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

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.