Matrix Factorization Framework for Community Detection under the Degree-Corrected Block Model

TL;DR

This work reframes community detection under the degree-corrected block model (DCBM) as a constrained nonnegative matrix factorization problem, introducing the orthogonal symmetric nonnegative matrix trifactorization (OtrisymNMF) with Frobenius reconstruction error. The authors present FROST, an alternating optimization algorithm for OtrisymNMF, and a robust initialization based on separable NMF (SVCA) that yields accurate starting estimates and accelerates convergence for DCBM inference. Empirical results on synthetic LFR benchmarks and real networks (including bipartite datasets) show that OtrisymNMF with FROST achieves comparable accuracy to DCBM-based methods while offering faster runtimes and scalability to large graphs; SVCA initialization consistently improves solution quality and reduces iterations across methods. Overall, the matrix-factorization perspective provides a scalable, structure-agnostic framework for community detection under the DCBM, with practical impact on large-scale network analysis.

Abstract

Community detection is a fundamental task in data analysis. Block models form a standard approach to partition nodes according to a graph model, facilitating the analysis and interpretation of the network structure. By grouping nodes with similar connection patterns, they enable the identification of a wide variety of underlying structures. The degree-corrected block model (DCBM) is an established model that accounts for the heterogeneity of node degrees. However, existing inference methods for the DCBM are heuristics that are highly sensitive to initialization, typically done randomly. In this work, we show that DCBM inference can be reformulated as a constrained nonnegative matrix factorization problem. Leveraging this insight, we propose a novel method for community detection and a theoretically well-grounded initialization strategy that provides an initial estimate of communities for inference algorithms. Our approach is agnostic to any specific network structure and applies to graphs with any structure representable by a DCBM, not only assortative ones. Experiments on synthetic and real benchmark networks show that our method detects communities comparable to those found by DCBM inference, while scaling linearly with the number of edges and communities; for instance, it processes a graph with 100,000 nodes and 2,000,000 edges in approximately 4 minutes. Moreover, the proposed initialization strategy significantly improves solution quality and reduces the number of iterations required by all tested inference algorithms. Overall, this work provides a scalable and robust framework for community detection and highlights the benefits of a matrix-factorization perspective for the DCBM.

Figures (10)

• Figure 1: Examples of structures detectable by an SBM with 3 blocks, illustrated with the matrix $\theta$, where entries with high values are shown in black.
• Figure 2: Geometric illustration of the separability property with $n=20$ and $r=3$. The extreme rays of $\text{cone}(A)$ are present in $A$ as columns and correspond to the columns of $W$ (up to scaling). For visualization, the example is represented in 3D, although the data is originally $n=20$ dimensional.
• Figure 3: Examples of LFR benchmark networks with $1{,}000$ nodes for different values of $\mu$.
• Figure 4: Average AMI and average runtime over 10 LFR benchmark graphs for $\mu$ ranging from $0$ to $0.6$. Each method is run 10 times per graph, and the solution with the best objective value is kept. (S) indicates SVCA initialization.
• Figure 5: Average AMI and runtime over 10 LFR graphs for different network sizes.