Interacting diffusions on random graphs with diverging degrees: hydrodynamics and large deviations

Abstract

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 Erdős-Rényi 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.

arXiv comment37 pages

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• Large deviations for marked sparse random graphs with applications to interacting diffusions
• Stochastic optimization on matrices and a graphon McKean-Vlasov limit
• Marked random graphs with given degree sequence: large deviations on the local topology