Poisson approximation of subgraph counts in stochastic block models and a graphon model

Small subgraph counts can be used as summary statistics for large random graphs. We use the Stein–Chen method to derive Poisson approximations for the distribution of the number of subgraphs in the stochastic block model which are isomorphic to some fixed graph. We also obtain Poisson approximations for subgraph counts in a graphon-type generalisation of the model in which the edge probabilities are (possibly dependent) random variables supported on a subset of [ 0,1 ]. Our results apply when the fixed graph is a member of the class of strictly balanced graphs.