed, particularly in both empirical and modeled complex networks. Unlike previous approaches that typically focused on growth models with preferential attachment, this work employs the configuration model to generate random networks. This methodology allows for a more accurate assessment of the inherent properties of these networks, detached from specific growth processes. The q-exponential distribution is characterized by its two parameters, q and λ, and serves as a generalized extension of the traditional power-law distribution. The thesis discusses how the distribution transitions from a power-law behavior for large values of k (node degree) to a plateau distribution for smaller values of k, with the parameter λ−1 playing a crucial role in this transition. This study highlights that deviations from the standard power-law behavior at smaller degrees can induce significant structural changes in networks. Such changes have the potential to profoundly affect the characteristics and fundamental processes within the networks. Networks generated under the configuration model reveal distinct topological properties, directly derived from the q-exponential nature of their degree distribution. Characteristics such as assortativity, shortest average path, and robustness against random failures and directed attacks are investigated in detail. The results suggest that q-exponential networks offer greater robustness compared to conventional scale-free networks, particularly in scenarios of malicious attacks. Furthermore, the analysis of the k-core decomposition reveals a more extensive and robust core structure in q-exponential networks than in pure power-law networks.