Homogeneous Boltzmann-type equations on dense graphs

Abstract

In kinetic theory, interactions between particles are typically assumed to be "all-to-all", meaning that any pair of randomly selected particles may, in principle, interact. This assumption originates from the theory of colliding gas molecules; however, it may be less appropriate for describing other forms of interaction, such as social interactions. These are more naturally characterised as "some-to-some", reflecting the existence of preferential connections between agents. In this paper, we consider homogeneous Boltzmann-type equations on finite graphs that model such networks of preferential interactions, and we rigorously derive their dense graph limit as the number of agents tends to infinity. We also investigate the long-time behaviour of the limiting equation in the case of linear pairwise interactions, characterising the emergent equilibrium distributions and relating them to their counterparts in the classical "all-to-all" setting.

Keywords: graphons, cut distance, Fourier and Wasserstein metrics, dense graph limit, asymptotic distributions