Community Detection in the Multi-View Stochastic Block Model

TL;DR

This paper introduces the multi-view stochastic block model (MVSBM) for correlated multi-graph community detection, establishing theoretical bounds for exact recovery and demonstrating the model's generality over previous SBMs.

Contribution

The paper proposes the MVSBM, providing the first information-theoretic bounds for community detection in correlated multi-graph settings, extending prior SBM results.

Findings

Exact recovery is achievable above a certain parameter threshold.

Below the threshold, any estimator misclassifies more than one node on average.

The model generalizes standard SBM and multiple independent SBMs.

Abstract

This paper considers the problem of community detection on multiple potentially correlated graphs from an information-theoretical perspective. We first put forth a random graph model, called the multi-view stochastic block model (MVSBM), designed to generate correlated graphs on the same set of nodes (with cardinality $n$). The $n$ nodes are partitioned into two disjoint communities of equal size. The presence or absence of edges in the graphs for each pair of nodes depends on whether the two nodes belong to the same community or not. The objective for the learner is to recover the hidden communities with observed graphs. Our technical contributions are two-fold: (i) We establish an information-theoretic upper bound (Theorem~1) showing that exact recovery of community is achievable when the model parameters of MVSBM exceed a certain threshold. (ii) Conversely, we derive an information-theoretic lower bound (Theorem~2) showing that when the model parameters of MVSBM fall below the aforementioned threshold, then for any estimator, the expected number of misclassified nodes will always be greater than one. Our results for the MVSBM recover several prior results for community detection in the standard SBM as well as in multiple independent SBMs as special cases.