  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
$\epsilon>0$. We combine the averaging ($\epsilon \rightarrow 0$) and the many
particles ($N \rightarrow \infty$) limits and prove that the evolution of the
particles' empirical density is described (after taking both limits) by a
non-linear Fokker-Planck equation; we moreover give conditions under which such
limits can be taken uniformly in time, hence providing a criterion under which
the limiting non-linear 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 in $N$ of the averaging estimates.
