The Weisfeiler-Leman algorithm ($1$-WL) is a well-studied heuristic for the graph isomorphism problem. Recently, the algorithm has played a prominent role in understanding the expressive power of message-passing graph neural networks (MPNNs) and being effective as a graph kernel. Despite its success, $1$-WL faces challenges in distinguishing non-isomorphic graphs, leading to the development of more expressive MPNN and kernel architectures. However, the relationship between enhanced expressivity and improved generalization performance remains unclear. Here, we show that an architecture's expressivity offers limited insights into its generalization performance when viewed through graph isomorphism. Moreover, we focus on augmenting $1$-WL and MPNNs with subgraph information and employ classical margin theory to investigate the conditions under which an architecture's increased expressivity aligns with improved generalization performance. In addition, we show that gradient flow pushes the MPNN's weights toward the maximum margin solution. Further, we introduce variations of expressive $1$-WL-based kernel and MPNN architectures with provable generalization properties. Our empirical study confirms the validity of our theoretical findings.

翻译：Weisfeiler-Leman算法（$1$-WL）是图同构问题中一个被深入研究的启发式方法。最近，该算法在理解消息传递图神经网络（MPNNs）的表达能力以及作为图核的有效性方面发挥了重要作用。尽管取得了成功，$1$-WL在区分非同构图方面仍面临挑战，这推动了更具表达力的MPNN和图核架构的发展。然而，增强的表达能力与改进的泛化性能之间的关系尚不明确。本文表明，当通过图同构的视角来看时，一个架构的表达能力对其泛化性能提供的见解有限。此外，我们专注于用子图信息增强$1$-WL和MPNNs，并运用经典的间隔理论来研究在何种条件下，架构表达能力的增强与泛化性能的提升相一致。此外，我们证明了梯度流将MPNN的权重推向最大间隔解。进一步地，我们引入了具有可证明泛化性质的、基于表达能力强的$1$-WL的图核和MPNN架构的变体。我们的实证研究证实了理论发现的有效性。