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.本研究聚焦随机效应随机块模型（Random Effects Stochastic Block Model）框架下的多层网络联合社区检测问题。该模型存在独特的建模挑战：其会使多层网络各层的社区结构产生变异，此类变异源自贯穿所有网络层的公共社区结构所产生的随机变换。对该模型进行精确拟合属于NP难（NP-hard）问题。为此我们提出拆分似然法（Split-Likelihood Method），该方法通过解耦社区分配的行、列标签，实现近似似然最大化流程，可兼顾检测精度与计算效率。我们为所提方法建立了收敛性理论，并推导了基于该方法得到的社区标签估计量的一致性理论。大量仿真结果表明，所提方法在检测精度与计算效率两方面均表现优异。最后，我们针对精神分裂症患者开展了静息态功能磁共振成像（resting-state fMRI）研究，以验证所提方法的实际应用价值。