A stochastic block Ising model for multi‐layer networks with inter‐layer dependence

Community detection has attracted tremendous interests in network analysis, which aims at finding group of nodes with similar characteristics. Various detection methods have been developed to detect homogeneous communities in multi‐layer networks, where inter‐layer dependence is a widely acknowledged but severely under‐investigated issue. In this paper, we propose a novel stochastic block Ising model (SBIM) to incorporate the inter‐layer dependence to help with community detection in multi‐layer networks. The community structure is modeled by the stochastic block model (SBM) and the inter‐layer dependence is incorporated via the popular Ising model. Furthermore, we develop an efficient variational EM algorithm to tackle the resultant optimization task and establish the asymptotic consistency of the proposed method. Extensive simulated examples and a real example on gene co‐expression multi‐layer network data are also provided to demonstrate the advantage of the proposed method.