A quadratic lower bound for the convergence rate in the one-dimensional Hegselmann-Krause bounded confidence dynamics

TL;DR

This paper establishes a quadratic lower bound on the convergence time for one-dimensional Hegselmann-Krause bounded confidence dynamics, matching the known bounds in higher dimensions and advancing understanding of the system's efficiency.

Contribution

The paper proves a quadratic lower bound for the convergence time in 1D, aligning it with bounds previously known for higher dimensions, thus closing a gap in the theoretical understanding.

Findings

f_1(n) = (n^2) lower bound established

Convergence time in 1D matches higher-dimensional bounds

Improves theoretical understanding of bounded confidence dynamics

Abstract

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