The 2R-Conjecture for the Hegselmann--Krause Model: A Proof in Expectation and New Directions Article Swipe
Hegselmann--Krause models are localized, distributed averaging dynamics on spatial data. A key aspect of these dynamics is that they lead to cluster formation, which has important applications in geographic information systems, dynamic clustering algorithms, opinion dynamics, and social networks. For these models, the key questions are whether a fixed point exists and, if so, characterizing it. In this work, we establish new results towards the "2R-Conjecture" for the Hegselmann--Krause model, for which no meaningful progress, or even any precise statement, has been made since its introduction in 2007. This conjecture relates to the structure of the fixed point when there are a large number of agents per unit space. We provide, among other results, a proof in expectation and a statement of a stronger result that is supported by simulation. The key methodological contribution is to consider the dynamics as an infinite-dimensional problem on the space of point processes, rather than on finitely many points. This enables us to leverage stationarity, shift invariance, and certain other symmetries to obtain the results. These techniques do not have finite-dimensional analogs.
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Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2508.08299
- https://arxiv.org/pdf/2508.08299
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W4416243643
All OpenAlex metadata
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https://doi.org/10.48550/arxiv.2508.08299Digital Object Identifier
- Title
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The 2R-Conjecture for the Hegselmann--Krause Model: A Proof in Expectation and New DirectionsWork title
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preprintOpenAlex work type
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enPrimary language
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2025Year of publication
- Publication date
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2025-08-06Full publication date if available
- Authors
-
S. Rasoul EtesamiList of authors in order
- Landing page
-
https://arxiv.org/abs/2508.08299Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2508.08299Direct link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2508.08299Direct OA link when available
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0Total citation count in OpenAlex
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