M. Ghadirian - Department of Control Engineering, Faculty of Technical and Engineering, Imam-Khomeini International University, Qazvin, Iran.
N. Bigdeli - Department of Control Engineering, Faculty of Technical and Engineering, Imam-Khomeini International University, Qazvin, Iran.

kground and Objectives: Community detection is a critical problem in ‎investigating complex networks. Community detection based on ‎modularity/general modularity density are the popular methods with the ‎advantage of using complex network features and the disadvantage of ‎being NP-hard problem for clustering. Moreover, Non-negative matrix ‎factorization (NMF)-based community detection methods are a family of ‎community detection tools that utilize network topology; but most of ‎them cannot thoroughly exploit network features. In this paper, a hybrid ‎NMF-based community detection infrastructure is developed, including ‎modularity/ general modularity density as more comprehensive indices of ‎networks. The proposed infrastructure enables to solve the challenges of ‎combining the NMF method with modularity/general modularity density ‎criteria and improves the community detection methods for complex ‎networks.‎Methods: First, new representations, similar to the model of symmetric ‎NMF, are derived for the model of community detection based on ‎modularity/general modularity density. Next, these indices are ‎innovatively augmented to the proposed hybrid NMF-based model as two ‎novel models called ‘general modularity density NMF (GMDNMF) and ‎mixed modularity NMF (MMNMF)’. In order to solve these two NP-hard ‎problems, two iterative optimization algorithms are developed.‎Results: it is proved that the modularity/general modularity density-‎based community detection can be consistently represented in the form ‎of SNMF-based community detection. The performances of the proposed ‎models are verified on various artificial and real-world networks of ‎different sizes. It is shown that MMNMF and GMDNMF models ‎outperform other community detection methods. Moreover, the ‎GMDNMF model has better performance with higher computational ‎complexity compared to the MMNMF model.‎Conclusion: The results show that the proposed MMNMF model improves ‎the performance of community detection based on NMF by employing ‎the modularity index as a network feature for the NMF model, and the ‎proposed GMDNMF model enhances NMF-based community detection by ‎using the general modularity density index.‎

Complex Networks, ‎Nonnegative Matrix ‎Factorization, Modularity, ‎General Modularity Density, ‎Graph Clustering