Discovering overlapping communities in multi-layer directed networks

TL;DR

This work advances overlapping community detection in multi-layer directed networks by introducing the multi-layer MM-ScBM, which models nodes as mixed members in sending and receiving patterns across layers. It then develops CSPDSoS, a spectral procedure that leverages debiased aggregation matrices and vertex hunting (SPA) to consistently estimate per-node memberships. The authors prove per-node error bounds that improve with higher sparsity, more nodes, or more layers, and validate the theory with synthetic data and a real-world FAO trade network, showing insightful asymmetries between import and export communities. The approach unifies directed-multi-layer modeling with mixed memberships and offers a scalable, theoretically grounded tool for uncovering complex community structure in real systems.

Abstract

Community detection in multi-layer undirected networks has attracted considerable attention in recent years. However, multi-layer directed networks are common in the real world, and existing community detection methods often either ignore the asymmetric structure in multi-layer directed networks or assume that every node solely belongs to a single community, significantly limiting their applicability to overlapping multi-layer directed networks, where nodes can belong to multiple communities simultaneously. To fill this gap, this article explores the challenging problem of detecting overlapping communities in multi-layer directed networks. Our goal is to understand the underlying asymmetric overlapping community structure by analyzing the mixed memberships of nodes. We introduce a novel multi-layer mixed membership stochastic co-block model (multi-layer MM-ScBM) to model overlapping multi-layer directed networks. We develop a spectral procedure to estimate nodes' memberships in both sending and receiving patterns. Our method uses a successive projection algorithm on a few leading eigenvectors of two debiased aggregation matrices. To our knowledge, this is the first work to detect asymmetric overlapping communities in multi-layer directed networks. We demonstrate the consistent estimation properties of our method by providing per-node error rates under the multi-layer MM-ScBM framework. Our theoretical analysis reveals that increasing the overall sparsity, the number of nodes, or the number of layers can improve the accuracy of overlapping community detection. Extensive numerical experiments validate these theoretical findings. We also apply our method to one real-world multi-layer directed network, gaining insightful results.

Figures (12)

• Figure 1: A toy multi-layer directed network with 5 nodes and 6 layers.
• Figure 2: A toy example for the settings of $\Pi_{r}$, $\Pi_{c}$, and $\{B_{l}\}^{L}_{l=1}$. The sparsity parameter $\rho$ is set as 1 here. In this example, we consider a 3-layer directed network with 5 nodes belonging to 3 row (and column) communities with different weights. The row and column membership matrices are $\Pi_{r}$ and $\Pi_{c}$, respectively. One can easily check that $\Omega_{l} = \rho \Pi_{r} B_{l} \Pi'_{c}$ for $l \in [L]$.
• Figure 3: Numerical results of Experiments 1-3.
• Figure 4: Comparison of different methods on the FAOMTN dataset using estimated row and column membership matrices from our CSPDSoS algorithm as ground truth. Plots show the Hamming error, Relative error, and ONMI for the remaining five methods as the number of communities increases.
• Figure 5: Overall asymmetric degree measured by Hamming error, Relative error, and ONMI between $\hat{\Pi}_{r}$ and $\hat{\Pi}_{c}$ returned by our CSPDSoS algorithm when $K$ increases for the FAOMTN dataset.

Theorems & Definitions (11)

• Definition 1
• Remark 1
• Lemma 1
• Remark 2
• Theorem 1
• Remark 3
• Remark 4
• proof
• proof
• Lemma 2