$K$-averaging agent-based model: propagation of chaos and convergence to equilibrium

Abstract

The paper treats an agent-based model with averaging dynamics to which we refer as the K-averaging model. Broadly speaking, our model can be added to the growing list of dynamics exhibiting self-organization such as the well-known Vicsek-type models [1, 2, 29]. In the $K$-averaging model, each of the $N$ particles updates their position by averaging over $K$ randomly selected particles with additional noise. To make the $K$-averaging dynamics more tractable, we first establish a propagation of chaos type result in the limit of infinite particle number (i.e. $N \to \infty$) using a martingale technique. Then, we prove the convergence of the limit equation toward a suitable Gaussian distribution in the sense of Wasserstein distance as well as relative entropy. We provide additional numerical simulations to illustrate both results.