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)).
Quenched asymptotics for interacting diffusions on inhomogeneous random graphs
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abstract
The aim of the paper is to address the behavior in large population of diffusions interacting on a random, possibly diluted and inhomogeneous graph. This is the natural continuation of a previous work, where the homogeneous Erd\H os-R\'enyi case was considered. The class of graphs we consider includes disordered $W$-random graphs, with possibly unbounded graphons. The main result concerns a quenched convergence (that is true for almost every realization of the random graph) of the empirical measure of the system towards the solution of a nonlinear Fokker-Planck PDE with spatial extension, also appearing in different contexts, especially in neuroscience. The convergence of the spatial profile associated to the diffusions is also considered, and one proves that the limit is described in terms of a nonlinear integro-differential equation which matches the neural field equation in certain particular cases.
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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)).