Symmetry and Graph Bi-Regularized Non-Negative Matrix Factorization for Precise Community Detection

Zhigang Liu, Xin Luo, Mengchu Zhou

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Community is a fundamental and highly desired pattern in a Large-scale Undirected Network (LUN). Community detection is a vital issue when LUN representation learning is performed. Owing to its good scalability and interpretability, a Symmetric and Non-negative Matrix Factorization model is frequently utilized to tackle this issue. It adopts a unique Latent Factor (LF) matrix for precisely representing LUN&#x2019;s symmetry, which, unfortunately, leads to a reduced LF space that decreases its representation learning ability to a target LUN. Motivated by this discovery, this study proposes a Symmetry and Graph Bi-regularized Non-negative Matrix Factorization (B-NMF) method that: a) leverages multiple LF matrices when representing LUN, thereby boosting the representation learning ability; b) constructs a symmetry regularization term that implies the equality constraint among its multiple LF matrices, thereby illustrating LUN&#x2019;s intrinsic symmetry; and c) incorporates graph regularization into its learning objective, thereby illustrating LUN&#x2019;s local geometry. A theoretical proof is given to theoretically validate B-NMF&#x2019;s convergence ability. The regularization hyperparameters are selected by validating model modularity, thereby guaranteeing B-NMF&#x2019;s practicability in addressing real application issues. Extensive experimental results on ten LUNs from real applications demonstrate that the proposed B-NMF-based community detector significantly outperforms several baseline and state-of-the-art models in achieving highly-accurate community detection results. <italic>Note to Practitioners</italic>&#x2014;LUNs are very-commonly seen in real applications like a social network system. Communities in LUNs are vital for various knowledge discovery-related applications. For accurately detecting them, a detector should guarantee its high representation learning ability to a target LUN. To do so, this paper presents a B-NMF model that is able to perform precise representation learning to LUNs, thereby achieving accurate community detection results. In comparison with conventional Symmetric and Non-negative Matrix Factorization-based community detectors, a B-NMF-based community detector enjoys its enlarged latent feature space, which ensures its higher representation ability to a target LUN. It depends on two regularization hyperparameters, which can be selected by performing grid-search on the target LUN via its modularity evaluation. This paper gives the empirical values of B-NMF&#x2019;s regularization hyperparameters based on the parametersensitivity tests on the involved experimental datasets. The proposed B-NMF model is shown to be highly suitable for addressing community detection and clustering tasks on LUNs from real applications.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalIEEE Transactions on Automation Science and Engineering
StateAccepted/In press - 2023

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Electrical and Electronic Engineering


  • Convergence
  • Detectors
  • Geometry
  • LUN
  • Large-scale undirected network
  • Representation learning
  • Symmetric matrices
  • Task analysis
  • Topology
  • community detection
  • network representation
  • non-negative matrix factorization
  • representation learning
  • symmetric model
  • symmetry and graph bi-regularization


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