Recently, many works studied the expressive power of graph neural networks (GNNs) by linking it to the $1$-dimensional Weisfeiler--Leman algorithm ($1\text{-}\mathsf{WL}$). Here, the $1\text{-}\mathsf{WL}$ is a well-studied heuristic for the graph isomorphism problem, which iteratively colors or partitions a graph's vertex set. While this connection has led to significant advances in understanding and enhancing GNNs' expressive power, it does not provide insights into their generalization performance, i.e., their ability to make meaningful predictions beyond the training set. In this paper, we study GNNs' generalization ability through the lens of Vapnik--Chervonenkis (VC) dimension theory in two settings, focusing on graph-level predictions. First, when no upper bound on the graphs' order is known, we show that the bitlength of GNNs' weights tightly bounds their VC dimension. Further, we derive an upper bound for GNNs' VC dimension using the number of colors produced by the $1\text{-}\mathsf{WL}$. Secondly, when an upper bound on the graphs' order is known, we show a tight connection between the number of graphs distinguishable by the $1\text{-}\mathsf{WL}$ and GNNs' VC dimension. Our empirical study confirms the validity of our theoretical findings.

翻译：最近，许多研究通过将图神经网络（GNN）的表达能力与一维Weisfeiler-Leman算法（$1\text{-}\mathsf{WL}$）相联系来探讨其局限性。$1\text{-}\mathsf{WL}$是为图同构问题设计的成熟启发式算法，它通过迭代对图的顶点集进行着色或划分。尽管这一关联在理解和增强GNN表达能力方面取得了重要进展，却未能揭示其泛化性能——即模型在训练集之外做出有意义预测的能力。本文从Vapnik-Chervonenkis（VC）维度理论出发，在两种设定下研究GNN在图表征预测中的泛化能力。首先，当图阶数无已知上界时，我们证明GNN权重位长度严格约束其VC维度，并进一步利用$1\text{-}\mathsf{WL}$产生的颜色数量推导出GNN的VC维度上界。其次，当图阶数存在已知上界时，我们揭示了$1\text{-}\mathsf{WL}$可区分的图数量与GNN的VC维度之间的紧密联系。实验研究证实了我们理论结果的有效性。