Graph limit for interacting particle systems on weighted random graphs

In this paper, we study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove quantitative convergence results in probability, as the number of particles tends to infinity, of the finite-dimensional system toward the solution of a deterministic graph limit equation. In this limit equation, the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We reveal the interplay between the large-population limit and the switching time. In particular, we show that for a fixed switching time, these systems converge as the number of particles tends to infinity to the same graph limit equation, in which the interaction is prescribed by the constant-in-time first moment of the weighted random graph law. Our results are illustrated by some numerical simulations.