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

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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 analyze in a formal way the formation of clusters in a model with a continuum of agents. Finally, 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

Suggested Citation

  • Wout De Vos & Michel Grabisch & Agnieszka Rusinowska, 2025. "When social networks polarize: On the number of clusters in the Hegselmann-Krause model," Documents de travail du Centre d'Economie de la Sorbonne 25018, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:25018
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    1. Bolletta, Ugo & Pin, Paolo, 2025. "Dynamic opinion updating with endogenous networks," European Economic Review, Elsevier, vol. 176(C).
    2. Michel Grabisch & Agnieszka Rusinowska, 2020. "A Survey on Nonstrategic Models of Opinion Dynamics," Games, MDPI, vol. 11(4), pages 1-29, December.
    3. Michel Grabisch & Antoine Mandel & Agnieszka Rusinowska, 2023. "On the Design of Public Debate in Social Networks," Operations Research, INFORMS, vol. 71(2), pages 626-648, March.
    4. Rainer Hegselmann & Ulrich Krause, 2002. "Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 5(3), pages 1-2.
    5. Christos Mavridis & Nikolas Tsakas, 2021. "Social Capital, Communication Channels and Opinion Formation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 56(4), pages 635-678, May.
    6. Garance Genicot, 2022. "Tolerance and Compromise in Social Networks," Journal of Political Economy, University of Chicago Press, vol. 130(1), pages 94-120.
    7. Rainer Hegselmann & Ulrich Krause, 2006. "Truth and Cognitive Division of Labour: First Steps Towards a Computer Aided Social Epistemology," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 9(3), pages 1-10.
    8. Antonio Parravano & Ascensión Andina-Díaz & Miguel A Meléndez-Jiménez, 2016. "Bounded Confidence under Preferential Flip: A Coupled Dynamics of Structural Balance and Opinions," PLOS ONE, Public Library of Science, vol. 11(10), pages 1-23, October.
    9. Lorenz, Jan, 2005. "A stabilization theorem for dynamics of continuous opinions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(1), pages 217-223.
    10. Jan Lorenz, 2006. "Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 9(1), pages 1-8.
    11. Azzimonti, Marina & Fernandes, Marcos, 2023. "Social media networks, fake news, and polarization," European Journal of Political Economy, Elsevier, vol. 76(C).
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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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