The Hegselmann-Krause dynamics for equally spaced agents. Issue 11 (1st November 2016) - Hegarty, Peter; Wedin, Edvin
Abstract : We consider the Hegselmann–Krause bounded confidence dynamics for n equally spaced opinions inR . We completely determine the evolution when the initial separation d equals the confidence boundr = 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 time5 n 6 + O ( 1 ) . We show that for valuesd ≈ 0.81, the evolution is similarly periodic but with a freezing time ofn + O ( 1 ), and conjecture that this is maximal for equidistant configurations. Finally, we consider the dynamics for arbitrary spacingsd ∈ [ 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 asd → 0 .