# A Symmetry and Graph Regularized Nonnegative Matrix Factorization Model for Community Detection

## TL;DR

This paper introduces SGNMF, a novel community detection model for large-scale undirected networks that enhances representation learning by using multiple latent matrices, enforces symmetry, and incorporates graph regularization.

## Contribution

It proposes a new SGNMF model that improves community detection by leveraging multiple latent matrices, symmetry regularization, and graph regularization, with proven convergence and superior experimental results.

## Key findings

- SGNMF outperforms baseline models in community detection accuracy.
- The model effectively captures network symmetry and intrinsic geometry.
- Theoretical convergence of SGNMF is established.

## Abstract

Community is a fundamental and critical characteristic of a Large-scale Undirected Network (LUN) like a social network, making community detection a vital yet thorny issue in LUN representation learning. Owing to its good scalability and interpretability, a Symmetric and Nonnegative Matrix Factorization (SNMF) model is commonly used to tackle this issue. However, it adopts a unique Latent Factor (LF) matrix for precisely representing an LUN's symmetry, which leads to a reduced LF space that impairs its representational learning ability. Motivated by this discovery, this study proposes a Symmetry and Graph-regularized Nonnegative Matrix Factorization (SGNMF) method that adopts three-fold ideas: a) leveraging multiple LF matrices to represent an LUN, thereby enhancing its representation learning ability; b) introducing a symmetry regularization term that implies the equality constraint between its multiple LF matrices, thereby illustrating an LUN's symmetry; and c) incorporating graph regularization into its learning objective, thereby illustrating an LUN's intrinsic geometry. A theoretical proof is provided to ensure SGNMF's convergence on an LUN. Extensive experiment results on ten LUNs from real applications demonstrate that our proposed SGNMF-based community detector significantly outperforms several baseline and state-of-the-art models in achieving highly-accurate results for community detection.

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Source: https://tomesphere.com/paper/2302.12122