Uniform-in-Time Boltzmann Mean-Field Limits for Anchored Binary Opinion Dynamics

Abstract

We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, any fixed number of agents become asymptotically independent, with a common law solving a nonlinear Boltzmann equation. The proof combines a graphical approximation on finite time intervals with exponential convergence of the finite system and its nonlinear limit to their respective stationary laws. We examine two applications. For an anchored variant of the Friedkin-Johnsen model with random coefficients, we give conditions for sub-Gaussian or power-law tails of stationary deviations from the anchors. Occasional overreaction can produce power-law tails with Gaussian anchors and noise, even as the nonlinear law converges exponentially fast to stationarity. We also study an anchored model of biased assimilation, determining when a nearby opinion moves toward or away from neutrality with the anchor and incoming evidence held at neutral values. Simulations illustrate the tail predictions and show how a single peak in the smoothed empirical opinion profile can split into two.