Consistent estimation of the number of communities via regularized network embedding

The network analysis plays an important role in numerous application domains including biomedicine. Estimation of the number of communities is a fundamental and critical issue in network analysis. Most existing studies assume that the number of communities is known a priori, or lack of rigorous theoretical guarantee on the estimation consistency. In this paper, we propose a regularized network embedding model to simultaneously estimate the community structure and the number of communities in a unified formulation. The proposed model equips network embedding with a novel composite regularization term, which pushes the embedding vector toward its center and pushes similar community centers collapsed with each other. A rigorous theoretical analysis is conducted, establishing asymptotic consistency in terms of community detection and estimation of the number of communities. Extensive numerical experiments have also been conducted on both synthetic networks and brain functional connectivity network, which demonstrate the superior performance of the proposed method compared with existing alternatives.