Graphon Particle Systems, Part I: Spatio-Temporal Approximation and Law of Large Numbers
Yan Chen, Tao Li, Xiaofeng Zong
TL;DR
The paper analyzes graphon particle systems with time-varying random coefficients, establishing existence and uniqueness of solutions and a spatio-temporal law of large numbers that link finite-network dynamics to a graphon mean-field limit. It introduces two-level approximation schemes to handle measurability and convergence in the 2-Wasserstein sense, proving that discrete-time particle systems on large networks converge to the graphon system as the number of particles grows and the discretization step shrinks. Furthermore, it shows that the limiting graphon dynamics describe the limiting behavior of distributed SGD on large networks when local gradients are Lipschitz, providing a principled bridge between networked stochastic optimization and graphon-based mean-field models. These results illuminate how large-scale heterogeneous networks can be analyzed via graphon limits and have potential implications for understanding distributed learning and consensus in expansive systems.
Abstract
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 exogenous stochastic processes. 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 a large-scale network converges to the graphon particle system. As a byproduct, we discover that the graphon particle system can describe the limiting dynamics 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.