Heterogeneous Hegselmann-Krause Dynamics with Environment and Communication Noise

TL;DR

This paper analyzes heterogeneous Hegselmann-Krause opinion dynamics models with environment and communication noise, revealing phase transitions, critical noise levels, and the impact of heterogeneity on synchronization.

Contribution

First analysis of heterogeneous HK models with noise, identifying phase transitions, critical noise thresholds, and effects of heterogeneity on opinion synchronization.

Findings

Phase transition for maximum opinion difference under environment noise

Critical noise amplitude depends on minimal confidence threshold

Heterogeneity impairs synchronization, making consensus harder to achieve

Abstract

The Hegselmann-Krause (HK) model is a wellknown opinion dynamics, attracting a significant amount of interest from a number of fields. However, the heterogeneous HK model is difficult to analyze - even the most basic property of convergence is still open to prove. For the first time, this paper takes into consideration heterogeneous HK models with environment or communication noise. Under environment noise, it has been revealed that the heterogeneous HK model with or without global information has a phase transition for the upper limit of the maximum opinion difference, and has a critical noise amplitude depending on the minimal confidence threshold for quasi-synchronization. In addition, the convergence time to the quasi-synchronization is bounded by a negative exponential distribution. The heterogeneous HK model with global information and communication noise is also analyzed. Finally, for the basic HK model with communication noise, we show that the heterogeneous case exhibits a different behavior regarding quasi-synchronization from the homogenous case. Interestingly, raising the confidence thresholds of constituent agents may break quasi-synchronization. Our results reveal that the heterogeneity of individuals is harmful to synchronization, which may be the reason why the synchronization of opinions is hard to reach in reality, even within that of a small group.