  The stochastic block model is widely used for detecting community structures
in network data. However, the research interest of much literature focuses on
the study of one sample of stochastic block models. How to detect the
difference of the 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. Further, we show that the
proposed test has asymptotic power guarantee against alternative models. Both
simulation studies and real-world data examples indicate that the proposed
method works well.
