We consider the graphon mean-field system introduced by Bayraktar et al. in Bayraktar, Chakraborty, Wu (AAP 2023)
which is the large-population limit of a heterogeneously interacting diffusive particle system.
The interaction is of mean-field type with weights characterized by an underlying graphon function.
Via continuous observations of the trajectories of the finite-population particle system,
we build plug-in estimators of the particle densities, drift coefficients, and graphon interaction weights of the mean-field system.
Our estimators for the densities and drifts are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function
We prove that the estimator converges to the true graphon function as the number of particles tends to infinity when all other parameters are properly chosen.
Besides, we also conduct a minimax analysis on the plug-in estimator of the particle densities within a particular class of particle systems, which justifies its pointwise optimality.

Joint work with Erhan Bayraktar