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

Local weak convergence for sparse networks of interacting processes