The Weisfeiler-Lehman (WL) test is a widely used algorithm in graph machine learning, including graph kernels, graph metrics, and graph neural networks. However, it focuses only on the consistency of the graph, which means that it is unable to detect slight structural differences. Consequently, this limits its ability to capture structural information, which also limits the performance of existing models that rely on the WL test. This limitation is particularly severe for traditional metrics defined by the WL test, which cannot precisely capture slight structural differences. In this paper, we propose a novel graph metric called the Wasserstein WL Subtree (WWLS) distance to address this problem. Our approach leverages the WL subtree as structural information for node neighborhoods and defines node metrics using the $L_1$-approximated tree edit distance ($L_1$-TED) between WL subtrees of nodes. Subsequently, we combine the Wasserstein distance and the $L_1$-TED to define the WWLS distance, which can capture slight structural differences that may be difficult to detect using conventional metrics. We demonstrate that the proposed WWLS distance outperforms baselines in both metric validation and graph classification experiments.

翻译：Weisfeiler-Lehman（WL）检验是图机器学习中广泛使用的算法，包括图核、图度量和图神经网络。然而，它仅关注图的一致性，这意味着它无法检测细微的结构差异。因此，这限制了其捕捉结构信息的能力，同时也限制了依赖WL检验的现有模型的性能。这一局限性对于由WL检验定义的传统度量尤为严重，因为它们无法精确捕捉细微的结构差异。在本文中，我们提出了一种名为Wasserstein WL子树（WWLS）距离的新型图度量来解决这一问题。我们的方法利用WL子树作为节点邻域的结构信息，并使用节点WL子树之间的$L_1$近似树编辑距离（$L_1$-TED）来定义节点度量。随后，我们将Wasserstein距离与$L_1$-TED结合，定义了WWLS距离，该距离能够捕捉传统度量难以检测的细微结构差异。我们证明，所提出的WWLS距离在度量验证和图分类实验中均优于基线方法。