The Hegselmann-Krause dynamics on the circle converge

Abstract

We consider the Hegselmann-Krause dynamics on a one-dimensional torus and provide the first proof of convergence of this system. The proof requires only fairly minor modifications of existing methods for proving convergence in Euclidean space.

Keywords:

• Hegselmann-Krause model
• circle
• convergence
• directed influence graph

AMS Subject Classifications:

• 39A60
• 93A14
• 91D10

Acknowledgements

We thank Bernadette Charron-Bost, Matthias Függer and Thomas Nowak for pointing out an error in an earlier version of the manuscript, in which we incorrectly stated that the matrix A in (19) was regular. Remark 2.3 presents the corrected assertions

Notes

No potential conflict of interest was reported by the authors.

1 Note that it is crucial that the directed influence graph be fixed for the same to be true of the transition matrix, it would not suffice in general for the underlying graph to be fixed.

2 Indeed, there is a solution for any $n\ge 12$ except if $n\in \{13,14,15,16,18,20,21,23,25,28,30,35\}$.

3 By a cluster we mean a set of agents in agreement.