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

Author(s): Wang, Chu; Li, Qianxiao; E, Weinan; Chazelle, Bernard

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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.
Publication Date: Mar-2017
Electronic Publication Date: 27-Jan-2017
Citation: Wang, Chu, Li, Qianxiao, E, Weinan, Chazelle, Bernard. (2017). Noisy Hegselmann-Krause Systems: Phase Transition and the 2R-Conjecture. JOURNAL OF STATISTICAL PHYSICS, 166 (1209 - 1225. doi:10.1007/s10955-017-1718-x
DOI: doi:10.1007/s10955-017-1718-x
ISSN: 0022-4715
EISSN: 1572-9613
Pages: 1209 - 1225
Type of Material: Journal Article
Version: Author's manuscript

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