Harnessing Multiple Correlated Networks for Exact Community Recovery

TL;DR

This paper establishes the exact information-theoretic threshold for community detection in multiple correlated networks, extending previous two-graph results to any constant number of graphs, even when pairwise matchings are unidentifiable.

Contribution

It derives the precise threshold for exact community recovery using multiple correlated graphs, addressing the challenge of aggregating information without exact vertex correspondences.

Findings

Threshold for community recovery with multiple graphs

Recovery possible even without pairwise vertex matching

Extension from two graphs to any constant number of graphs

Abstract

We study the problem of learning latent community structure from multiple correlated networks, focusing on edge-correlated stochastic block models with two balanced communities. Recent work of Gaudio, R\'acz, and Sridhar (COLT 2022) determined the precise information-theoretic threshold for exact community recovery using two correlated graphs; in particular, this showcased the subtle interplay between community recovery and graph matching. Here we study the natural setting of more than two graphs. The main challenge lies in understanding how to aggregate information across several graphs when none of the pairwise latent vertex correspondences can be exactly recovered. Our main result derives the precise information-theoretic threshold for exact community recovery using any constant number of correlated graphs, answering a question of Gaudio, R\'acz, and Sridhar (COLT 2022). In particular, for every $K \geq 3$ we uncover and characterize a region of the parameter space where exact community recovery is possible using $K$ correlated graphs, even though (1) this is information-theoretically impossible using any $K-1$ of them and (2) none of the latent matchings can be exactly recovered.