Community detection in multi-layer networks by regularized debiased spectral clustering

Community detection is a crucial problem in the analysis of multi-layer networks. While regularized spectral clustering methods using the classical regularized Laplacian matrix have shown great potential in handling sparse single-layer networks, to our knowledge, their potential in multi-layer network community detection remains unexplored. To address this gap, in this work, we introduce a new method, called regularized debiased sum of squared adjacency matrices (RDSoS), to detect communities in multi-layer networks. RDSoS is developed based on a novel regularized Laplacian matrix that regularizes the debiased sum of squared adjacency matrices. In contrast, the classical regularized Laplacian matrix typically regularizes the adjacency matrix of a single-layer network. Therefore, at a high level, our regularized Laplacian matrix extends the classical one to multi-layer networks. We establish the consistency property of RDSoS under the multi-layer stochastic block model (MLSBM) and further extend RDSoS and its theoretical results to the degree-corrected version of the MLSBM model. Additionally, we introduce a sum of squared adjacency matrices modularity (SoS-modularity) to measure the quality of community partitions in multi-layer networks and estimate the number of communities by maximizing this metric. Our methods offer promising applications for predicting gene functions, improving recommender systems, detecting medical insurance fraud, and facilitating link prediction. Experimental results demonstrate that our methods exhibit insensitivity to the selection of the regularizer, generally outperform state-of-the-art techniques, uncover the assortative property of real networks, and that our SoS-modularity provides a more accurate assessment of community quality compared to the average of the Newman-Girvan modularity across layers.