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A semi-orthogonal nonnegative matrix tri-factorization algorithm for overlapping community detection

STATISTICAL PAPERS 2024年第6期65卷 3601-3619页

作者： Li, Zhaoyang Yang, Yuehan Fudan Univ Sch Publ Hlth Dept Biostat Shanghai 200433 Peoples R China Cent Univ Finance & Econ Sch Stat & Math Beijing 100081 Peoples R China

In this paper, we focus on overlapping community detection and propose an efficient semi-orthogonal nonnegative matrix tri-factorization (semi-ONMTF) algorithm. This method factorizes a matrix X into an orthogonal matrix U, a nonnegative matrix B, and a transposed matrix UT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U<^>\mathrm {\scriptscriptstyle T} $$\end{document}. We use the Cayley Transformation to maintain strict orthogonality of U that each iteration stays on the Stiefel Manifold. This algorithm is computationally efficient because the solutions of U and B are simplified into a matrix-wise update algorithm. Applying this method, we detect overlapping communities by the belonging coefficient vector and analyse associations between communities by the unweighted network of communities. We conduct simulations and applications to show that the proposed method has wide applicability. In a real data example, we apply the semi-ONMTF to a stock data set and construct a directed association network of companies. Based on the modularity for directed and overlapping communities, we obtain five overlapping communities, 17 overlapping nodes, and five outlier nodes in the network. We also discuss the associations between communities, providing insights into the overlapping community detection on the stock market network.

关键词： Semi-ONMTF Overlapping community detection Cayley transformation matrix-wise update algorithm Stock market

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