The stochastic block model (SBM) has been widely used to analyze network data. Various goodness-of-fit tests have been proposed to assess the adequacy of model structures. To the best of our knowledge, however, none of the existing approaches are applicable for sparse networks in which the connection probability of any two communities is of order log(n)/n, and the number of communities is divergent. To fill this gap, we propose a novel goodness-of-fit test for the stochastic block model. The key idea is to construct statistics by sampling the maximum entry-deviations of the adjacency matrix that the negative impacts of network sparsity are alleviated by the sampling process. We demonstrate theoretically that the proposed test statistic converges to the Type-I extreme value distribution under the null hypothesis regardless of the network structure. Accordingly, it can be applied to both dense and sparse networks. In addition, we obtain the asymptotic power against alternatives. Moreover, we introduce a bootstrap-corrected test statistic to improve the finite sample performance, recommend an augmented test statistic to increase the power, and extend the proposed test to the degree-corrected SBM. Simulation studies and two empirical examples with both dense and sparse networks indicate that the proposed method performs well.
Wu et al. (Sat,) studied this question.