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. Observing continuous-time trajectories of a 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 justify the pointwise optimality of the density estimator via a minimax analysis over a particular class of particle systems.

翻译：本文考虑Bayraktar、Chakraborty和Wu所提出的图论平均场系统。该模型是异质交互扩散粒子系统在大规模种群极限下的表现形式，其中交互作用具有平均场特征，其权重由底层图论函数刻画。通过观测有限规模种群粒子系统的连续时间轨迹，我们构建了该平均场系统中粒子密度、漂移系数以及图论交互权重的插入式估计量。密度与漂移的估计量直接源于对经验数据进行核插值的结果，而解卷积方法则推导出底层图论函数的估计量。我们证明，当所有其他参数选取适当时，随着粒子数量趋于无穷，该估计量收敛至真实图论函数。此外，我们通过对特定粒子系统类别的极小化极大分析，论证了密度估计量的逐点最优性。