Community detection in multiplex networks based on orthogonal nonnegative matrix tri-factorization

TL;DR

This work addresses heterogeneity in multiplex networks by proposing MX-ONMTF, which represents each layer’s adjacency as the sum of a common-community component and a layer-specific private component via Orthogonal Nonnegative Matrix Tri-Factorization. It introduces a two-stage method to determine the numbers of common $k_c$ and private $k_{p_l}$ communities, provides convergence and recovery guarantees under MLSBM, and validates the approach on synthetic and real networks as well as multiview clustering tasks. The method outperforms state-of-the-art multiplex techniques by capturing both shared and layer-specific structures, demonstrating robustness to noise and variation across layers, and enabling meaningful alignment with metadata in real-world data. These contributions advance community detection in heterogeneous multiplex settings and offer a scalable, interpretable framework for joint and private community discovery across layers.

Abstract

Networks are commonly used to model complex systems. The different entities in the system are represented by nodes of the network and their interactions by edges. In most real life systems, the different entities may interact in different ways necessitating the use of multiplex networks where multiple links are used to model the interactions. One of the major tools for inferring network topology is community detection. Although there are numerous works on community detection in single-layer networks, existing community detection methods for multiplex networks mostly learn a common community structure across layers and do not take the heterogeneity across layers into account. In this paper, we introduce a new multiplex community detection method that identifies communities that are common across layers as well as those that are unique to each layer. The proposed method, Multiplex Orthogonal Nonnegative Matrix Tri-Factorization, represents the adjacency matrix of each layer as the sum of two low-rank matrix factorizations corresponding to the common and private communities, respectively. Unlike most of the existing methods, which require the number of communities to be pre-determined, the proposed method also introduces a two stage method to determine the number of common and private communities. The proposed algorithm is evaluated on synthetic and real multiplex networks, as well as for multiview clustering applications, and compared to state-of-the-art techniques.

Figures (7)

• Figure 1: Illustration of the proposed community detection algorithm.
• Figure 2: Example dendrogram illustrating the hierarchical clustering of the columns of the embedding matrices $\mathbf{U}_1,\mathbf{U}_2,\mathbf{U}_3$ for a 3-layer network with 3 common communities. The red line indicates where the algorithm stops.
• Figure 3: Mean NMI over 100 realizations of (a)-(c) 3-layer, 4-layer and 5-layer benchmark networks, respectively, for the scenario with 2 common communities across all layers; (d)-(f) 3-layer, 4-layer and 5-layer benchmark networks, respectively, for the scenario with 3 common communities across different subsets of layers. All networks are generated with 8 different values of the mixing parameter $\mu$ and $n=256$.
• Figure 4: 5-layer network generated with 6 different values of the interlayer dependency probability $p_1$, with $\mu=0.1$, and $n=256$
• Figure 5: 5-layer network generated with different values of common communities $k_c$ across layers.

Theorems & Definitions (7)

• Lemma 1
• Theorem 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof