Large deviations for marked sparse random graphs with applications to interacting diffusions

TL;DR

This paper establishes a large deviation principle for the empirical neighborhood distribution of marked sparse Erd"os-Rényi graphs, with applications to interacting diffusions like the stochastic Kuramoto model, extending previous work to Polish space marks.

Contribution

It introduces an approximation method to extend large deviation principles from discrete to Polish space marks in sparse random graphs, enabling new applications.

Findings

Proves a large deviation principle for marked sparse Erd"os-Rényi graphs.

Extends results to Polish space marks via approximation techniques.

Applies findings to interacting diffusions, including the stochastic Kuramoto model.

Abstract

We consider the empirical neighborhood distribution of marked sparse Erd\"os-R\'enyi random graphs, obtained by decorating edges and vertices of a sparse Erd\"os-R\'enyi random graph with i.i.d.\ random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam~(2019) which is inspired on the previous work by Bordenave and Caputo~(2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erd\"os-R\'enyi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.