Fast Non-mean-field Networks: Uniform in Time Averaging

We study a population of N particles, which evolve according to a diffusion process and interact through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e., with $O(N)$ particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with $O(1)$ particles. We also assume that the network's dynamics is much faster than the particles' dynamics, with the time-scale of the network described by a parameter ε>0. We combine the averaging (ε → 0) and the many-particles (N → ∞) limits and prove that the evolution of the particles' empirical density is described (after taking both limits) by a nonlinear Fokker--Planck equation; moreover, we give conditions under which such limits can be taken uniformly in time, hence providing a criterion under which the limiting nonlinear Fokker--Planck equation is a good approximation of the original system uniformly in time. The heart of our proof consists of controlling precisely the dependence on N of the averaging estimates.