Summary: 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 Erdős-Rényi 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 Erdős-Rényi 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 the 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.