Abstract: We study large systems of diffusions (partices) in which each particle is associated with a vertex in a graph and interacts only with its neighbors. The complete graph case is well understood, and the inifinite-population limit (as the size of the graph grows to infinity) gives rise to the McKean-Vlasov equation, which describes the behavior of one typical particle. For general (sparse) graphs, the system is no longer exchangeable, and mean field arguments do not apply. We identify a useful Gibbs (or Markov random field) structure and study some of its implications. Most notably, when the graphs approximate a regular tree or a Galton-Watson tree, the dynamics of a single typical particle and its nearest neighbors are summarized by a peculiar but autonomous system of equations. The situation is clearest for linear models, as distributions can often be identified explicitly in terms of the spectrum of the graph Laplacian. This work is motivated in part by recent mean field models of interbank lending, which successfully illustrate several features of systemic risk but thus far lack realistic network structure.

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Financial Mathematics Seminar