Mixed Hegselmann-Krause Dynamics--Nondeterministic Case

TL;DR

This paper investigates a nondeterministic variant of the mixed Hegselmann-Krause model, analyzing its stability and contrasting its properties with synchronous and asynchronous models in opinion dynamics.

Contribution

It extends previous deterministic analysis by studying the nondeterministic mixed HK model, identifying conditions for asymptotic stability and differences from synchronous and asynchronous models.

Findings

Finite-time convergence does not hold for the mixed model.

Certain properties of the asynchronous model are not valid for the mixed model.

Conditions for asymptotic stability are established.

Abstract

The original Hegselmann-Krause (HK) model is composed of a finite number of agents characterized by their opinion, a number in $[0,1]$. An agent updates its opinion via taking the average opinion of its neighbors whose opinion differs by at most $\epsilon$ for $\epsilon>0$ a confidence bound. An agent is absolutely stubborn if it does not change its opinion while update, and absolutely open-minded if its update is the average opinion of its neighbors. There are two types of HK models--the synchronous HK model and the asynchronous HK model. The paper is about a variant of the HK dynamics, called the mixed model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at all times. The mixed model reduces to the synchronous HK model if all agents are absolutely open-minded all the time, and the asynchronous HK model if only one uniformly randomly selected agent is absolutely open-minded and the others are absolutely stubborn at all times. In \cite{mhk}, we discuss the mixed model deterministically. Point out some properties of the synchronous HK model, such as finite-time convergence, do not hold for the mixed model. In this topic, we study the mixed model nondeterministically. List some properties of the asynchronous model which do not hold for the mixed model. Then, study circumstances under which the asymptotic stability holds.