Estimating the number of communities in the stochastic block model with outliers

The stochastic block model (SBM) is a popular model for community detecting problems. Many community detecting approaches have been proposed, and most of them assume that the number of communities is given previously. However, in practice, the number of communities is often unknown. Plenty of approaches were proposed to estimate the number of communities, but most of them were computationally intensive. Moreover, when outliers exist, there are no approaches to consistently estimate the number of communities. In this article, we propose a fast method based on the eigenvalues of the regularized and normalized adjacency matrix to estimate the number of communities under the SBM with outliers. We show that our method can consistently estimate the number of communities when outliers exist. Moreover, we extend our method to the degree-corrected SBM. We show that our approach is comparable to the other existing approaches in simulations. We also illustrate our approach on four real-world networks.