A periodic high-density limit for bounded-confidence opinion clusters
Let the points of a homogeneous Poisson point process of intensity $\lambda > 0$ on $\mathbb{R}$ be the opinions of agents, and run the Hegselmann–Krause dynamics with confidence radius one: in synchronous discrete time, every opinion is replaced by the arithmetic mean of all current opinions at distance at most one from it, including itself, agents counted with multiplicity. Almost surely the initial gaps longer than one split the process into finite isolated groups, so every agent has a final opinion. Let $C_\lambda$ be the simple counting measure on the distinct final opinions, with one atom per cluster.
Blondel, Hendrickx and Tsitsiklis observed in simulations that dense agents end in clusters roughly two confidence radii apart, and noted that clusters must be separated by at least one radius. Dey, Etesami and Gopalan proved that for this model the expected gap between consecutive clusters tends to $2$ as $\lambda \to \infty$.
Problem. As $\lambda \to \infty$, does $C_\lambda$ converge in distribution, in the vague topology on locally finite measures, to $\sum_{j \in \mathbb{Z}} \delta_{U + 2j}$, where $U$ is uniform on $[0, 2)$?
Source: Blondel, Hendrickx and Tsitsiklis (2007), the 2R conjecture, from simulations; in this form, Dey, Etesami and Gopalan (2025), Conjecture 2 (strong 2R conjecture), part (1). Stated by AIM — Open Applied Problems for Poisson initial opinions, with convergence in distribution in the vague topology.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, Hegselmann–Krause model, opinion dynamics, Poisson point processes
References
@inproceedings{blondel2007,
author = {Blondel, Vincent D. and Hendrickx, Julien M. and Tsitsiklis, John N.},
title = {On the 2{R} conjecture for multi-agent systems},
booktitle = {2007 European Control Conference (ECC)},
publisher = {IEEE},
year = {2007},
pages = {874--881},
doi = {10.23919/ECC.2007.7068885},
}
@misc{dey2025,
author = {Dey, Partha S. and Etesami, S. Rasoul and Gopalan, Aditya S.},
title = {The 2{R}-Conjecture for the {H}egselmann--{K}rause Model: A Proof in Expectation and New Directions},
year = {2025},
eprint = {2508.08299},
archivePrefix = {arXiv},
}
@article{bernardo2024,
author = {Bernardo, Carmela and Altafini, Claudio and Proskurnikov, Anton and Vasca, Francesco},
title = {Bounded confidence opinion dynamics: A survey},
journal = {Automatica},
volume = {159},
year = {2024},
pages = {111302},
doi = {10.1016/j.automatica.2023.111302},
}
@misc{aim-hegselmann-krause-cluster-lattice,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A periodic high-density limit for bounded-confidence opinion clusters}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/284-hegselmann-krause-cluster-lattice.md},
note = {Entry 284 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00440, "A periodic high-density limit for bounded-confidence opinion clusters", clopen.solutions.