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

TL;DR

This paper investigates how systems of interacting diffusions on sparse random graphs compare to dense systems, establishing conditions under which they share the same hydrodynamic limit and large deviation principles.

Contribution

It identifies the optimal sparsity conditions for equivalence in hydrodynamic limits and large deviations between sparse and dense interaction systems.

Findings

Hydrodynamic limit matches for certain sparsity levels

Large deviation principles are established for sparse systems

First LDP results for systems with sparse random interactions

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 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.