  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.
