Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

TL;DR

This paper develops compound Poisson approximation techniques for counting subgraphs in generalized stochastic block models that include multiple edges, extending classical results to more complex network structures.

Contribution

It introduces Stein-Chen method-based compound Poisson approximations for subgraph counts in models with multiple edges, including pseudo-graphs, broadening the scope of probabilistic network analysis.

Findings

Validates Poisson approximations in multiple edge regimes

Extends classical models to include self-loops and multiple edges

Provides theoretical bounds for approximation accuracy

Abstract

We use the Stein-Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. Both the cases that the fixed graph is a simple graph and that it has multiple edges are treated. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman \cite{kn11}.