The Hegselmann-Krause dynamics for equally spaced agents

TL;DR

This paper rigorously analyzes the evolution of equally spaced opinions under Hegselmann-Krause dynamics, revealing predictable clustering patterns and periodic behaviors depending on agent spacing and size.

Contribution

It provides the first rigorous proof of the evolution pattern for equally spaced agents and explores the effects of inter-agent spacing on dynamics and convergence time.

Findings

Agents form clusters every 5 steps, with a predictable pattern.

The final configuration depends on the number of agents modulo 6.

Time to convergence varies with inter-agent spacing, sometimes slower or faster.

Abstract

We consider the Hegselmann-Krause bounded confidence dynamics for n equally spaced opinions on the real line, with gaps equal to the confidence bound r, which we take to be 1. We prove rigorous results on the evolution of this configuration, which confirm hypotheses previously made based on simulations for small values of n. Namely, for every n, the system evolves as follows: after every 5 time steps, a group of 3 agents become disconnected at either end and collapse to a cluster at the subsequent step. This continues until there are fewer than 6 agents left in the middle, and these finally collapse to a cluster, if n is not a multiple of 6. In particular, the final configuration consists of 2*[n/6] clusters of size 3, plus one cluster in the middle of size n (mod 6), if n is not a multiple of 6, and the number of time steps before freezing is 5n/6 + O(1). We also consider the dynamics for arbitrary, but constant, inter-agent spacings d \in [0, 1] and present three main findings. Firstly we prove that the evolution is periodic also at some other, but not all, values of d, and present numerical evidence that for all d something "close" to periodicity nevertheless holds. Secondly, we exhibit a value of d at which the behaviour is periodic and the time to freezing is n + O(1), hence slower than that for d = 1. Thirdly, we present numerical evidence that, as d --> 0, the time to freezing may be closer, in order of magnitude, to the diameter d(n-1) of the configuration rather than the number of agents n.