Graphons have traditionally served as limit objects for dense graph sequences, with the cut distance serving as the metric for convergence. However, sparse graph sequences converge to the trivial graphon under the conventional definition of cut distance, which make this framework inadequate for many practical applications. In this paper, we utilize the concepts of generalized graphons and stretched cut distance to describe the convergence of sparse graph sequences. Specifically, we consider a random graph process generated from a generalized graphon. This random graph process converges to the generalized graphon in stretched cut distance. We use this random graph process to model the growing sparse graph, and prove the convergence of the adjacency matrices' eigenvalues. We supplement our findings with experimental validation. Our results indicate the possibility of transfer learning between sparse graphs.

翻译：图历来被用作稠密图序列的极限对象，切割距离作为收敛的度量。然而，传统切割距离定义下稀疏图序列收敛到平凡图，这导致该框架在许多实际应用中存在不足。本文利用广义图和拉伸切割距离的概念描述稀疏图序列的收敛性。具体而言，我们考虑由广义图生成的随机图过程。该随机图过程在拉伸切割距离下收敛到广义图。我们利用此随机图过程对增长稀疏图进行建模，并证明了邻接矩阵特征值的收敛性。我们通过实验验证了研究结果。我们的发现揭示了稀疏图之间迁移学习的可能性。