On approximate equivalence of modularity, D and non-negative matrix factorization

TL;DR

This paper explores the theoretical connections between modularity measures and non-negative matrix factorization (NMF) in community detection, proposing new NMF-based methods and validating them on synthetic networks.

Contribution

It establishes the first formal links between modularity criteria and NMF, enabling new community detection approaches based on these insights.

Findings

Q maximization approximates NMF with Frobenius norm for large networks

D maximization can be reformulated as NMF

Proposed methods are effective on synthetic networks

Abstract

Community structures detection is one of the fundamental problems in complex network analysis towards understanding the topology structures of the network and the functions of it. Nonnegative matrix factorization (NMF) is a widely used method for community detection, and modularity Q and modularity density D are criteria to evaluate the quality of community structures. In this paper, we establish the connections between Q, D and NMF for the first time. Q maximization can be approximately reformulated under the framework of NMF with Frobenius norm, especially when $n$ is large, and D maximization can also be reformulated under the framework of NMF. Q minimization can be reformulated under the framework of NMF with Kullback-Leibler divergence. We propose new methods for community structures detection based on the above findings, and the experimental results on synthetic networks demonstrate their effectiveness.