On Equivalence of Likelihood Maximization of Stochastic Block Model and Constrained Nonnegative Matrix Factorization

TL;DR

This paper explores the relationship between stochastic block models and nonnegative matrix factorization in community detection, showing their likelihood functions are mathematically related and comparing their algorithmic behaviors.

Contribution

It demonstrates the equivalence of likelihood maximization in stochastic block models and constrained nonnegative matrix factorization, linking two popular community detection methods.

Findings

Likelihood functions are mathematically equivalent under certain reformulations.

Algorithms for the two methods exhibit different behaviors in practice.

Preliminary experiments compare algorithm performances.

Abstract

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. Besides the model equivalence, the algorithms employed by the two methods are different. Preliminary numerical experiments are carried out to compare the behaviors of the algorithms.