Network Cross-Validation for Determining the Number of Communities in   Network Data
  The stochastic block model and its variants have been a popular tool in analyzing large network data with community structures. In this paper we develop an efficient network cross-validation (NCV) approach to determine the number of communities, as well as to choose between the regular stochastic block model and the degree corrected block model. The proposed NCV method is based on a block-wise node-pair splitting technique, combined with an integrated step of community recovery using sub-blocks of the adjacency matrix. We prove that the probability of under selection vanishes as the number of node increases, under mild conditions satisfied by a wide range of popular community recovery algorithms. The solid performance of our method is also demonstrated in extensive simulations and a data example. 
Kehui Chen and Jing Lei
stat.ME