Concentration of measure for Graphon particle system
Erhan Bayraktar, Donghan Kim
TL;DR
The paper analyzes concentration of measures for graphon-based diffusive particle systems and their finite-particle approximations over a finite time horizon, extending prior dense-graph results to general sparsity sequences $p(n)$. It develops a mean-field framework leveraging transportation inequalities and a DLR:AOP-style approach to obtain exponential concentration bounds for both $W_1$ and $W_2$ distances between empirical measures and the graphon-system law, including cases where the interaction function lies in the $L^1$-Fourier class. Key contributions include (i) LLN-type results showing convergence of empirical measures to the graphon-averaged law, (ii) exponential concentration for Lipschitz observables of the finite-particle system, and (iii) explicit concentration toward the graphon system in the $W_1$ and $W_2$ metrics under not-too-dense graph regimes, with improvements when the interaction is in the $L^1$- Fourier class. The work advances finite-time, sparse-graph mean-field analysis with practical implications for heterogeneous networked diffusions and mean-field games on graphs. The results extend existing dense-graph theories and provide a toolkit for sharp probabilistic control of graphon particle systems in applied settings.
Abstract
We study heterogeneously interacting diffusive particle systems with mean-field type interaction characterized by an underlying graphon and their finite particle approximations. Under suitable conditions, we obtain exponential concentration estimates over a finite time horizon for both 1 and 2 Wasserstein distances between the empirical measures of the finite particle systems and the averaged law of the graphon system, extending the work of Bayraktar-Wu.