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Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks

Huan Qing, Jingli Wang

TL;DR

BiMMDF delivers a unified distribution-free framework for overlapping bipartite weighted networks by letting edge weights come from an arbitrary distribution with mean $\Omega=\rho\Pi_r P \Pi_c^\top$. The proposed DiSP spectral algorithm achieves consistent mixed-membership estimation under distribution-specific separation conditions, with a rigorous analysis of how $\gamma$ and $\rho$ shape error rates. A missing-edge extension enables scalable handling of sparse networks by coupling BiMMDF with bipartite unweighted models. Empirical results on synthetic data and eight real networks demonstrate robust performance across diverse distributions, including signed weights, and show practical utility for uncovering complex, overlapping community structure. Together, these contributions offer a flexible, scalable approach that extends classic blockmodels to general, overlapping bipartite weighted networks with theoretical guarantees.

Abstract

Modeling and estimating mixed memberships for overlapping unipartite un-weighted networks has been well studied in recent years. However, to our knowledge, there is no model for a more general case, the overlapping bipartite weighted networks. To close this gap, we introduce a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an adjacency matrix to follow any distribution as long as its expectation has a block structure related to node membership. In particular, BiMMDF can model overlapping bipartite signed networks and it is an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for different distributions. Furthermore, we also consider missing edges for sparse networks. The advantage of BiMMDF is demonstrated in extensive synthetic networks and eight real-world networks.

Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks

TL;DR

BiMMDF delivers a unified distribution-free framework for overlapping bipartite weighted networks by letting edge weights come from an arbitrary distribution with mean . The proposed DiSP spectral algorithm achieves consistent mixed-membership estimation under distribution-specific separation conditions, with a rigorous analysis of how and shape error rates. A missing-edge extension enables scalable handling of sparse networks by coupling BiMMDF with bipartite unweighted models. Empirical results on synthetic data and eight real networks demonstrate robust performance across diverse distributions, including signed weights, and show practical utility for uncovering complex, overlapping community structure. Together, these contributions offer a flexible, scalable approach that extends classic blockmodels to general, overlapping bipartite weighted networks with theoretical guarantees.

Abstract

Modeling and estimating mixed memberships for overlapping unipartite un-weighted networks has been well studied in recent years. However, to our knowledge, there is no model for a more general case, the overlapping bipartite weighted networks. To close this gap, we introduce a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an adjacency matrix to follow any distribution as long as its expectation has a block structure related to node membership. In particular, BiMMDF can model overlapping bipartite signed networks and it is an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for different distributions. Furthermore, we also consider missing edges for sparse networks. The advantage of BiMMDF is demonstrated in extensive synthetic networks and eight real-world networks.
Paper Structure (24 sections, 2 theorems, 28 equations, 16 figures, 7 tables, 1 algorithm)

This paper contains 24 sections, 2 theorems, 28 equations, 16 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Bipartite Mixed Membership Distribution-Free model
  3. A spectral algorithm for fitting the model
  4. Main results for DiSP
  5. Missing edge
  6. Experimental Results
  7. Baseline methods
  8. Evaluation metric
  9. Synthetic bipartite weighted networks
  10. Bernoulli distribution
  11. Poisson distribution
  12. Binomial distribution
  13. Normal distribution
  14. Exponential distribution
  15. Uniform distribution
  16. ...and 9 more sections

Key Result

Theorem 1

(Error of DiSP) Under $BiMMDF(n_{r}, n_{c}, K, P,\rho, \Pi_{r}, \Pi_{c},\mathcal{F})$, let $\hat{\Pi}_{r}$ an $\hat{\Pi}_{c}$ be obtained from Algorithm alg:DiSP, suppose Assumption a1 and Condition c1 hold, and furthermore, $\sigma_{K}(\Omega)\gg \sqrt{\rho\gamma(n_{r}+n_{c})\mathrm{log}(n_{r}+n_{ Especially, when $n_{r}=O(n), n_{c}=O(n)$, we have

Figures (16)

  • Figure 1: Illustrations for networks modeled by BiMMDF. Panel (a): un-directed un-weighted network. Panel (b): un-directed weighted network with positive weights. Panel (c): un-directed weighted network with positive and negative weights. Panel (d): un-directed signed network. Panel (e): directed un-weighted network. Panel (f): directed weighted network with positive weights. Panel (g): directed weighted network with positive and negative weights. Panel (h): directed signed network. Panel (i): bipartite un-weighted network. Panel (j): bipartite weighted network with positive weights. Panel (k): bipartite weighted network with positive and negative weights. Panel (l): bipartite signed network.
  • Figure 2: Bernoulli distribution. For panels (e) and (f): the darker pixel represents a lower Hamming Error of DiSP.
  • Figure 3: Poisson distribution. For panels (e) and (f): the darker pixel represents a lower Hamming Error.
  • Figure 4: Binomial distribution. For panels (e) and (f): the darker pixel represents a lower Hamming Error.
  • Figure 5: Normal distribution. For panels (e) and (f): the darker pixel represents a lower Hamming Error.
  • ...and 11 more figures

Theorems & Definitions (19)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Definition 2
  • Corollary 1
  • Remark 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • ...and 9 more