Tests of graph homogeneity, subgraph counts, and quasirandomness

Abstract

Homogeneous random graphs, also known as Erd{\H os}-Rényi graphs, are a subset of the family of dense random graphs, specified by a graphon. We analyze several goodness-of-fit tests for these models that are based on subgraph counts. To obtain the limiting null distribution of the test statistics we use a decomposition of graph functionals, which reveals a cancellation effect. Motivated by a quasirandomness result we obtain a test that is consistent against all alternatives, and we use two popular parametric subfamilies to evaluate the other tests. The theoretical results refer to the limit $n\to \infty$ of the size $n$ of the graph, the behavior for finite $n$ is illustrated by simulations.