The aim of the talk it to consider a system of N diffusions interacting on a (possibly random) graph. An easy instance corresponds to the case of full-connectivity, that is when the graph of interaction is complete (mean-field case). In this case, the empirical measure of the system converges as $N\to\infty$ to the solution of a nonlinear Fokker-Planck equation. The question is then: what happens if one considers an nontrivial graph of interaction, that is no longer complete, possibly (very) diluted ? the coupling is no longer a functional of the empirical measure, but rather of local empirical measures around each vertex. What is the proper condition on the graph of interaction under which the behavior of the system remains as in the mean-field case? We address this question of universality both at the level of the law of large numbers and fluctuations, for a large class of possibly random graphs, including the Erdös-Rényi class. We will show in particular that the dependence of the initial condition w.r.t. the graph is crucial. This is based on joint works with S. Delattre, G. Giacomin, F. Coppini and C. Poquet.