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Noisy Hegselmann-Krause systems: Phase transition and the 2R-conjecture

Author(s): Wang, Chu; Li, Qianxiao; Weinan, E; 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: 2016
Electronic Publication Date: 27-Jan-2017
Citation: Wang, C, Li, Q, Weinan, E, Chazelle, B. (2016). Noisy Hegselmann-Krause systems: Phase transition and the 2R-conjecture. 2632 - 2637. doi:10.1109/CDC.2016.7798659
DOI: doi:10.1109/CDC.2016.7798659
Pages: 2632 - 2637
Type of Material: Journal Article
Journal/Proceeding Title: Journal of Statistical Physics
Version: Author's manuscript

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