Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the $k$-dimensional Weisfeiler-Leman ($k$WL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the $k$WL test. A central focus of research in this field revolves around determining the least dimensionality $k$, for which $k$WL can discern graphs with different number of occurrences of a pattern graph $P$. We refer to such a least $k$ as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern $P$. We additionally demonstrate that in cases where the $k$WL test distinguishes between graphs with varying occurrences of a pattern $P$, the exact number of occurrences of $P$ can be computed uniformly using only local information of the last layer of a corresponding GNN. We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern $P$, answering an open question from previous work.

翻译：图神经网络领域的开创性研究揭示了GNN的表达能力与$k$维Weisfeiler-Leman测试（一种广泛认可的图同构验证方法）之间的直接对应关系。这一联系重新激发了人们对理解$k$WL测试能够有效区分的具体图属性的兴趣。该领域研究的核心焦点在于确定最小的维度$k$，使得$k$WL能够区分具有不同模式图$P$出现次数的图。我们将此最小$k$称为模式计数问题的WL维数。传统上，这一研究涉及与模式相关的两类不同计数问题：子图计数和诱导子图计数。有趣的是，尽管它们最初看起来是采用看似不同方法的独立挑战，但这两个问题都是更综合问题——“图模参数”的相互关联组成部分。在本文中，我们精确刻画了带标签图模参数的WL维数。作为该结果的具体实例，我们获得了每个带标签模式$P$的子图计数和诱导子图计数问题的WL维数特征。我们还证明，在$k$WL测试能够区分具有不同模式$P$出现次数的图的情况下，可以仅使用对应GNN最后一层的局部信息统一计算$P$的确切出现次数。最后，我们深入探讨了识别各类图参数WL维数的挑战。我们提出了一个多项式时间算法，用于确定给定模式$P$的子图计数问题的WL维数，从而回答了先前工作中的一个开放问题。