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-Lehman算法（$1$-WL）是图同构问题中经广泛研究的启发式方法。该算法近年来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$1$-WL取得了成功，但在区分非同构图时仍面临挑战，这促使了更具表达能力的MPNN与核架构的发展。然而，增强的表达能力与改进的泛化性能之间的关系仍不明确。本文表明，当通过图同构视角审视时，架构的表达能力对其泛化性能提供的洞察有限。此外，我们聚焦于通过子图信息增强$1$-WL与MPNN，并利用经典间隔理论研究架构增强表达能力与提升泛化性能相一致的条件。同时，我们证明梯度流会将MPNN的权重推向最大间隔解。进一步地，我们引入了基于$1$-WL的具有可证明泛化特性的表达性核与MPNN架构变体。我们的实证研究证实了理论发现的有效性。