In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of $k-$WL for any $k$. We analyze the power of Local $k-$WL and prove that it is more expressive than $k-$WL and at most as expressive as $(k+1)-$WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local $k-$WL equivalent. We also introduce two variants of $k-$WL: Layer $k-$WL and recursive $k-$WL. These methods are more time and space efficient than applying $k-$WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just $1-$WL. The same idea can be extended further for larger patterns using $k>1$. We also compare the expressive power of Local $k-$WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a].

翻译：本文提出韦斯费勒-莱曼（WL）算法的局部化版本，旨在增强其表达能力并降低计算开销。我们聚焦子图计数这一具体问题，并针对任意k值给出k-WL的局部化版本。我们分析了局部k-WL的效能，证明其表达能力优于k-WL，且至多与(k+1)-WL相当。我们刻画了模式的特征：若两图在局部k-WL意义下等价，则这些模式作为子图及导出子图的计数保持不变。此外，我们引入k-WL的两种变体：层级k-WL与递归k-WL，这些方法在时间和空间效率上均优于在全图上应用k-WL。我们还提出一种碎片化技术，该技术仅使用1-WL即可保证精确计数所有大小不超过4的导出子图。当k>1时，该思路可进一步扩展至更大模式。我们还将局部k-WL与其他图神经网络层级结构进行表达能力对比，结果表明：在时间复杂度受限条件下，我们的方法比Papp与Wattenhofer[2022a]中提及的方法更具表达能力。