The stochastic block model is a popular tool for detecting community structures in network data. Detecting the difference between two community structures is an important issue for stochastic block models. However, the two-sample test has been a largely under-explored domain, and too little work has been devoted to it. In this article, based on the maximum entry--wise deviation of the two centered and rescaled adjacency matrices, we propose a novel test statistic to test two samples of stochastic block models. We prove that the null distribution of the proposed test statistic converges in distribution to a Gumbel distribution, and we show the change of the two samples from stochastic block models can be tested via the proposed method. Then, we show that the proposed test has an asymptotic power guarantee against alternative models. One noticeable advantage of the proposed test statistic is that the number of communities can be allowed to grow linearly up to a logarithmic factor. 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.
翻译:随机区块模型是探测网络数据中社区结构的流行工具。 检测两个社区结构之间的差异是随机区块模型的一个重要问题。 但是, 两样样测试是一个基本上探索不足的领域, 也很少为此做任何工作。 在本条中, 以两个中枢和重新测量的相邻矩阵的最大切入偏差为基础, 我们提出了一个新颖的测试统计来测试两个随机区块模型的样本。 我们证明, 拟议的测试统计数据的无效分布在分布到 Gumbel 分布中, 我们展示了两个样本从随机区块模型中的变化可以通过拟议方法进行测试。 然后, 我们显示, 拟议的测试对替代模型具有一种无药力保证。 拟议测试统计的一个显著优点是, 允许社区数量成线性增长到对数。 此外, 我们将拟议方法的无效分布扩展到经度校正区块模型。 模拟研究和现实世界数据示例都表明, 拟议的方法工作很成功。