  We consider a class of particle systems described by differential equations
(both stochastic and deterministic), in which the interaction network is
determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter
$p_n\in (0, 1]$, where $n$ is the size of the graph (i.e., the number of
particles). If $p_n\equiv 1$ the graph is the complete graph (mean field model)
and it is well known that, under suitable hypotheses, the empirical measure
converges as $n\to \infty$ to the solution of a PDE: a McKean-Vlasov (or
Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the
deterministic one. It has already been shown that this holds for rather general
interaction networks, that include Erd\H{o}s-R\'enyi graphs with $\lim_n p_n n
=\infty$, and properly rescaling the interaction to account for the dilution
introduced by $p_n$. However, these results have been proven under strong
assumptions on that initial datum which has to be chaotic, i.e. a sequence of
independent identically distributed random variables. The aim of our
contribution is to present results -- Law of Large Numbers and Large Deviation
Principle -- assuming only the convergence of the empirical measure of the
initial condition.
