Non-parametric estimates for graphon mean-field particle systems

TL;DR

This paper develops non-parametric estimators for the underlying graphon and system parameters in a large-population mean-field particle system, demonstrating their convergence and optimality.

Contribution

It introduces kernel-based and deconvolution estimators for the graphon mean-field system and proves their consistency and optimality.

Findings

Graphon estimator converges pointwise to the true graphon.

Density and drift estimators are consistent and optimal.

Establishes convergence in the cut metric for the graphon estimator.

Abstract

We consider the graphon mean-field system introduced in the work of Bayraktar, Chakraborty, and Wu. It is the large-population limit of a heterogeneously interacting diffusive particle system, where the interaction is of mean-field type with weights characterized by an underlying graphon function. Through observation of continuous-time trajectories within the particle system, we construct plug-in estimators of the particle density, the drift coefficient, and thus the graphon interaction weights of the mean-field system. Our estimators for the density and drift are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function. We show that, as the number of particles increases, the graphon estimator converges to the true graphon function pointwisely, and as a consequence, in the cut metric. Besides, we conduct a minimax analysis within a particular class of particle systems to justify the pointwise optimality of the density and drift estimators.