On the Convergence of the Hegselmann-Krause System

TL;DR

This paper proves the first polynomial time convergence bound for the Hegselmann-Krause opinion dynamics model in arbitrary dimensions, improving previous bounds and providing new insights into its convergence behavior.

Contribution

It establishes the first polynomial time convergence bound for the system in any dimension and refines bounds for one-dimensional cases.

Findings

First polynomial time convergence bound in arbitrary dimensions

Quadratic lower bound for convergence time

Improved upper bound for one-dimensional systems to O(n^3)

Abstract

We study convergence of the following discrete-time non-linear dynamical system: n agents are located in R^d and at every time step, each moves synchronously to the average location of all agents within a unit distance of it. This popularly studied system was introduced by Krause to model the dynamics of opinion formation and is often referred to as the Hegselmann-Krause model. We prove the first polynomial time bound for the convergence of this system in arbitrary dimensions. This improves on the bound of n^{O(n)} resulting from a more general theorem of Chazelle. Also, we show a quadratic lower bound and improve the upper bound for one-dimensional systems to O(n^3).