Monotonicity in the averaging process
E. Ben-Naim and P.L. Krapivsky
We investigate an averaging process that describes how interacting
agents approach consensus through binary interactions. In each
elementary step, two agents are selected at random and they reach
compromise by adopting their opinion average. We show that the
fraction of agents with a monotonically decreasing opinion decays as
$e^{-\alpha t}$, and that the exponent
$\alpha=\tfrac{1}{2}-\tfrac{1+\ln \ln 2}{4\ln 2}$ is selected as the
extremum from a continuous spectrum of possible values. The opinion
distribution of monotonic agents is asymmetric, and it becomes
self-similar at large times. Furthermore, the tails of the opinion
distribution are algebraic, and they are characterized by two distinct
and nontrivial exponents. We also explore statistical properties of
agents with an opinion strictly above average.
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