# Thermodynamic Limit of Interacting Particle Systems over Time-varying Sparse Random Networks

## TL;DR

This paper proves that the macroscopic behavior of large interacting particle systems on fast time-varying sparse networks converges to a deterministic differential equation, despite the microscopic processes being non-Markovian.

## Contribution

It establishes a weak law of large numbers for particle systems on dynamic sparse networks and introduces techniques for proving convergence of non-Markovian macrostate processes.

## Key findings

- Proportion of agents at each state converges to an ODE solution as N grows large.
- Prelimit processes are non-Markovian due to microscopic state dependence.
- Techniques for weak convergence of high-dimensional, non-Markovian processes are developed.

## Abstract

We establish a functional weak law of large numbers for observable macroscopic state variables of interacting particle systems (e.g., voter and contact processes) over fast time-varying sparse random networks of interactions. We show that, as the number of agents $N$ grows large, the proportion of agents $\left(\overline{Y}_{k}^{N}(t)\right)$ at a certain state $k$ converges in distribution -- or, more precisely, weakly with respect to the uniform topology on the space of \emph{c\`adl\`ag} sample paths -- to the solution of an ordinary differential equation over any compact interval $\left[0,T\right]$. Although the limiting process is Markov, the prelimit processes, i.e., the normalized macrostate vector processes $\left(\mathbf{\overline{Y}}^{N}(t)\right)=\left(\overline{Y}_{1}^{N}(t),\ldots,\overline{Y}_{K}^{N}(t)\right)$, are non-Markov as they are tied to the \emph{high-dimensional} microscopic state of the system, which precludes the direct application of standard arguments for establishing weak convergence. The techniques developed in the paper for establishing weak convergence might be of independent interest.