Learning Communities from Equilibria of Nonlinear Opinion Dynamics

TL;DR

This paper introduces community detection methods based on equilibria analysis of nonlinear opinion dynamics models, achieving near-perfect recovery under various network configurations and influence conditions.

Contribution

It develops novel algorithms leveraging equilibria and spectral methods for community detection in nonlinear opinion models, including cases with positive influence weights.

Findings

Clustering from a single equilibrium detects most communities with size and link probability differences.

The proposed algorithm detects communities with negative influence weights when communities are identical in size and density.

Multiple equilibria-based detection works for positive influence weights, confirmed by numerical experiments.

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.