Two-sample test for stochastic block models via the largest singular value

The stochastic block model is widely used for detecting community structures in network data. However, the research interest in much of the literature focuses on the study of one sample of stochastic block models. Detecting the difference between the two community structures is a less studied issue for stochastic block models. In this article, we propose a novel test statistic based on the largest singular value of a residual matrix obtained by subtracting the geometric mean of two estimated block mean effects from the sum of two observed adjacency matrices. We prove that the null distribution of the proposed test statistic converges in distribution to the Tracy-Widom distribution with index 1, and we show the difference between the two samples for stochastic block models can be tested via the proposed method. We show that the proposed test is asymptotically powerful against alternative models. Further, we extend the proposed method to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.