Core-periphery Detection Based on Masked Bayesian Non-negative Matrix Factorization

TL;DR

The paper addresses core-periphery detection in networks by introducing a generative framework called masked Bayesian NMF, which factorizes the adjacency matrix into latent core-periphery affiliations while leveraging a mask to emphasize core nodes. It jointly models W, H, M, and hyperparameters with a Poisson likelihood and half-normal/Gamma priors, derives multiplicative update rules, and proves convergence to a local optimum. The approach handles overlapping core-periphery pairs and yields soft core scores, demonstrated on synthetic and real networks where it outperforms traditional methods in terms of $NMI_{cp}$ and scalability, with GPU acceleration providing substantial speedups. The work contributes a principled, scalable method for identifying complex CP structures and overlaps, with practical implications for analyzing mesoscale organization in various networks; code is publicly available.

Abstract

Core-periphery structure is an essential mesoscale feature in complex networks. Previous researches mostly focus on discriminative approaches while in this work, we propose a generative model called masked Bayesian non-negative matrix factorization. We build the model using two pair affiliation matrices to indicate core-periphery pair associations and using a mask matrix to highlight connections to core nodes. We propose an approach to infer the model parameters, and prove the convergence of variables with our approach. Besides the abilities as traditional approaches, it is able to identify core scores with overlapping core-periphery pairs. We verify the effectiveness of our method using randomly generated networks and real-world networks. Experimental results demonstrate that the proposed method outperforms traditional approaches.

Figures (11)

• Figure 1: A visual illustration of a simple network with two core-periphery structures. Yellow nodes are core nodes ,while blue nodes are periphery nodes.
• Figure 2: Comparison between the traditional non-negative matrix factorization and the proposed factorization method. $\hat{V}$ is an approximation to the adjacency matrix, Factors $W$ and $H$ can indicate core-periphery pair affiliations. In our proposed method, a mask matrix $M$ is multiplied with $W$ and $H$ to highlight core nodes.
• Figure 3: A graphical illustration of our masked Bayesian non-negative matrix factorization model. The observed value $V$ depends on $W$, $H$, and $M$, and these variables further depend on $\beta$ and $\mu$. $a$, $b$, $\overline{\sigma}$, $\hat{\mu}$, and $\hat{sigma}$ are hyper-parameters.
• Figure 4: Sensitivity analyze of hyperparameters. For each hyperparameter, we apply a variation on it and run experiments to record $NMI_{cp}$. We report the mean and standard deviation of the recorded results.
• Figure 5: Runtime of different algorithms

Theorems & Definitions (5)

• Definition 1
• Lemma 1
• Proof 3.1
• Theorem 1
• Proof 3.2