On equivalence of likelihood maximization of stochastic block model and constrained nonnegative matrix factorization

Community structures detection in complex network is important for understanding not only the topological structures of the network, but also the functions of it. Stochastic block model and nonnegative matrix factorization are two widely used methods for community detection, which are proposed from different perspectives. In this paper, the relations between them are studied. The logarithm of likelihood function for stochastic block model can be reformulated under the framework of nonnegative matrix factorization. Despite model equivalence, the algorithms employed by the two methods are different. Preliminary numerical experiments are carried out to compare the behaviors of the algorithms, demonstrating that the multiplicative update rules for NMF are more effective.