Community detection by spectral methods in multi-layer networks

TL;DR

The paper tackles community detection in multi-layer networks by proposing two spectral clustering methods under the multi-layer degree-corrected SBM (MLDCSBM): NSoA, which operates on the sum of adjacency matrices, and NDSoSA, which uses a debiased sum of squared adjacencies. It provides rigorous consistency results showing that leveraging multiple layers improves detection accuracy and often favors the debiased method, with accelerated subsampling enabling scalability to large networks. A modularity-based approach for estimating the number of communities is proposed and validated through simulations and real-data experiments, where NDSoSA generally outperforms competing methods in both accuracy and efficiency. The work highlights the value of multi-layer information for community structure discovery and offers practical, scalable tools for analyzing complex multiplex datasets.

Abstract

Community detection in multi-layer networks is a crucial problem in network analysis. In this paper, we analyze the performance of two spectral clustering algorithms for community detection within the framework of the multi-layer degree-corrected stochastic block model (MLDCSBM) framework. One algorithm is based on the sum of adjacency matrices, while the other utilizes the debiased sum of squared adjacency matrices. We also provide their accelerated versions through subsampling to handle large-scale multi-layer networks. We establish consistency results for community detection of the two proposed methods under MLDCSBM as the size of the network and/or the number of layers increases. Our theorems demonstrate the advantages of utilizing multiple layers for community detection. Our analysis also indicates that spectral clustering with the debiased sum of squared adjacency matrices is generally superior to spectral clustering with the sum of adjacency matrices. Furthermore, we provide a strategy to estimate the number of communities in multi-layer networks by maximizing the averaged modularity. Substantial numerical simulations demonstrate the superiority of our algorithm employing the debiased sum of squared adjacency matrices over existing methods for community detection in multi-layer networks, the high computational efficiency of our accelerated algorithms for large-scale multi-layer networks, and the high accuracy of our strategy in estimating the number of communities. Finally, the analysis of several real-world multi-layer networks yields meaningful insights.

Figures (26)

• Figure 1: Adjacency matrices of a simple example of multi-layer network with 10 nodes and 3 layers.
• Figure 2: In this simulated example, we set $n=300$, $K=3$, and $L=20$, where each community comprises 100 nodes. Each entry of $B_{l}$ (as well as $\theta$) is a random value from a Uniform distribution on $[0,1]$ for $l\in [L]$. After configuring $Z$, $\mathcal{B}$, and $\Theta$, we calculate $\Omega_{l}$ using Equation (\ref{['AlMLDCSBM']}), and subsequently derive $\Omega_{\mathrm{sum}}$ and $\tilde{S}_{\mathrm{sum}}$. Then we compute $U$ and $V$, as well as their normalized counterparts $U_{*}$ and $V_{*}$, from $\Omega_{\mathrm{sum}}$ and $\tilde{S}_{\mathrm{sum}}$, respectively. Panels (a)-(d) illustrate points, where each point represents a row from $U$, $U_{*}$, $V$, and $V_{*}$, respectively. Across all panels, points of the same color indicate nodes belonging to the same community. The observations from Panels (a) and (c) reveal that, before normalization, nodes within the same community exhibit identical directions in the projected space. Conversely, Panels (b) and (d) demonstrate that, after normalization, nodes within the same community occupy identical positions in the projected space.
• Figure 3: The overall technical roadmap of the NSoA algorithm.
• Figure 4: The overall technical roadmap of the NDSoSA algorithm.
• Figure 6: Numerical results of Experiment 1.

Theorems & Definitions (34)

• Lemma 1
• Lemma 2
• Remark 1
• Remark 2
• Lemma 3
• Remark 3
• Theorem 1
• Corollary 1
• Remark 4
• Lemma 4