Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture

TL;DR

This paper studies the noisy Hegselmann-Krause opinion dynamics model, revealing phase transitions between order and disorder, and provides a theoretical explanation for the 2R conjecture, supported by stability analysis and simulations.

Contribution

It introduces a phase diagram for the noisy HK model, analyzes phase transitions via mean-field stability, and explains the 2R conjecture theoretically.

Findings

System exhibits phase transition from order to disorder with increasing noise.

Ordered phase clusters have widths depending only on noise level.

Theoretical analysis confirms the 2R conjecture and predicts higher-dimensional properties.

Abstract

The classic Hegselmann-Krause (HK) model for opinion dynam- ics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of all the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field limiting Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. The ordered phase features clusters whose width depends only on the noise level. We introduce an order parameter to track the phase transition and resolve the corresponding phase dia- gram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis also confirms previous simulations and predicts properties of the noisy HK model in higher dimension.