Spectral and matrix factorization methods for consistent community detection in multi-layer networks

TL;DR

This paper analyzes the theoretical consistency of spectral and matrix factorization methods for community detection in multi-layer networks, providing new asymptotic results and demonstrating their advantages over other fusion techniques.

Contribution

It offers the first theoretical guarantees for intermediate fusion methods in multi-layer community detection under a high-dimensional stochastic blockmodel.

Findings

Intermediate fusion methods outperform late fusion in sparse networks.

They are more effective in networks with mixed homophilic and heterophilic communities.

The paper provides asymptotic consistency results for these methods.

Abstract

We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these "intermediate fusion" methods involve obtaining a low column rank matrix by optimizing an objective function and then using the columns of the matrix for clustering. However, the theoretical properties of these methods remain largely unexplored. In the absence of statistical guarantees on the objective functions, it is difficult to determine if the algorithms optimizing the objectives will return good community structures. We investigate the consistency properties of the global optimizer of some of these objective functions under the multi-layer stochastic blockmodel. For this purpose, we derive several new asymptotic results showing consistency of the intermediate fusion techniques along with the spectral clustering of mean adjacency matrix under a high dimensional setup, where the number of nodes, the number of layers and the number of communities of the multi-layer graph grow. Our numerical study shows that the intermediate fusion techniques outperform late fusion methods, namely spectral clustering on aggregate spectral kernel and module allegiance matrix in sparse networks, while they outperform the spectral clustering of mean adjacency matrix in multi-layer networks that contain layers with both homophilic and heterophilic communities.