Solução analítica do modelo 2-estrelas com graus correlacionados

Abstract

In this work, we study an important tool to model complex networks built from empirical data, wich are exponential random graphs. We propose a model to generate random graphs, whose hamiltonian enables to impose generic constraints that involve correlations between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end- points of two-stars (next nearest neighbors). A two-star is formed by a pair of edges that share a node. In the present work, we will show how to solve this model analytically in the sparse regime, that is, when the average number of neighbors per node remains finite in the thermodynamic limit. From the analytic solution, we obtain the phase diagram which reveals the existence of a first order transition. The line of first order phase transition finishes at the critical point, whose location is precisely determined on the phase diagram. The first order transition marks a transition to a condensed phase, where the degree distribution exhibits one or two maximum. In particular, we show that the degree distribution depends strongly on the degree correlations between next nearest neighbors. For positive degree correlations between next nearest neighbors, the degree distribution inside the condensed phase shows a maximum at the maximum degree, while for negative degree correlations the condensed phase is characterized by a bimodal degree distribution. We show that the assortativities are non-monotonic functions of the model parameters, with a discontinuous behavior at the firs-torder transition. Some of our theoretical results are independently confirmed through Monte Carlo simulations. The results of this thesis are useful to model empirical networks with correlated degrees, since one can identify, base on the phase diagram, the metaestable region in which generating random graphs becames a difficult problem.