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：本文提出了一种局部化的Weisfeiler-Leman（WL）算法，旨在提升其表达能力并降低计算开销。我们聚焦于子图计数的具体问题，并给出了任意k的局部化k−WL版本。分析了局部k−WL的能力，证明其比k−WL更具表达力，且至多与(k+1)−WL等价。我们刻画了若两个图满足局部k−WL等价性时，其子图和诱导子图计数为不变量的模式。此外，引入了k−WL的两种变体：层级k−WL和递归k−WL。这些方法比在全图上应用k−WL更节省时间和空间。我们还提出了一种碎片化技术，仅使用1−WL即可保证精确计数所有大小不超过4的诱导子图。对于更大的模式，该思路可基于k>1进一步扩展。最后，将局部k−WL与其他GNN层级体系进行表达能力对比，表明在时间复杂度受限的前提下，我们的方法比Papp和Wattenhofer[2022a]中提及的方法更具表达力。