Title: Thermodynamic Limit of Interacting Particle Systems over Time-varying Sparse Random Networks,
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{càdlàg} 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.