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When social networks polarize: On the number of clusters in the Hegselmann–Krause model

Author

Listed:
  • Wout de Vos

    (Tilburg University - Department of Econometrics and Operations Research - Tilburg University [Netherlands])

  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, Charles University - Department of Applied Mathematics and Institute of Theoretical Computer Science - UK - Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic] = Université Charles [Prague, Republique tchèque], UP1 - Université Paris 1 Panthéon-Sorbonne, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris)

  • Agnieszka Rusinowska

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, UP1 - Université Paris 1 Panthéon-Sorbonne, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris)

Abstract

In the present paper, we study opinion dynamics in a social network, where individuals only listen to those with opinion not farther away than a given threshold from their own opinion (known as bounded confidence models, proposed by Hegselmann and Krause). It is well known that in bounded confidence models consensus does not always exist, and that agents split in clusters (polarization), with convergence to consensus in each cluster. We are precisely interested in the effect of bounded confidence on polarization in the network. Our main focus concerns the formation of clusters and their number, as well as its non-monotonicity with respect to the value of the threshold. First, a framework with a finite number of agents is considered. We study analytically disintegration of various types of opinion chains (clusters), and investigate by simulation the likelihood of chains of a certain length and their disintegration. Next, we examine the (non-)monotonicity of the number of clusters with respect to the threshold for a given initial vector of opinions and in expectation. Finally, we analyse in a formal way the formation of clusters in a model with a continuum of agents.

Suggested Citation

  • Wout de Vos & Michel Grabisch & Agnieszka Rusinowska, 2026. "When social networks polarize: On the number of clusters in the Hegselmann–Krause model," PSE-Ecole d'économie de Paris (Postprint) hal-05611642, HAL.
    • Handle: RePEc:hal:pseptp:hal-05611642
    DOI: 10.1016/j.mathsocsci.2026.102530
    Note: View the original document on HAL open archive server: https://hal.science/hal-05611642v1
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    JEL classification:

    • D85 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Network Formation
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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