IDEAS home Printed from https://ideas.repec.org/p/mse/cesdoc/25018.html

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

Author

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
    DOI: 10.1016/j.mathsocsci.2026.102530
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-05274032
    Download Restriction: no
    File URL: https://doi.org/10.1016/j.mathsocsci.2026.102530
    Download Restriction: no
    File URL: https://libkey.io/10.1016/j.mathsocsci.2026.102530?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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. 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.
    5. Garance Genicot, 2022. "Tolerance and Compromise in Social Networks," Journal of Political Economy, University of Chicago Press, vol. 130(1), pages 94-120.
    6. 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.
    7. Azzimonti, Marina & Fernandes, Marcos, 2023. "Social media networks, fake news, and polarization," European Journal of Political Economy, Elsevier, vol. 76(C).
    8. 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.
    9. Pan, Zhengzheng, 2010. "Trust, influence, and convergence of behavior in social networks," Mathematical Social Sciences, Elsevier, vol. 60(1), pages 69-78, July.
    10. 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.
    11. 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.
    12. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rainer Hegselmann & Stefan König & Sascha Kurz & Christoph Niemann & Jörg Rambau, 2015. "Optimal Opinion Control: The Campaign Problem," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 18(3), pages 1-18.
    2. Buechel, Berno & Hellmann, Tim & Klößner, Stefan, 2015. "Opinion dynamics and wisdom under conformity," Journal of Economic Dynamics and Control, Elsevier, vol. 52(C), pages 240-257.
    3. Liu, Qipeng & Wang, Xiaofan, 2013. "Social learning with bounded confidence and heterogeneous agents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2368-2374.
    4. Marco D'Errico & Gulnur Muradoglu & Silvana Stefani & Giovanni Zambruno, 2014. "Opinion Dynamics and Price Formation: a Nonlinear Network Model," Papers 1408.0308, arXiv.org.
    5. Kevin JS. Zollman, 2012. "Social network structure and the achievement of consensus," Politics, Philosophy & Economics, , vol. 11(1), pages 26-44, February.
    6. Börgers, Christoph & Dragovic, Natasa & Kirshtein, Arkadz, 2026. "Candidate voter dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 682(C).
    7. Buechel, Berno & Klößner, Stefan & Meng, Fanyuan & Nassar, Anis, 2023. "Misinformation due to asymmetric information sharing," Journal of Economic Dynamics and Control, Elsevier, vol. 150(C).
    8. Zhengzheng Pan, 2012. "Opinions and Networks: How Do They Effect Each Other," Computational Economics, Springer;Society for Computational Economics, vol. 39(2), pages 157-171, February.
    9. Michel Grabisch & Agnieszka Rusinowska, 2020. "A Survey on Nonstrategic Models of Opinion Dynamics," Games, MDPI, vol. 11(4), pages 1-29, December.
    10. Michel Grabisch & Antoine Mandel & Agnieszka Rusinowska & Emily Tanimura, 2015. "Strategic influence in social networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01158168, HAL.
    11. Benjamin Cabrera & Björn Ross & Daniel Röchert & Felix Brünker & Stefan Stieglitz, 2021. "The influence of community structure on opinion expression: an agent-based model," Journal of Business Economics, Springer, vol. 91(9), pages 1331-1355, November.
    12. 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.
    13. Phillip Monin & Richard Bookstaber, 2017. "Information Flows, the Accuracy of Opinions, and Crashes in a Dynamic Network," Staff Discussion Papers 17-01, Office of Financial Research, US Department of the Treasury.
    14. Hou, Jian & Li, Wenshan & Jiang, Mingyue, 2021. "Opinion dynamics in modified expressed and private model with bounded confidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 574(C).
    15. Zhang, Jiangbo & Hong, Yiguang, 2013. "Opinion evolution analysis for short-range and long-range Deffuant–Weisbuch models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(21), pages 5289-5297.
    16. Michael Hatcher & Tim Hellmann, 2024. "Communication, networks and asset price dynamics: a survey," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 19(1), pages 1-58, January.
    17. Fernandes, Marcos R., 2023. "Confirmation bias in social networks," Mathematical Social Sciences, Elsevier, vol. 123(C), pages 59-76.
    18. M. T. Agieva & A. V. Korolev & G. A. Ougolnitsky, 2020. "Modeling and Simulation of Impact and Control in Social Networks with Application to Marketing," Mathematics, MDPI, vol. 8(9), pages 1-28, September.
    19. 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.
    20. Zafer Kanik & Zaruhi Hakobyan, 2026. "Strategic Expression, Popularity Traps, and Welfare in Social Media," Papers 2601.01370, arXiv.org, revised Mar 2026.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:mse:cesdoc:25018. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Lucie Label (email available below). General contact details of provider: https://edirc.repec.org/data/cenp1fr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.