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Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture

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Abstract

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.

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Acknowledgements

We wish to thank the anonymous referees for many useful comments and suggestions. Bernard Chazelle’s work is supported in part by NSF Grants CCF-0832797, CCF-0963825, and CCF-1016250; Weinan E’s work is supported by DOE grant DE-SC0009248 and ONR grant N00014-13-1-0338; Qianxiao Li’s work is supported by Agency for Science, Technology and Research, Singapore.

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Authors and Affiliations

  1. The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08540, USA

    Chu Wang & Qianxiao Li

  2. Nokia Bell Labs, 600 Mountain Avenue, Murray Hill, NJ, 07974, USA

    Chu Wang

  3. Department of Mathematics and the Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08540, USA

    Weinan E

  4. Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

    Bernard Chazelle

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  1. Chu Wang
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  2. Qianxiao Li
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  3. Weinan E
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  4. Bernard Chazelle
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Correspondence to Chu Wang.

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Wang, C., Li, Q., E, W. et al. Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture. J Stat Phys 166, 1209–1225 (2017). https://doi.org/10.1007/s10955-017-1718-x

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  • DOI: https://doi.org/10.1007/s10955-017-1718-x

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