Estimação do número de comunidades no modelo estocástico de blocos com correção de grau

Abstract

The stochastic block model (SBM) is a random graph model that splits the set of vertices into blocks, and the probability connection between each pair of vertices depends on the blocks to which the vertices belong. The SBM was introduced by Holland et al. (1983) and it is traditionally applied to simple graphs, with each entry in the adjacency matrix following the Bernoulli distribution. Karrer and Newman (2011) extended the model in two directions: they defined the multigraph model (Poisson SBM), in which the entries of the adjacency matrix follow the Poisson distribution, and introduced the degree corrected stochastic block model (DCSBM) that allows the degree distribution of vertices also depend on the vertices, and not just on the blocks they belong to. This thesis is devoted to the problem of estimating the number of communities in the Poisson SBM and DCSBM. We consider the dense regime, in which the probability of connection between pairs of vertices does not depend on the size of the graph, or even the semi-sparse regime, in which the probability of connection between pairs of vertices can decay to 0 (at a certain rate) with the size of the graph. In this general context, we prove that the estimator of the number of communities introduced by Cerqueira and Leonardi (2020) (with the necessary changes) is still strongly consistent.