  We consider systems of mean-field interacting diffusions, where the pairwise
interaction structure is described by a sparse (and potentially inhomogeneous)
random graph. Examples include the stochastic Kuramoto model with pairwise
interactions given by an Erd\H{o}s-R\'{e}nyi graph. Our problem is to compare
the bulk behavior of such systems with that of corresponding systems with dense
nonrandom interactions. For a broad class of interaction functions, we find the
optimal sparsity condition that implies that the two systems have the same
hydrodynamic limit, which is given by a McKean-Vlasov diffusion. Moreover, we
also prove matching behavior of the two systems at the level of large
deviations. Our results extend classical results of dai Pra and den Hollander
and provide the first examples of LDPs for systems with sparse random
interactions.

Probability;Disordered Systems and Neural Networks