Local dynamics for large sparse networks of interacting diffusions

This talk is an overview of a recent and ongoing line of work on large sparse networks of interacting diffusion processes. Each process is associated with a vertex in a graph and interacts only with its neighbors. When the graph is complete and the size grows to infinity, the system is well-approximated by its mean field limit, which describes the behavior of one typical process. For general graphs, however, the mean field approximation can fail, most dramatically when the graph is sparse. Nevertheless, if the underlying graph is locally tree-like (as is the case for many canonical sparse random graph models), we show that a single process and its nearest neighbors are characterized by an autonomous evolution which we call the "local dynamics." This can be viewed as a sparse counterpart of the usual McKean-Vlasov equation. The structure of the local dynamics depend heavily on the symmetries of the underlying graph and the conditional independence structure of the solution process. In the time-stationary case, the local dynamics take a particular tractable form. Based on joint works with Kavita Ramanan, Ruoyu Wu, and Jiacheng Zhang.