A Law of Large Numbers and Large Deviations for interacting diffusions on Erd\H{o}s-R\'enyi graphs

TL;DR

This paper establishes Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs, relaxing initial data assumptions and extending mean field convergence results to more general network structures.

Contribution

It proves LLN and LDP for particle systems on Erdős-Rényi graphs with minimal initial data assumptions, broadening the scope of mean field limit results.

Findings

LLN holds under weak initial empirical measure convergence.

LDP characterizes the probability of deviations from the mean field limit.

Results extend to sparse Erdős-Rényi graphs with appropriate scaling.

Abstract

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter $p_n\in (0, 1]$, where $n$ is the size of the graph (i.e., the number of particles). If $p_n\equiv 1$ the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as $n\to \infty$ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erd\H{o}s-R\'enyi graphs with $\lim_n p_n n =\infty$, and properly rescaling the interaction to account for the dilution introduced by $p_n$. However, these results have been proven under strong assumptions on that initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results -- Law of Large Numbers and Large Deviation Principle -- assuming only the convergence of the empirical measure of the initial condition.