Joint Community Detection in Random Effects Stochastic Block Models via the Split-likelihood Method

In this study, we tackle the joint community detection in multi-layer networks under a random effects stochastic block model. This model presents a unique challenge as it induces variability in the community structure across each layer of the multi-layer network. This variability is a random transformation originating from a common community structure that permeates all layers. The exact fit for this model is an NP-hard problem. We propose a solution, the split-likelihood method, which balances detection accuracy and computational efficiency. It employs an approximate likelihood maximization process by decoupling the row and column labels of community assignment. We establish the convergence theory for our proposed method, along with the consistency theories for the estimated community labels derived from it. Extensive simulation results suggest that the proposed method excels in both detection accuracy and computational efficiency. Finally, we conducted a resting state fMRI study on schizophrenia, to demonstrate the practical applicability of the proposed method.