The Hegselmann-Krause dynamics for equally spaced agents

Journal article, 2016

We consider the Hegselmann–Krause bounded confidence dynamics for n equally spaced opinions in R. We completely determine the evolution when the initial separation d equals the confidence bound r=1. Every fifth time step, three agents disconnect at either end before collapsing to a cluster. This continues until fewer than 6 agents remain in the middle, and these finally collapse to a cluster, if n is not a multiple of 6. The configuration thus freezes in time 5n6+O(1). We show that for values d≈0.81, the evolution is similarly periodic but with a freezing time of n+O(1), and conjecture that this is maximal for equidistant configurations. Finally, we consider the dynamics for arbitrary spacings d \in [0,1]. Based on a mix of rigorous analysis and simulations, we propose hypotheses concerning the regularity of the evolution for arbitrary d, and a limiting behaviour as d→0.