Universality and Invariance in Hegselmann-Krause Opinion Dynamics: Proof of Three Conjectures

TL;DR

This paper proves three conjectures related to the Hegselmann-Krause opinion dynamics, establishing finiteness, structural relationships, and invariance properties of opinion switches, thus providing a formal foundation for empirical observations.

Contribution

It offers the first formal proofs of three key conjectures about opinion dynamics, enhancing understanding of their structural and invariance properties.

Findings

Number of epsilon-switches is always finite.

Opinion evolution is identical up to switch time for consecutive switches.

Dynamics are invariant under positive-affine transformations of initial opinions.

Abstract

Three conjectures from [R. Hegselmann, The Journal of Artificial Societies and Social Simulations 26(4), 11 (2023)] about the Hegselmann-Krause opinion dynamics and the structure of $\epsilon$-switches are proved. The first conjecture states that the number of $\epsilon$-switches for any given initial opinion distribution is always finite, guaranteeing that the algorithm for enumerating them terminates. The second conjecture concerns the relationship between the dynamics of two consecutive $\epsilon$-switches, showing that the opinion evolution is identical up to the switch time. The third conjecture establishes the invariance of the dynamics under positive-affine transformations of the initial distribution, with a corresponding rescaling of all $\epsilon$-switch values. Together, these results provide a formal foundation for the empirical observations reported in the literature and offer a step towards a systematic classification of BC-processes based on their initial conditions.