Community detection by spectral methods in multi-layer networks

TL;DR

This paper compares spectral clustering algorithms for community detection in multi-layer networks, demonstrating the advantages of using debiased squared adjacency matrices and providing scalable methods with proven consistency and accurate community number estimation.

Contribution

The paper introduces and analyzes two spectral clustering algorithms for multi-layer networks, establishing their consistency, efficiency, and superiority, along with a novel strategy for estimating the number of communities.

Findings

Debiased squared adjacency matrices outperform simple adjacency sums.

Accelerated algorithms enable large-scale network analysis.

The community number estimation strategy is highly accurate.

Abstract

Community detection in multi-layer networks is a crucial problem in network analysis. In this paper, we analyze the performance of two spectral clustering algorithms for community detection within the framework of the multi-layer degree-corrected stochastic block model (MLDCSBM) framework. One algorithm is based on the sum of adjacency matrices, while the other utilizes the debiased sum of squared adjacency matrices. We also provide their accelerated versions through subsampling to handle large-scale multi-layer networks. We establish consistency results for community detection of the two proposed methods under MLDCSBM as the size of the network and/or the number of layers increases. Our theorems demonstrate the advantages of utilizing multiple layers for community detection. Our analysis also indicates that spectral clustering with the debiased sum of squared adjacency matrices is generally superior to spectral clustering with the sum of adjacency matrices. Furthermore, we provide a strategy to estimate the number of communities in multi-layer networks by maximizing the averaged modularity. Substantial numerical simulations demonstrate the superiority of our algorithm employing the debiased sum of squared adjacency matrices over existing methods for community detection in multi-layer networks, the high computational efficiency of our accelerated algorithms for large-scale multi-layer networks, and the high accuracy of our strategy in estimating the number of communities. Finally, the analysis of several real-world multi-layer networks yields meaningful insights.