  We study the limiting behavior of interacting particle systems indexed by
large sparse graphs, which evolve either according to a discrete time Markov
chain or a diffusion, in which particles interact directly only with their
nearest neighbors in the graph. To encode sparsity we work in the framework of
local weak convergence of marked (random) graphs. We show that the joint law of
the particle system varies continuously with respect to local weak convergence
of the underlying graph marked with the initial conditions. In addition, we
show that the global empirical measure converges to a non-random limit for a
large class of graph sequences including sparse Erd\"{o}s-R\'{e}nyi graphs and
configuration models, whereas the empirical measure of the connected component
of a uniformly random vertex converges to a random limit. Along the way, we
develop some related results on the time-propagation of ergodicity and
empirical field convergence, as well as some general results on local weak
convergence of Gibbs measures in the uniqueness regime which appear to be new.
The results obtained here are also useful for obtaining autonomous descriptions
of marginal dynamics of interacting diffusions and Markov chains on sparse
graphs. While limits of interacting particle systems on dense graphs have been
extensively studied, there are relatively few works that have studied the
sparse regime in generality.
