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 \(\varOmega (n) = f_{1}(n) = O(n^3)\). Here we show that \(f_{1}(n) = \varOmega (n^2)\), which matches the best-known lower bound in all dimensions \(k \ge 2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Other terms used in the literature are “in equilibrium” or “has converged”. We think our term captures the point with the least possible room for misinterpretation, however.
In the general theory of irreducible Markov chains on graphs, dumbbell-like graphs are known to have the longest mixing times. See, for example, [8].
In fact, in the Markov chain literature, this configuration is commonly termed a dumbbell, whereas ours would be referred to as a “dumbbell with a chain in between”. We hope the reader is not confused!
An important fact which makes the one-dimensional model much simpler to analyse is that, as soon as an agent becomes isolated, he will remain so forever. This is not always the case in higher dimensions. As an example in \(\mathbb {R}^2\), consider three agents \(a, b, c\) initially placed at \((0, -0.5), (0, 0.5)\) and \((1, 0)\) respectively. At \(t=0\), only \(a\) and \(b\) will interact, but this first interaction will bring them both to \((0, 0)\) where they are close enough to \(c\) to interact at \(t=1\).
By symmetry, it is clear that all agents will end up in agreement in this case.
References
Bhattacharya, A., Braverman, M., Chazelle, B., Nguyen, H.L.: On the convergence of the Hegselmann–Krause system. In: Proceedings of the 4th Innovations in Theoretical Computer Science Conference (ICTS 2013), Berkeley, CA (2013)
Chazelle, B.: The total \(s\)-energy of a multiagent system. SIAM J. Control Optim. 49(4), 1680–1706 (2011)
Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv. Complex. Syst. 3, 87–98 (2000)
Hegarty, P., Wedin, E.: The Hegselmann–Krause dynamics for equally spaced agents. Preprint at http://arxiv.org/abs/1406.0819
Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence: models, analysis and simulations. J. Artif. Soc. Soc. Simul. 5(3), (2002). http://jasss.soc.surrey.ac.uk/5/3/2/2
Krause, U.: Soziale Dynamiken mit vielen Interakteuren, eine Problemskizze. In: Krause, U., Stöckler, M. (eds.) Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren. Universität Bremen, Bremen (1997)
Kurz, S.: How long does it take to consensus in the Hegselmann–Krause model? Preprint at http://arxiv.org/abs/1405.5757
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times (with a chapter by James G. Propp and David B. Wilson), pp. xviii + 371. American Mathematical Society, Providence, RI (2009)
Martinez, S., Bullo, F., Cortes, J., Frazzoli, E.: On synchronous robotic networks—part II: time complexity of rendezvous and deployment algorithms. IEEE Trans. Automat. Contr. 52(12), 2214–2226 (2007)
Wedin, E., Hegarty, P.: The Hegselmann–Krause dynamics for continuous agents and a regular opinion function do not always lead to consensus. To appear in IEEE Trans. Automat. Control. Preprint at http://arxiv.org/abs/1402.7184
Acknowledgments
We thank Sascha Kurz and Anders Martinsson for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wedin, E., Hegarty, P. A Quadratic Lower Bound for the Convergence Rate in the One-Dimensional Hegselmann–Krause Bounded Confidence Dynamics. Discrete Comput Geom 53, 478–486 (2015). https://doi.org/10.1007/s00454-014-9657-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-014-9657-7