Large deviations for empirical measures of generalized random graphs

Abstract

In a generalized random graph with random vertex weights, we investigate the asymptotic behaviors for two crucial empirical measures: The empirical pair measure, which represents the number of edges connecting each pair of weights, and the empirical neighborhood measure, which interprets the number of vertices of a given weight connected to a given number of vertices of each weight. By some mixing approaches, we obtain the large deviation principles for these empirical measures in the weak topology. Through these large deviation results, the large deviation principle for the number of edges in a generalized random graph is obtained, as well as the large deviation principle for the empirical degree distribution.