Regularized spectral clustering under the mixed membership stochastic block model

Abstract

Mixed membership community detection is a challenging problem in network analysis. Previous spectral clustering algorithms for this problem are developed based on the adjacency matrix instead of regularized Laplacian matrix. To close this gap, under the popular mixed membership stochastic blockmodels (MMSB), this article proposes two efficient spectral clustering algorithms based on an application of the regularized Laplacian matrix, the Simplex Regularized Spectral Clustering (SRSC) algorithm, and the Cone Regularized Spectral Clustering (CRSC) algorithm. SRSC and CRSC are developed based on the simplex structure and the cone structure in the variants of the eigendecomposition of the regularized Laplacian matrix. We show that these two approaches SRSC and CRSC are asymptotically consistent under mild conditions by providing error bounds for the estimated membership vector of each node under MMSB. These two proposed approaches are successfully applied to synthetic and empirical networks with encouraging results compared with some benchmark methods.