Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery

TL;DR

This paper introduces a correlated contextual stochastic block model that leverages both edge and attribute correlations across multiple networks to improve community detection, especially in regimes where single-graph methods fail.

Contribution

It develops a novel two-step algorithm combining graph matching and community detection, demonstrating conditions for exact node matching and enhanced community recovery using correlated multi-graph data.

Findings

The proposed method achieves exact node matching under certain conditions.

Combining graphs improves community detection beyond single-graph capabilities.

The approach broadens the applicability of multi-graph community detection with attributes.

Abstract

We study community detection in multiple networks with jointly correlated node attributes and edges. This setting arises naturally in applications such as social platforms, where a shared set of users may exhibit both correlated friendship patterns and correlated attributes across different platforms. Extending the classical Stochastic Block Model (SBM) and its contextual counterpart (Contextual SBM or CSBM), we introduce the correlated CSBM, which incorporates structural and attribute correlations across graphs. To build intuition, we first analyze correlated Gaussian Mixture Models, wherein only correlated node attributes are available without edges, and identify the conditions under which an estimator minimizing the distance between attributes achieves exact matching of nodes across the two databases. For the correlated CSBMs, we develop a two-step procedure that first applies $k$-core matching to most nodes using edge information, then refines the matching for the remaining unmatched nodes by leveraging their attributes with a distance-based estimator. We identify the conditions under which the algorithm recovers the exact node correspondence, enabling us to merge the correlated edges and average the correlated attributes for enhanced community detection. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. Our results illustrate how the interplay between graph matching and community recovery can boost performance, broadening the scope of multi-graph, attribute-based community detection.