Graphon mean field systems

Abstract

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system.

34 pages. To appear in Annals of Applied Probability

References in corpus (3)

• Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach
• Weakly interacting oscillators on dense random graphs
• Large subgraphs in pseudo-random graphs

Cited by in corpus (6)

• Weakly interacting oscillators on dense random graphs
• A note on Fokker-Planck equations and graphons
• Large Deviations of Non-Stochastic Interacting Particles on Sparse Random Graphs
• Large population limits of Markov processes on random networks
• LQG Graphon Mean Field Games: Analysis via Graphon Invariant Subspaces
• Graphon particle system: Uniform-in-time concentration bounds