Probability of Consensus of Hegselmann-Krause Dynamics

TL;DR

This paper extends the Hegselmann-Krause opinion dynamics model to multi-dimensional spaces and derives bounds for the probability of reaching consensus, providing new theoretical insights into opinion formation.

Contribution

It introduces bounds for consensus probability in multi-dimensional opinion spaces, including improved bounds in one-dimensional cases, advancing understanding of opinion dynamics.

Findings

Positive lower bound for consensus probability in general cases

Lower bound for consensus probability on a unit cube

Improved bounds for one-dimensional consensus probability

Abstract

The original Hegselmann-Krause (HK) model comprises a set of $n$ agents characterized by their opinion, a number in $[0,1]$. Agent $i$ updates its opinion $x_i$ via taking the average opinion of its neighbors whose opinion differs by at most $\epsilon$ from $x_i$. In the article, the opinion space is extended to $\mathbf{R^d}.$ The main result is to derive bounds for the probability of consensus. In general, we have a positive lower bound for the probability of consensus and demonstrate a lower bound for the probability of consensus on a unit cube. In particular for one dimensional case, we derive an upper bound and a better lower bound for the probability of consensus and demonstrate them on a unit interval.