Local weak convergence for sparse networks of interacting processes

Understand

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.