  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.
