  The classic Hegselmann-Krause (HK) model for opinion dynam- ics consists of a
set of agents on the real line, each one instructed to move, at every time
step, to the mass center of all 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 limiting 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. The ordered phase features
clusters whose width depends only on the noise level. We introduce an order
parameter to track the phase transition and resolve the corresponding phase
dia- gram. 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 also confirms previous simulations and predicts properties
of the noisy HK model in higher dimension.
