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.

翻译：本文考虑Bayraktar、Chakraborty和Wu提出的图论均值场系统。该系统是异质相互作用扩散粒子系统在大种群极限下的结果，其中相互作用为均值场类型，权重由底层图论函数刻画。通过观测粒子系统中的连续时间轨迹，我们构建了粒子密度、漂移系数以及均值场系统图论交互权重的插件估计量。密度和漂移的估计量是对经验数据进行核插值的直接结果，而反卷积方法则给出了底层图论函数的估计量。我们证明，随着粒子数量的增加，图论估计量在点态意义下收敛于真实图论函数，进而也在割度量意义下收敛。此外，我们在特定类别的粒子系统中进行极小化极大分析，以证明密度和漂移估计量的点态最优性。