Abstract: This Community detection is one of the most interesting problems in the study of social networks. Most of the recent studies focused on how to design algorithms to find the communities without knowing the number of communities in advance. In this paper, we define the k-path graph, and generalize Newman’s modularity as weighted modularity. It is also highlight the relationship between eigenvalues and the number of communities of social networks in this paper.
DOI: *As the DOI is a unique identifier, it is already available in the pdf version. **The DOI link will be activated in the first midst of January 2026.