On the mathematics of the one-dimensional Hegselmann-Krause model

Abstract

This thesis contains three papers, which all in different ways concern the asymptotics of the Hegselmann-Krause model in opinion dynamics. In this model a set of conformist agents repeatedly and synchronously replace their real-valued opinions by the average of those opinions that are within unit distance of their own. The resulting dynamics become exceedingly subtle, and are sensitive to the precise initial configuration.

In Paper I, my supervisor and I investigate the case when the system is initialised with a chain of equidistant opinions. This includes determining the precise evolution for every (possibly infinite) chain with inter-agent distance 1 or approximately 0.81.

In Paper II, I expand the current theory for the evolution of systems where a very large number of opinions are independently drawn at random from the uniform distribution on an interval. I develop a method for approximating the updates of configurations with arbitrarily many agents by updates of smaller ones which can be explicitly computed. I use this method to show rigorously for the first time that for some interval lengths, one asymptotically almost surely reaches a consensus. This gives theoretical support for an observation by Lorenz, that sometimes a group that would not otherwise reach a consensus could do so by individually becoming more closed-minded.

In Paper III, we show that if opinions are taken to be points on a circle with perimeter larger than 2 instead of being real numbers, the resulting sequence of updates must converge pointwise from any possible initial configuration.