**Weakly interacting particle systems on random graphs: from dense to sparse**

We consider the asymptotic behaviors of weakly interacting (mean-field) particle systems on

random graphs that could be dense or sparse. The system consists of a large number of nodes in

which the state of each node is governed by a stochastic process that has a mean-field type

interaction with the neighboring nodes. Such systems arise in many areas, including but not

limited to neuroscience, queueing theory and social sciences, which we will discuss in this talk.

In the dense graph case, the limiting system is described by the classic McKean–Vlasov

equation. Law of large numbers, propagation of chaos, and central limit theorems are

established and turn out to be the same as those in the complete graph case.

In the sparse case, the limiting system is quite different and depends heavily on the graph

structure. We obtain an autonomous characterization of the local dynamics of a typical node

and its neighbors, when the limiting graph is a D-regular tree or a Galton–Watson tree.

If time permits, certain queueing systems with non-mean-field interactions will be discussed.