2025/08/21 by Molignini, Paolo
#Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Physics (math-ph) #Physics and Society (physics.soc-ph)
paper · doi:10.48550/arxiv.2508.15982
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 ε-switches are proved. The first conjecture states that the number of ε-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 ε-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 ε-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.