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Learning Communities from Equilibria of Nonlinear Opinion Dynamics

Yu Xing, Anastasia Bizyaeva, Karl H. Johansson

TL;DR

The study addresses detecting communities from equilibria of a nonlinear opinion dynamics model on stochastic block models. It develops two algorithms: a single-equilibrium method using k-means with theoretical recovery guarantees under specific SBM conditions, and a multiple-equilibria method that reconstructs the adjacency via fixed-point relations and applies spectral clustering. The results show that nonlinear saturation and the sign of the influence weights critically affect detectability, with almost exact recovery achievable in diverse regimes, especially for disassortative networks with negative influence when multiple equilibria are used. Numerically, the proposed methods outperform traditional dynamical clustering approaches and demonstrate how nonlinearity can be leveraged for structure discovery in networks. The findings advance understanding of how equilibrium properties in nonlinear dynamics reveal underlying community structure and suggest directions for extending to more complex, labeled, or jointly learned models.

Abstract

This paper studies community detection for a nonlinear opinion dynamics model from its equilibria. It is assumed that the underlying network is generated from a stochastic block model with two communities, where agents are assigned with community labels and edges are added independently based on these labels. Agents update their opinions following a nonlinear rule that incorporates saturation effects on interactions. It is shown that clustering based on a single equilibrium can detect most community labels (i.e., achieving almost exact recovery), if the two communities differ in size and link probabilities. When the two communities are identical in size and link probabilities, and the inter-community connections are denser than intra-community ones, the algorithm can achieve almost exact recovery under negative influence weights but fails under positive influence weights. Utilizing fixed point equations and spectral methods, we also propose a detection algorithm based on multiple equilibria, which can detect communities with positive influence weights. Numerical experiments demonstrate the performance of the proposed algorithms.

Learning Communities from Equilibria of Nonlinear Opinion Dynamics

TL;DR

The study addresses detecting communities from equilibria of a nonlinear opinion dynamics model on stochastic block models. It develops two algorithms: a single-equilibrium method using k-means with theoretical recovery guarantees under specific SBM conditions, and a multiple-equilibria method that reconstructs the adjacency via fixed-point relations and applies spectral clustering. The results show that nonlinear saturation and the sign of the influence weights critically affect detectability, with almost exact recovery achievable in diverse regimes, especially for disassortative networks with negative influence when multiple equilibria are used. Numerically, the proposed methods outperform traditional dynamical clustering approaches and demonstrate how nonlinearity can be leveraged for structure discovery in networks. The findings advance understanding of how equilibrium properties in nonlinear dynamics reveal underlying community structure and suggest directions for extending to more complex, labeled, or jointly learned models.

Abstract

This paper studies community detection for a nonlinear opinion dynamics model from its equilibria. It is assumed that the underlying network is generated from a stochastic block model with two communities, where agents are assigned with community labels and edges are added independently based on these labels. Agents update their opinions following a nonlinear rule that incorporates saturation effects on interactions. It is shown that clustering based on a single equilibrium can detect most community labels (i.e., achieving almost exact recovery), if the two communities differ in size and link probabilities. When the two communities are identical in size and link probabilities, and the inter-community connections are denser than intra-community ones, the algorithm can achieve almost exact recovery under negative influence weights but fails under positive influence weights. Utilizing fixed point equations and spectral methods, we also propose a detection algorithm based on multiple equilibria, which can detect communities with positive influence weights. Numerical experiments demonstrate the performance of the proposed algorithms.
Paper Structure (17 sections, 5 theorems, 24 equations, 3 figures, 2 algorithms)

This paper contains 17 sections, 5 theorems, 24 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Outline
  5. Preliminaries
  6. Nonlinear Opinion Dynamics
  7. Stochastic Block Model
  8. Problem Formulation
  9. Detection Algorithms and Main Results
  10. Detection from Single Equilibrium
  11. Detection from Multiple Equilibria
  12. Numerical Experiments
  13. Conclusion
  14. Proof of Theorem \ref{['thm_main']}
  15. Proof of (i.a)
  16. ...and 2 more sections

Key Result

Proposition 1

Suppose that $\mathcal{G}$ is connected, $u\ge 0$, and $\mathbf{b} = \mathbf{0}$.

Figures (3)

  • Figure 1: Detection accuracy for an SBM with unequal-sized communities and $\gamma > 0$.
  • Figure 2: Detection accuracy for the SSBM.
  • Figure 3: Detection accuracy from multiple input-equilibrium pairs.

Theorems & Definitions (9)

  • Proposition 1
  • Definition 1: SBM
  • Definition 2
  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3