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

TL;DR

This paper develops Poisson approximation techniques for counting specific subgraphs in stochastic block models and graphon models, providing a probabilistic understanding of subgraph distributions in large random graphs.

Contribution

It introduces the use of the Stein-Chen method to derive Poisson approximations for subgraph counts in both stochastic block and graphon models, extending previous results to more general settings.

Findings

Poisson approximations are valid for subgraph counts in the models studied.

Results apply to strictly balanced graphs within the models.

The methods handle dependent edge probabilities in graphon models.

Abstract

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.