Detection of communities by modularity matrix

Hosseini, Akram; Fath-Tabar, Gholamhosein; Shabani, Hossein

doi:10.22061/jdma.2025.11684.1112

Detection of communities by modularity matrix

Document Type : Full Length Article

Authors

¹ Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran

² Faculty of Computer, Network and Communication , Imam Hossein Comprehensive University, Tehran, I. R. Iran

10.22061/jdma.2025.11684.1112

Abstract

One of the most discussed topics in social networks is community detection. As these networks become more complex, spectral graph properties and graph-related structures are increasingly used for community detection. In this paper, we examine these properties of the modularity matrix, such as the eigenvalues of the modularity matrix structure of some specific graphs, modularity energy, and the Estrada modularity index. Additionally, we study the bounds for the energy and Estrada indices. Furthermore, considering the significant issue of estimating the number of communities in some community detection algorithms in networks, we focus on the modularity eigenvalues.

Graphical Abstract