Uniqueness of communities in regular stochastic block models

TL;DR

This paper investigates the structure of regular stochastic block models with multiple communities, demonstrating that under certain conditions, the communities are uniquely identifiable with high probability as the network size grows.

Contribution

It introduces a detailed analysis of community uniqueness in regular stochastic block models with multiple communities, extending understanding of their structural properties.

Findings

The probability measure of the model closely matches the uniform measure on regular graphs.

Communities are asymptotically almost surely unique under weak assumptions.

The results hold for models with three or more communities as the number of vertices increases.

Abstract

This paper studies the regular stochastic block model comprising \emph{several} communities: each of the $k$ non-overlapping communities, for $k \geqslant 3$, possesses $n$ vertices, each of which has total degree $d$. The values of the intra-cluster degrees (i.e.\ the number of neighbours of a vertex inside the cluster it belongs to) and the inter-cluster degrees (i.e.\ the number of neighbours of a vertex inside a cluster different from its own) are allowed to vary across clusters. We discuss two main results: the first compares the probability measure induced by our model with the uniform measure on the space of $d$-regular graphs on $kn$ vertices, and the second establishes that the clusters, under rather weak assumptions, are unique asymptotically almost surely as $n \rightarrow \infty$.