Can graph neural networks generalize to graphs that are different from the graphs they were trained on, e.g., in size? In this work, we study this question from a theoretical perspective. While recent work established such transferability and approximation results via graph limits, e.g., via graphons, these only apply non-trivially to dense graphs. To include frequently encountered sparse graphs such as bounded-degree or power law graphs, we take a perspective of taking limits of operators derived from graphs, such as the aggregation operation that makes up GNNs. This leads to the recently introduced limit notion of graphops (Backhausz and Szegedy, 2022). We demonstrate how the operator perspective allows us to develop quantitative bounds on the distance between a finite GNN and its limit on an infinite graph, as well as the distance between the GNN on graphs of different sizes that share structural properties, under a regularity assumption verified for various graph sequences. Our results hold for dense and sparse graphs, and various notions of graph limits.

翻译：图神经网络能否泛化至与训练图结构不同的图上（例如尺寸差异）？本文从理论视角探究该问题。近期研究虽通过图极限（如图函数）建立了此类可迁移性与近似结果，但仅非平凡地适用于稠密图。为涵盖常见稀疏图（如有限度图或幂律图），我们采用从图中推导算子极限的视角，例如构成图神经网络的聚合操作。这引出了最近提出的图算子极限概念（Backhausz 和 Szegedy, 2022）。我们证明算子视角如何使我们能够：在正则性假设下（该假设对多种图序列均成立），为有限图神经网络与其无限图极限之间的距离，以及共享结构属性的不同尺寸图神经网络之间的距离，建立定量界。我们的结果适用于稠密图与稀疏图，并涵盖多种图极限概念。