A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs

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 Erdős-Rényi 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 Erdős-Rényi 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.

arXiv comment16 pages

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• A note on Fokker-Planck equations and graphons
• Network-based kinetic models: Emergence of a statistical description of the graph topology
• Opinion formation on evolving network. The DPA method applied to a nonlocal cross-diffusion PDE-ODE system
• Gamma Convergence Approach For The Large Deviations Of The Density In Systems Of Interacting Diffusion Processes
• Continuum limit for interacting systems on adaptive networks