Noisy Hegselmann-Krause Systems: Phase Transition and the 2 R -Conjecture
conference on decision and control(2017)
摘要
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 2 R 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 2 R . Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.
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关键词
Collective behavior,Opinion dynamics,Cluster formation,Phase transition,Dynamic networks
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