How long does it take to consensus in the Hegselmann-Krause model?

TL;DR

This paper determines the exact maximum time for consensus or its impossibility in the Hegselmann-Krause opinion dynamics model for small groups, using integer linear programming to close the gap between known bounds.

Contribution

It provides exact values for the worst-case convergence time in the Hegselmann-Krause model for small agent groups, a novel computational approach.

Findings

Exact maximum convergence times for small groups

Integer linear programming effectively computes these bounds

Closes the gap between known lower and upper bounds

Abstract

Hegselmann and Krause introduced a discrete-time model of opinion dynamics with agents having limit confidence. It is well known that the dynamics reaches a stable state in a polynomial number of time steps. However, the gap between the known lower and upper bounds for the worst case is still immense. In this paper exact values for the maximum time, needed to reach consensus or to discover that consensus is impossible, are determined for small number of agents using an integer linear programming approach.