Large Deviation Principles for Empirical Measures of Coloured Random Graphs

Abstract. For any finite coloured graph we define the empirical neighbourhood measure, which counts the number of vertices of a given colour connected to a given number of vertices of each colour, and the empirical pair measure, which counts the number of edges connecting each pair of colours. For a class of models of sparse coloured random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erdős-Rényi graphs near criticality.