Degree-Corrected Joint Matrix Factorization for Multilayer Community Detection

Alexandra Dache; Manon Rustin; Arnaud Vandaele; Nicolas Gillis

cs.SIOct 1, 2026

Degree-Corrected Joint Matrix Factorization for Multilayer Community Detection

Authors: Alexandra Dache, Manon Rustin, Arnaud Vandaele, Nicolas Gillis

Organizations: Department of Mathematics and Operational Research, University of Mons, Mons, Belgium

Abstract

Multilayer networks allow the modeling of interactions between the same entities across different contexts, such as temporal observations, varying settings, or interactions of different types. The goal of community detection in multilayer networks is to identify groups of nodes exhibiting similar connectivity patterns, which may vary across layers. We propose a method based on a joint nonnegative symmetric matrix trifactorization for community detection in multilayer networks, where each graph is approximated by a nonnegative symmetric matrix trifactorization. Our approach enforces constraints on the factor matrices so that communities are disjoint and shared across layers, while allowing each layer to have its own connectivity patterns and node degrees. This flexibility enables the model to capture both local and global structural variations across layers. We also develop an algorithm to efficiently solve this problem. We evaluate multilayer community detection methods using the multilayer degree-corrected stochastic block model (MDCBM), a flexible framework for generating realistic multilayer graphs with heterogeneous degrees and varying connectivity patterns. Experiments show that our method reliably detects communities across diverse regimes, whereas existing state-of-the-art approaches are often limited by restrictive structural assumptions.

Figures & tables

Panel of Figure 1 — Fig. 1: Examples of connectivity patterns that can be represented by OtrisymNMF with 3 communities, illustrated with the matrix $\theta$ , where entries with high values are shown in black.

Fig. 2: Average misclustering error as a function of the number of layers under different configurations of the connectivity matrices $\theta_{l}$ and node degree parameters $D_{l}$ .

Fig. 3: Average misclustering error as a function of the edge density of the networks under different configurations of the connectivity matrices $\theta_{l}$ and node degree parameters $D_{l}$ . The number of layers is set to $L=20$ .

Network	$n$	$L$	$r$	Min./max. community size	Edges	Density (%)	Avg. degree	Isolated nodes (%)
AUCS	61	5	8	$1/14$	620	$6.8\pm 4.0$	$4.1\pm 2.4$	$26.7\pm 26.2$
HS11	126	4	4	$8/45$	2797	$8.9\pm 1.6$	$11.1\pm 2.0$	$6.5\pm 2.9$
Cora	1662	2	3	$418/818$	27359	$1.0\pm 1.1$	$16.5\pm 18.4$	$1.1\pm 1.6$
Caltech	9144	6	102	$31/800$	2621175	$1.1\pm 0.0$	$95.6\pm 2.1$	$0$

TABLE I: Characteristics of the multilayer networks. For layer-dependent statistics, values are reported as mean

\pm

standard deviation across layers.

Method	AUCS		HS11		Cora		Caltech
Method	NMI	ARI	NMI	ARI	NMI	ARI	NMI	ARI
DC-MASE	$76.5\pm 0.6$	$61.5\pm 0.4$	$68.6\pm 0.0$	$75.4\pm 0.0$	$38.5\pm 0.0$	$32.7\pm 0.0$	$32.0\pm 0.2$	$9.6\pm 0.3$
Sum A	$80.4\pm 0.0$	$61.8\pm 0.0$	$78.8\pm 0.0$	$83.0\pm 0.0$	$44.6\pm 0.0$	$37.0\pm 0.0$	$35.3\pm 0.3$	$9.0\pm 0.2$
Bias-adjusted SoS	$76.3\pm 0.0$	$59.2\pm 0.0$	$78.6\pm 0.0$	$83.0\pm 0.0$	$39.2\pm 0.0$	$31.2\pm 0.0$	$33.0\pm 0.0$	$7.6\pm 0.1$
MASE	$77.7\pm 0.0$	$63.2\pm 0.0$	$80.1\pm 0.0$	$\underline{83.0}\pm 0.0$	$44.9\pm 0.0$	$38.3\pm 0.0$	$32.4\pm 0.0$	$9.5\pm 0.1$
OLMF	$53.5\pm 0.0$	$18.1\pm 0.0$	$78.1\pm 0.0$	$82.4\pm 0.0$	$\underline{53.5}\pm 0.0$	$\underline{43.5}\pm 0.0$	OOM	OOM

TABLE II: Mean

\pm

standard deviation of NMI and ARI (both in %) on real-world multilayer networks. Bold and underlined means indicate the highest and second-highest scores in each column, respectively. OOM: out of memory.

Fig. 4: Average misclustering error as a function of the number of layers under different configurations of the connectivity matrices $\theta_{l}$ and degree-correction parameters $D_{l}$ .

Panel of Figure 7 — Fig. 5: Relative error (top) and NMI (bottom) at each iteration of mFROST on AUCS, HS11, Cora, and Caltech, using the proposed and random initialization. Solid and dashed lines show the mean values over 10 restarts, respectively, while shaded areas represent one standard deviation.

Network	Edges	$r$	Time (s)	Iterations	NMI (%)
AUCS	620	8	$0.19\pm 0.08$	$11.0\pm 3.5$	$78.0\pm 4.8$
HS11	2797	4	$0.17\pm 0.05$	$5.3\pm 1.0$	$81.2\pm 0.6$
Cora	27359	3	$1.07\pm 0.28$	$10.9\pm 3.4$	$44.1\pm 1.3$
Caltech	2621175	102	$232\pm 43$	$13.7\pm 2.4$	$35.2\pm 0.3$

TABLE III: Mean

\pm

standard deviation of runtime, NMI, and number of iterations over 10 restarts of mFROST on real-world datasets.

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