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Community Detection on Inhomogeneous Multilayer Networks with Extreme Sparsity

Tao Shen, Wanjie Wang

Abstract

We study layer-specific community detection in an $L$-layer network $\{A^{(l)}\}_{l\in[L]}$ on a common set of $n$ nodes. Because modern networks are constructed from multi-modal data or with different contexts, the community labels $π^{(l)}\in[K]^n$ are layer-dependent and the degree heterogeneity parameters $θ_i^{(l)}$ vary widely across nodes and layers. The inhomogeneity and extreme sparsity raise a challenge for classical community detection methods. We propose a multilayer-assisted regularized spectral method (MARS-CD) to address this challenge. For layer $l$, MARS-CD first constructs $X^{(l)}$ from the remaining layers, so that the problem is transformed into a network-with-covariates clustering problem on $(A^{(l)}, X^{(l)})$. Then we recover $π^{(l)}$ by NAC in Hu and Wang (2024) that allows misalignment. The key component is to construct $X^{(l)}$, where we stack regularized embeddings. Building upon this, we establish the first theoretical guarantees for the quality of $X^{(l)}$ under multilayer networks with extreme sparsity. These further lead to weak and strong consistency for recovering $π^{(l)}$. We further develop an optional label alignment step to interpret the shared community structure across layers. Simulations demonstrate the superior performance of our MARS-CD method. Applying MARS-CD to international food trading networks provides an interpretable product-specific community structure.

Community Detection on Inhomogeneous Multilayer Networks with Extreme Sparsity

Abstract

We study layer-specific community detection in an -layer network on a common set of nodes. Because modern networks are constructed from multi-modal data or with different contexts, the community labels are layer-dependent and the degree heterogeneity parameters vary widely across nodes and layers. The inhomogeneity and extreme sparsity raise a challenge for classical community detection methods. We propose a multilayer-assisted regularized spectral method (MARS-CD) to address this challenge. For layer , MARS-CD first constructs from the remaining layers, so that the problem is transformed into a network-with-covariates clustering problem on . Then we recover by NAC in Hu and Wang (2024) that allows misalignment. The key component is to construct , where we stack regularized embeddings. Building upon this, we establish the first theoretical guarantees for the quality of under multilayer networks with extreme sparsity. These further lead to weak and strong consistency for recovering . We further develop an optional label alignment step to interpret the shared community structure across layers. Simulations demonstrate the superior performance of our MARS-CD method. Applying MARS-CD to international food trading networks provides an interpretable product-specific community structure.
Paper Structure (19 sections, 6 theorems, 26 equations, 2 figures, 3 tables, 3 algorithms)

This paper contains 19 sections, 6 theorems, 26 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Our Contribution
  4. Notation and Organization
  5. Methodology: Multilayer Adjacency Regularized Stacking and Community Detection
  6. Multilayer Networks with Inhomogeneity and Sparsity
  7. MARS: Cross-layer Covariate Construction
  8. MARS-CD: Layerwise Community Detection
  9. Optional Refinement: Label Alignment
  10. Theoretical Analysis
  11. Model Assumptions
  12. MARS-CD under the Oracle Setting
  13. Consistency of Covariates by MARS
  14. Consistency of MARS-CD
  15. Consistency of Label Alignment
  16. ...and 4 more sections

Key Result

Lemma 3.1

Assume $\hat{K}\ge K$. Let $\bar{\bm Y}^{(l)}=\bar{\bm\Xi}^{(l)}\bar{\bm\Lambda}^{(l)}\bar{\bm V}^{(l)\top}$ be a rank-$K$ SVD. Then, the $i$-row of $\bar{\bm\Xi}^{(l)}$ only depends on the community-specific terms: Consequently, row-wise normalization of $\bm\Xi^{(l)}$ yields exact recovery of $\bm \pi^{(l)}$, up to label permutation.

Figures (2)

  • Figure 1: Misclustering rate per layer versus transition probability $q_1$.
  • Figure 2: Connections (top row) and estimated community structures (bottom row) for main countries on three products.

Theorems & Definitions (9)

  • Definition 2.1: Community-level sparsity and node sets
  • Definition 3.1: Relative vs. extreme sparsity
  • Definition 3.2: Layer-specific Bad Nodes
  • Lemma 3.1: Spectral clustering
  • Proposition 3.1
  • Corollary 3.1
  • Theorem 3.1: Strong consistency
  • Theorem 3.2: Weak consistency
  • Theorem 3.3