Large deviations for empirical measures of dense stochastic block graphs

Abstract

Stochastic block model has been proved to be the most successful tool for detecting community structure in various networks as well as synthesizing networks with different communities. In this paper, we discuss dense stochastic block model, and suppose that every vertex’s block membership is randomly taken from a finite set independently according to a fixed law. To capture the asymptotic behavior of the stochastic block graphs when the number of vertices tends to infinity, we first define the empirical pair measure for any dense stochastic block graph, exploring the number of edges connecting any given pair of blocks. Then we put forward the empirical block measure, which computes the number of vertices with given block membership. Our concern here is the large deviation principles for these crucial empirical measures in the corresponding weak topology, we first derive the large deviation principle for the empirical pair measure under the empirical block measure, then through a mixing approach, we obtain the joint large deviation principle for these two empirical measures.