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Estimating mixed memberships in multi-layer networks

Huan Qing

TL;DR

The theoretical results reveal that the method leveraging the sum of adjacency matrices generally performs poorer than the other two methods for mixed membership estimation in multi-layer networks, and two novel modularity metrics are introduced to quantify the quality of mixed membership community detection.

Abstract

Community detection in multi-layer networks has emerged as a crucial area of modern network analysis. However, conventional approaches often assume that nodes belong exclusively to a single community, which fails to capture the complex structure of real-world networks where nodes may belong to multiple communities simultaneously. To address this limitation, we propose novel spectral methods to estimate the common mixed memberships in the multi-layer mixed membership stochastic block model. The proposed methods leverage the eigen-decomposition of three aggregate matrices: the sum of adjacency matrices, the debiased sum of squared adjacency matrices, and the sum of squared adjacency matrices. We establish rigorous theoretical guarantees for the consistency of our methods. Specifically, we derive per-node error rates under mild conditions on network sparsity, demonstrating their consistency as the number of nodes and/or layers increases under the multi-layer mixed membership stochastic block model. Our theoretical results reveal that the method leveraging the sum of adjacency matrices generally performs poorer than the other two methods for mixed membership estimation in multi-layer networks. We conduct extensive numerical experiments to empirically validate our theoretical findings. For real-world multi-layer networks with unknown community information, we introduce two novel modularity metrics to quantify the quality of mixed membership community detection. Finally, we demonstrate the practical applications of our algorithms and modularity metrics by applying them to real-world multi-layer networks, demonstrating their effectiveness in extracting meaningful community structures.

Estimating mixed memberships in multi-layer networks

TL;DR

The theoretical results reveal that the method leveraging the sum of adjacency matrices generally performs poorer than the other two methods for mixed membership estimation in multi-layer networks, and two novel modularity metrics are introduced to quantify the quality of mixed membership community detection.

Abstract

Community detection in multi-layer networks has emerged as a crucial area of modern network analysis. However, conventional approaches often assume that nodes belong exclusively to a single community, which fails to capture the complex structure of real-world networks where nodes may belong to multiple communities simultaneously. To address this limitation, we propose novel spectral methods to estimate the common mixed memberships in the multi-layer mixed membership stochastic block model. The proposed methods leverage the eigen-decomposition of three aggregate matrices: the sum of adjacency matrices, the debiased sum of squared adjacency matrices, and the sum of squared adjacency matrices. We establish rigorous theoretical guarantees for the consistency of our methods. Specifically, we derive per-node error rates under mild conditions on network sparsity, demonstrating their consistency as the number of nodes and/or layers increases under the multi-layer mixed membership stochastic block model. Our theoretical results reveal that the method leveraging the sum of adjacency matrices generally performs poorer than the other two methods for mixed membership estimation in multi-layer networks. We conduct extensive numerical experiments to empirically validate our theoretical findings. For real-world multi-layer networks with unknown community information, we introduce two novel modularity metrics to quantify the quality of mixed membership community detection. Finally, we demonstrate the practical applications of our algorithms and modularity metrics by applying them to real-world multi-layer networks, demonstrating their effectiveness in extracting meaningful community structures.
Paper Structure (13 sections, 8 theorems, 47 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 8 theorems, 47 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multi-layer mixed membership stochastic block model
  3. Spectral methods for mixed membership estimation
  4. Main results
  5. Numerical results on synthetic multi-layer networks
  6. Real data applications
  7. Conclusion
  8. Proofs
  9. Proof of Lemma \ref{['IS']}
  10. Proof of Theorem \ref{['mainSPSum']}
  11. An alternative proof of Theorem \ref{['mainSPSum']}
  12. Proof of Theorem \ref{['mainSPDSoS']}
  13. Proof of Theorem \ref{['mainSPSoS']}

Key Result

Lemma 1

(Ideal Simplexes) Under the model $\mathrm{MLMMSB}(\Pi,\rho,\mathcal{B})$, depending on the conditions imposed on the set $\{B_{l}\}^{L}_{l=1}$, we arrive at the following conclusions:

Figures (7)

  • Figure 1: Networks of the first three products of the FAO-trade multi-layer network.
  • Figure 2: Numerical results.
  • Figure 3: Ternary diagram of the $71\times3$ estimated membership matrix $\hat{\Pi}$ for Lazega Law Firm. Each dot represents a company partner, and its location within the triangle corresponds to its membership scores.
  • Figure 4: Illustration of the 3 layers of Lazega Law Firm. Colors (shapes) indicate home based communities and the black square represents highly mixed nodes detected by SPSum with 3 communities. For all layers, we only plot the largest connected graph.
  • Figure 5: Illustration of the 3 layers of C.Elegans. Colors (shapes) indicate home based communities and the black square represents highly mixed nodes detected by SPSum with 2 communities. For all layers, we only plot the largest connected graph.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • proof
  • Lemma 2
  • proof
  • ...and 8 more