Weakly interacting particle systems and their mean field limits given by McKean—Vlasov processes have been extensively studied and arise in many areas, such as neuroscience, queueing theory, social sciences, etc. A typical feature of such systems is that the evolution of a particle’s state depends on the empirical distribution of all particles’ states. In recent years, there have been a growing interest in mean field systems on random graphs. Such systems consist 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.

In this mini course, we will first review the classic mean field theory, including the propagation of chaos, law of large numbers, and central limit theorems. Then we will present how dense graphs, which could be random/deterministic/homogeneous/heterogeneous, affect the large population asymptotic behavior while sharing many similarities with the classic case. Finally we will present the significant impact of sparse graphs on the large population asymptotics of both a fixed particle and the empirical measure of the whole system.

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