Recovering the random graph model from an observed collection of networks is known to present significant challenges in the setting, where the networks do not share a common node set and have different sizes. More specifically, the goal is the estimation of the graphon function that parametrizes the nonparametric exchangeable random graph model. Existing methods typically suffer from either limited accuracy or high computational complexity. We introduce a new histogram-based estimator with low algorithmic complexity that achieves high accuracy by jointly aligning the nodes of all graphs, in contrast to most conventional methods that order nodes graph by graph. Consistency results of the proposed graphon estimator are established. A numerical study shows that the proposed estimator outperforms existing methods in terms of accuracy, especially when the dataset comprises only small and variable-size networks. Moreover, the computing time of the new method is considerably shorter than that of other consistent methodologies. Additionally, when applied to a graph neural network classification task, the proposed estimator enables more effective data augmentation, yielding improved performance across diverse real-world datasets.

翻译：在观测到的网络集合不共享公共节点集且具有不同规模的设定下，从这些网络恢复随机图模型已知存在显著挑战。具体而言，目标在于估计参数化非参数可交换随机图模型的图函数。现有方法通常存在精度有限或计算复杂度高的问题。我们提出了一种新的基于直方图的估计器，其算法复杂度低，并通过联合对齐所有图的节点来实现高精度，这与大多数传统方法逐图排序节点的做法形成对比。本文建立了所提图函数估计器的一致性结果。数值研究表明，所提估计器在精度上优于现有方法，尤其是在数据集仅包含小型且规模可变的网络时。此外，新方法的计算时间显著短于其他一致性方法。进一步地，当应用于图神经网络分类任务时，所提估计器能够实现更有效的数据增强，从而在多样化的现实世界数据集中获得更好的性能。