On consistency of model selection for stochastic block models

Estimating the number of communities is one of the fundamental problems in the stochastic block model. We propose a penalized likelihood model selection criterion to decide the number of communities and show that the produced estimator is consistent. The novel penalty function improves that used by Wang and Bickel (2016) which leads to the estimator that is always underfit. Along the way, we derive the limiting distribution of the likelihood ratio test statistic in the cases of underfitting and overfitting. We also establish the Wilks theorem for the stochastic block model. Numerical studies demonstrate our theoretical results.