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Community detection in heterogeneous signed networks

Yuwen Wang, Shiwen Ye, Jingnan Zhang, Junhui Wang

TL;DR

This paper formally defines strong and weak balance in signed networks, and proposes a signed block $\beta$-model, which is capable of modeling strong- and weak-balanced signed networks simultaneously.

Abstract

Network data has attracted growing interest across scientific domains, prompting the development of various network models. Existing network analysis methods mainly focus on unsigned networks, whereas signed networks, consisting of both positive and negative edges, have been frequently encountered in practice but much less investigated. In this paper, we formally define strong and weak balance in signed networks, and propose a signed block $β$-model, which is capable of modeling strong- and weak-balanced signed networks simultaneously. We establish the identifiability of the proposed model by leveraging properties of bipartite graphs, and develop an efficient alternating updating algorithm to optimize the resulting log-likelihood function. More importantly, we establish the asymptotic consistencies of the proposed model in terms of both probability estimation and community detection. Its advantages are also demonstrated through extensive numerical experiments and the application to a real-world international relationship network.

Community detection in heterogeneous signed networks

TL;DR

This paper formally defines strong and weak balance in signed networks, and proposes a signed block -model, which is capable of modeling strong- and weak-balanced signed networks simultaneously.

Abstract

Network data has attracted growing interest across scientific domains, prompting the development of various network models. Existing network analysis methods mainly focus on unsigned networks, whereas signed networks, consisting of both positive and negative edges, have been frequently encountered in practice but much less investigated. In this paper, we formally define strong and weak balance in signed networks, and propose a signed block -model, which is capable of modeling strong- and weak-balanced signed networks simultaneously. We establish the identifiability of the proposed model by leveraging properties of bipartite graphs, and develop an efficient alternating updating algorithm to optimize the resulting log-likelihood function. More importantly, we establish the asymptotic consistencies of the proposed model in terms of both probability estimation and community detection. Its advantages are also demonstrated through extensive numerical experiments and the application to a real-world international relationship network.

Paper Structure

This paper contains 13 sections, 6 theorems, 14 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notations.
  3. Signed network and balance structure
  4. Signed block $\beta$-model
  5. Model identifiability
  6. Computing algorithm
  7. Theory
  8. Consistency in parameter estimation
  9. Consistency in community detection
  10. Numerical experiments
  11. Simulated examples
  12. International relation network
  13. Summary

Key Result

Proposition 1

A signed network $\mathcal{G}$ is strong-balanced if and only if (i) For any $(i, j, k)$, $(p_{ij}^+-p_{ij}^-)(p_{jk}^+-p_{jk}^-)(p_{ki}^+-p_{ki}^-)>0$; or (ii) There exists a partition $[n] = C_1 \cup C_2$ such that for each $i,j\in[n]$, $p_{ij}^+ > p_{ij}^-$ if $i,j$ belong to the same subset, and

Figures (3)

  • Figure 1: Configurations of four possible triads.
  • Figure 2: Heatmap of the international relation network.
  • Figure 3: The three estimated communities by SBBM on the world map, where economics in different communities are dyed with different colors.

Theorems & Definitions (8)

  • Definition 1: Strong balance
  • Definition 2: Weak balance
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Theorem 2
  • Theorem 3