For interacting oscillators on graphs whose normalized adjacency matrix is close to the all-ones matrix in the infinity-to-one norm, the empirical measure follows the McKean-Vlasov equation and stays near its stable stationary states for times up to exp(o(n)).
Interacting diffusions on sparse graphs: hydrodynamics from local weak limits
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abstract
We prove limit theorems for systems of interacting diffusions on sparse graphs. For example, we deduce a hydrodynamic limit and the propagation of chaos property for the stochastic Kuramoto model with interactions determined by Erd\H{o}s-R\'{e}nyi graphs with constant mean degree. The limiting object is related to a potentially infinite system of SDEs defined over a Galton-Watson tree. Our theorems apply more generally, when the sequence of graphs ("decorated" with edge and vertex parameters) converges in the local weak sense. Our main technical result is a locality estimate bounding the influence of far-away diffusions on one another. We also numerically explore the emergence of synchronization phenomena on Galton-Watson random trees, observing rich phase transitions from synchronized to desynchronized activity among nodes at different distances from the root.
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Long time dynamics for interacting oscillators on graphs
For interacting oscillators on graphs whose normalized adjacency matrix is close to the all-ones matrix in the infinity-to-one norm, the empirical measure follows the McKean-Vlasov equation and stays near its stable stationary states for times up to exp(o(n)).