Message passing neural networks (MPNNs) have emerged as the most popular framework of graph neural networks (GNNs) in recent years. However, their expressive power is limited by the 1-dimensional Weisfeiler-Lehman (1-WL) test. Some works are inspired by $k$-WL/FWL (Folklore WL) and design the corresponding neural versions. Despite the high expressive power, there are serious limitations in this line of research. In particular, (1) $k$-WL/FWL requires at least $O(n^k)$ space complexity, which is impractical for large graphs even when $k=3$; (2) The design space of $k$-WL/FWL is rigid, with the only adjustable hyper-parameter being $k$. To tackle the first limitation, we propose an extension, $(k,t)$-FWL. We theoretically prove that even if we fix the space complexity to $O(n^k)$ (for any $k\geq 2$) in $(k,t)$-FWL, we can construct an expressiveness hierarchy up to solving the graph isomorphism problem. To tackle the second problem, we propose $k$-FWL+, which considers any equivariant set as neighbors instead of all nodes, thereby greatly expanding the design space of $k$-FWL. Combining these two modifications results in a flexible and powerful framework $(k,t)$-FWL+. We demonstrate $(k,t)$-FWL+ can implement most existing models with matching expressiveness. We then introduce an instance of $(k,t)$-FWL+ called Neighborhood$^2$-FWL (N$^2$-FWL), which is practically and theoretically sound. We prove that N$^2$-FWL is no less powerful than 3-WL, and can encode many substructures while only requiring $O(n^2)$ space. Finally, we design its neural version named N$^2$-GNN and evaluate its performance on various tasks. N$^2$-GNN achieves record-breaking results on ZINC-Subset (0.059), outperforming previous SOTA results by 10.6%. Moreover, N$^2$-GNN achieves new SOTA results on the BREC dataset (71.8%) among all existing high-expressive GNN methods.

翻译：消息传递神经网络（MPNNs）近年来已成为图神经网络（GNNs）最流行的框架。然而，其表达能力受限于一维韦费勒-莱曼（1-WL）测试。部分研究受$k$-WL/FWL（民间WL）启发，设计了相应神经网络版本。尽管具有高表达能力，该研究路线存在严重局限：（1）$k$-WL/FWL至少需要$O(n^k)$空间复杂度，即便$k=3$时对大规模图也不实用；（2）$k$-WL/FWL的设计空间僵化，唯一可调超参数仅为$k$。针对第一个局限，我们提出扩展版本$(k,t)$-FWL。理论证明，即使在$(k,t)$-FWL中将空间复杂度固定为$O(n^k)$（$k\geq 2$），仍可构建出能解决图同构问题的表达能力层级结构。针对第二个问题，我们提出$k$-FWL+，其将任意等变集合而非所有节点视为邻域，从而大幅扩展了$k$-FWL的设计空间。结合这两项改进得到灵活强大的框架$(k,t)$-FWL+。我们证明$(k,t)$-FWL+能以匹配的表达能力实现大多数现有模型。随后引入$(k,t)$-FWL+的实例——邻域平方FWL（N$^2$-FWL），该模型在理论与实践上均具合理性。我们证明N$^2$-FWL的表达能力不低于3-WL，能够编码多种子结构，且仅需$O(n^2)$空间。最后，设计其神经网络版本N$^2$-GNN并在多项任务上评估性能。N$^2$-GNN在ZINC-Subset数据集上取得破纪录结果（0.059），较此前最优方法提升10.6%。此外，在所有现有高表达能力GNN方法中，N$^2$-GNN在BREC数据集上取得新最优结果（71.8%）。