A Quadratic Lower Bound for the Convergence Rate in the One-Dimensional Hegselmann–Krause Bounded Confidence Dynamics

Abstract

Let $$f_{k}(n)$$fk(n) be the maximum number of time steps taken to reach equilibrium by a system of $$n$$n agents obeying the $$k$$k-dimensional Hegselmann---Krause bounded confidence dynamics. Previously, it was known that $$\varOmega (n) = f_{1}(n) = O(n^3)$$Ω(n)=f1(n)=O(n3). Here we show that $$f_{1}(n) = \varOmega (n^2)$$f1(n)=Ω(n2), which matches the best-known lower bound in all dimensions $$k \ge 2$$k?2.