Stochastic gradient descents on graphs and a new McKean-Vlasov limit

It is well-known that hydrodynamical limits of large particle systems with mean-field interactions are given by McKean-Vlasov equations where a particle evolves by an SDE whose parameters are functions of its law that itself satisfies a PDE such as the Granular Media Equation. Motivated by network optimization problems, we consider stochastic gradient descents of functions of adjacency matrices of weighted graphs that are invariant under labelings of vertices. The talk will describe hydrodynamic limits of such random curves as the number of vertices go to infinity. The limiting space is that of graphons, a notion introduced by Lovasz and Szegedy to describe limits of dense graph sequences. The limiting curves are given by a novel notion of McKean-Vlasov equations on graphons and a corresponding notion of propagation of chaos holds. In the asymptotically zero-noise case, the limit is a notion of gradient flow on the space of graphons. This is an attempt to generalize Wasserstein calculus to higher-order exchangeable structures.