Conexões entre redes complexas geométricas e a q-estatística

Abstract

Networks are abound in nature, therefore, Network science is a very interdisciplinary theory and has been widely successfully used to study huge connected systems. The nonextensive statistical mechanics naturally emerge from the limitations of the BoltzmannGibbs statistic, being capable to describe systems in the regimes where the standard statistical mechanics fails. Nowadays the connections between these two areas are well known. In this thesis we study a d-dimensional geographically located network (characterized by the index αG ≥ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely P(w). In this model, each site has an evolving degree ki and a local energy εi ≡ Pki j=1 wij/2 (i = 1, 2, ..., N) that depend on the weights of the links connected to it. At the thermodynamic limit, the energy distribution is the form p(ε) ∝ e−βqεq, where ezq is the q-exponential defined by ezq ≡ [1 + (1−q)z] 1/(1−q) which optimizes the non-additive entropy Sq and when q → 1 the Boltzmann-Gibbs entropy is recovered. The parameters q and βq depends only on αA/d, thus exhibiting universality. Also, we provide here strong numerical evidence that a isomorphism appears to emerge connecting the energy q-exponential distribution (with q = 4/3 and βqω0 = 10/3) with a specific geographic growth random model based on preferential attachment through exponentially-distributed weighted links.