Graphon Particle Systems, Part I: Spatio-Temporal Approximation and Law of Large Numbers

We study a class of graphon particle systems with time-varying random coefficients. In a graphon particle system, the interactions among particles are characterized by the coupled mean field terms through an underlying graphon and the randomness of the coefficients comes from the stochastic processes associated with the particle labels. By constructing two-level approximated sequences converging in 2-Wasserstein distance, we prove the existence and uniqueness of the solution to the system. Besides, by constructing two-level approximated functions converging to the graphon mean field terms, we establish the law of large numbers, which reveals that if the number of particles tends to infinity and the discretization step tends to zero, then the discrete-time interacting particle system over the large-scale network converges to the graphon particle system. As a byproduct, we discover that the graphon particle system can describe the dynamic evolution of the distributed stochastic gradient descent algorithm over the large-scale network and prove that if the gradients of the local cost functions are Lipschitz continuous, then the graphon particle system can be regarded as the spatio-temporal approximation of the discrete-time distributed stochastic gradient descent algorithm as the number of network nodes tends to infinity and the algorithm step size tends to zero.