[Paper Review] Community detection in multiplex networks based on orthogonal nonnegative matrix tri-factorization

Meiby Ortiz-Bouza, Selin Aviyente|arXiv (Cornell University)|May 2, 2022

Complex Network Analysis Techniques50 references4 citations

TL;DR

This paper proposes MX-ONMTF, a novel multiplex community detection method that jointly identifies common communities across multiple network layers and layer-specific private communities using orthogonal nonnegative matrix tri-factorization. By decomposing each layer's adjacency matrix into shared and private low-rank components and employing an eigengap-based two-stage approach to determine community counts, the method outperforms state-of-the-art techniques on synthetic, real-world, and multiview clustering datasets.

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.

Motivation & Objective

• To address the limitation of existing multiplex community detection methods that assume identical community structures across layers and fail to capture heterogeneity.
• To develop a method capable of identifying both common communities (present in multiple layers) and private communities (unique to individual layers).
• To introduce a data-driven approach for determining the number of common and private communities without requiring prior knowledge.
• To evaluate the method on diverse network types, including synthetic, social, biological, and multiview clustering applications.

Proposed method

• The method models each layer's adjacency matrix as the sum of two low-rank nonnegative matrix factorizations: one for common communities and one for private communities.
• It employs orthogonal nonnegative matrix tri-factorization (ONMTF) to enforce orthogonality and improve interpretability of the community structures.
• An iterative multiplicative update algorithm is used to solve the optimization problem, ensuring convergence to a stationary point.
• A two-stage eigengap-based criterion is introduced to automatically estimate the number of common and private communities.
• The framework is generalized to L-layer multiplex networks, allowing common communities to appear in any subset of two or more layers.
• Theoretical convergence and recovery guarantees are provided, supporting the robustness of the algorithm.

Experimental results

Research questions

• RQ1 Can a multiplex community detection method effectively distinguish between communities that are common across multiple layers and those that are unique to individual layers?
• RQ2 How can the number of common and private communities be determined automatically without prior specification?
• RQ3 Does the proposed method outperform existing state-of-the-art techniques in detecting heterogeneous community structures across layers?
• RQ4 How well does the method generalize to real-world multiplex networks and multiview clustering tasks?

Key findings

• MX-ONMTF achieved the highest normalized mutual information (NMI) of 0.9432 on handwritten digit multiview data, outperforming all baselines.
• On Caltech-20 and Caltech-101 datasets, the method achieved NMI scores of 0.6861 and 0.5660, respectively, surpassing all compared methods.
• In synthetic network evaluations, the method demonstrated robust performance across varying noise levels, inter-layer dependency probabilities, and network sizes.
• The eigengap-based community count estimation method successfully identified the correct number of communities in both synthetic and real-world settings.
• The algorithm showed strong performance on real multiplex networks, including social and biological networks, confirming its practical utility.
• Theoretical analysis confirmed the convergence of the multiplicative update algorithm and provided recovery guarantees under low-rank assumptions.