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.

增强Weisfeiler-Leman算法和消息传递图神经网络的表达力的关系对于改进概括能力的条件是不明确的。通过引入子图信息和经典边界理论来探索这种表达力增强和概括能力改善的条件，并提出具有可证明概括性质的基于$1$-WL的核和消息传递图神经网络架构的变体。

边缘上的Weisfeiler-Leman：当更多的表达能力很重要