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：本文提出局部化版本的Weisfeiler-Leman（WL）算法，旨在同时提升表示能力并降低计算开销。我们聚焦于子图计数这一特定问题，给出了任意k值下的局部化$k-$WL版本。我们分析了局部$k-$WL的能力，证明其表达能力优于$k-$WL，且至多与$(k+1)-$WL相当。我们刻画了当两个图满足局部$k-$WL等价时，其子图计数与诱导子图计数保持不变的模式特征。此外，我们引入$k-$WL的两种变体：分层$k-$WL与递归$k-$WL。这些方法相比在全图上应用$k-$WL更具时间和空间效率。我们还提出一种碎片化技术，仅使用$1-$WL即可精确计数所有规模不超过4的诱导子图。该思想可进一步扩展到使用$k>1$的更大模式。我们还将局部$k-$WL与其他GNN层次结构的表示能力进行比较，证明在时间复杂度受限条件下，本文方法的表达能力优于Papp和Wattenhofer[2022a]所述方法。