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Community detection in multi-layer bipartite networks

Huan Qing

TL;DR

This paper introduces a multi-layer degree-corrected stochastic co-block model (multi-layer DC-ScBM) for community detection in bipartite networks and proposes a debiased spectral co-clustering algorithm, NcDSoS, to recover the shared row and column communities across layers. It provides rigorous high-probability error bounds showing consistency that improve as the number of layers grows, highlighting the benefit of multi-layer structure. The authors establish a series of lemmas and theorems, including concentration bounds via rectangular matrix Bernstein and perturbation analyses, and validate the method through extensive simulations and real-data applications. The work advances cross-layer community detection in bipartite networks and lays groundwork for scalable, theoretically grounded techniques applicable to complex multi-layer systems.

Abstract

The problem of community detection in multi-layer undirected networks has received considerable attention in recent years. However, practical scenarios often involve multi-layer bipartite networks, where each layer consists of two distinct types of nodes. Existing community detection algorithms tailored for multi-layer undirected networks are not directly applicable to multi-layer bipartite networks. To address this challenge, this paper introduces a novel multi-layer degree-corrected stochastic co-block model specifically designed to capture the underlying community structure within multi-layer bipartite networks. Within this framework, we propose an efficient debiased spectral co-clustering algorithm for detecting nodes' communities. We establish the consistent estimation property of our proposed algorithm and demonstrate that an increased number of layers in bipartite networks improves the accuracy of community detection. Through extensive numerical experiments, we showcase the superior performance of our algorithm compared to existing methods. Additionally, we validate our algorithm by applying it to real-world multi-layer network datasets, yielding meaningful and insightful results.

Community detection in multi-layer bipartite networks

TL;DR

This paper introduces a multi-layer degree-corrected stochastic co-block model (multi-layer DC-ScBM) for community detection in bipartite networks and proposes a debiased spectral co-clustering algorithm, NcDSoS, to recover the shared row and column communities across layers. It provides rigorous high-probability error bounds showing consistency that improve as the number of layers grows, highlighting the benefit of multi-layer structure. The authors establish a series of lemmas and theorems, including concentration bounds via rectangular matrix Bernstein and perturbation analyses, and validate the method through extensive simulations and real-data applications. The work advances cross-layer community detection in bipartite networks and lays groundwork for scalable, theoretically grounded techniques applicable to complex multi-layer systems.

Abstract

The problem of community detection in multi-layer undirected networks has received considerable attention in recent years. However, practical scenarios often involve multi-layer bipartite networks, where each layer consists of two distinct types of nodes. Existing community detection algorithms tailored for multi-layer undirected networks are not directly applicable to multi-layer bipartite networks. To address this challenge, this paper introduces a novel multi-layer degree-corrected stochastic co-block model specifically designed to capture the underlying community structure within multi-layer bipartite networks. Within this framework, we propose an efficient debiased spectral co-clustering algorithm for detecting nodes' communities. We establish the consistent estimation property of our proposed algorithm and demonstrate that an increased number of layers in bipartite networks improves the accuracy of community detection. Through extensive numerical experiments, we showcase the superior performance of our algorithm compared to existing methods. Additionally, we validate our algorithm by applying it to real-world multi-layer network datasets, yielding meaningful and insightful results.
Paper Structure (11 sections, 5 theorems, 23 equations, 13 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 5 theorems, 23 equations, 13 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The multi-layer degree-corrected stochastic co-block model
  3. Algorithm
  4. Theoretical properties
  5. Simulations
  6. Real data applications
  7. Conclusion
  8. Proofs
  9. Proof of Lemma \ref{['EigOsum2']}
  10. Proof of Lemma \ref{['boundSsum']}
  11. Proof of Theorem \ref{['mainNcDSoS']}

Key Result

Lemma 1

Consider the multi-layer DC-ScBM parameterized by $(Z_{r},Z_{c},\Theta_{r},\Theta_{c},\{B_{l}\}^{L}_{l=1})$.

Figures (13)

  • Figure 1: A toy example of transforming a directed network in panel (a) to a bipartite network in panel (b). In both panels, red dots and blue dots represent row nodes and column nodes, respectively.
  • Figure 2: A multi-layer bipartite network with 12 row nodes, 8 column nodes, and 3 layers.
  • Figure 3: The adjacency matrices corresponding to the 3 distinct layers of the multi-layer bipartite network shown in Figure \ref{['BipartiteNLayer3']}, where black square represents 1.
  • Figure 4: Experiment 1. Panel (a): Clustering error against increasing $\rho$. Panel (b): Hamming error against increasing $\rho$. Panel (c): NMI against increasing $\rho$. Panel (d): ARI against increasing $\rho$.
  • Figure 5: Experiment 2. Panel (a): Clustering error against increasing $n$. Panel (b): Hamming error against increasing $n$. Panel (c): NMI against increasing $n$. Panel (d): ARI against increasing $n$.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Theorem 1
  • Corollary 1
  • proof
  • proof
  • Theorem 2
  • proof