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

Chu Wang, Qianxiao Li, E. Weinan, Bernard Chazelle

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34 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1209-1225
Number of pages17
JournalJournal of Statistical Physics
Issue number5
StatePublished - Mar 1 2017

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Cluster formation
  • Collective behavior
  • Dynamic networks
  • Opinion dynamics
  • Phase transition


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