Scaling limits for continuous opinion dynamics systems

A class of large-scale stochastic discrete-time continuous-opinion dynamical systems is analyzed. Agents have pairwise random interactions in which their vector-valued opinions are updated to a weighted average of their current values. The intensity of the interactions is allowed to depend on the agents´ opinions themselves through an interaction kernel. This class of models includes as a special case the bounded-confidence opinion dynamics models recently introduced by Deffuant et al., in which agents interact only when their opinions differ by less than a given threshold, as well as more general interaction kernels. It is shown that, in the limit as the population size increases, upon a proper rescaling of the time index, the trajectories of such stochastic processes concentrate, at an exponential rate, around the solution of a measure-valued differential equation. The asymptotic properties of the solution of such a differential equation are then studied, and convergence is proven to a convex combination of delta measures whose number depends on the interaction kernel.