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 $O(n^2)$ 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, 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 superior performance on almost all tasks, with record-breaking results on ZINC-Subset (0.059) and ZINC-Full (0.013), outperforming previous state-of-the-art results by 10.6% and 40.9%, respectively.

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Weisfeiler-Lehman（1-WL）测试。部分工作受$k$-WL/FWL（Folklore WL）启发并设计了相应的神经版本。尽管具有高表达能力，该研究方向仍存在严重缺陷：（1）$k$-WL/FWL至少需要$O(n^k)$空间复杂度，即使当$k=3$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，我们提出扩展方案$(k,t)$-FWL。理论证明，即使将$(k,t)$-FWL的空间复杂度固定为$O(n^2)$，仍可构建直至解决图同构问题的表达能力层次结构。针对第二个问题，我们提出$k$-FWL+，该方案将任意等变集合（而非全部节点）视为邻居，极大扩展了$k$-FWL的设计空间。将这两项改进结合，形成灵活而强大的框架$(k,t)$-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，我们引入$(k,t)$-FWL+的实例——邻域平方FWL（N$^2$-FWL），其在理论和实践上均具有优势。我们证明N$^2$-FWL的表达能力不低于3-WL，能编码多种子结构且仅需$O(n^2)$空间。最后，我们设计其神经版本N$^2$-GNN，并在多种任务上评估其性能。N$^2$-GNN在几乎所有任务中均取得卓越表现，在ZINC-Subset（0.059）和ZINC-Full（0.013）上创下新纪录，分别相较此前最优结果提升10.6%和40.9%。