Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this hierarchy has propelled significant advances in GNN analysis and architecture developments, it suffers from several significant limitations. These include a complex definition that lacks direct guidance for model improvement and a WL hierarchy that is too coarse to study current GNNs. This paper introduces an alternative expressive power hierarchy based on the ability of GNNs to calculate equivariant polynomials of a certain degree. As a first step, we provide a full characterization of all equivariant graph polynomials by introducing a concrete basis, significantly generalizing previous results. Each basis element corresponds to a specific multi-graph, and its computation over some graph data input corresponds to a tensor contraction problem. Second, we propose algorithmic tools for evaluating the expressiveness of GNNs using tensor contraction sequences, and calculate the expressive power of popular GNNs. Finally, we enhance the expressivity of common GNN architectures by adding polynomial features or additional operations / aggregations inspired by our theory. These enhanced GNNs demonstrate state-of-the-art results in experiments across multiple graph learning benchmarks.

翻译：图神经网络（GNN）在其表达能力上存在固有局限。近年来的开创性工作（Xu等，2019；Morris等，2019b）引入了Weisfeiler-Lehman（WL）层次结构作为表达能力的一种度量标准。尽管这一层次结构推动了GNN分析与架构设计的重要进展，但它仍存在若干显著局限，包括缺乏对模型改进的直接指导的复杂定义，以及难以精细刻画当前GNN的过于粗粒度的WL层次结构。本文基于GNN计算特定次数等变多项式的能力，提出了一种替代性的表达能力层次结构。首先，我们通过引入具体基向量全面刻画了所有等变图多项式，显著推广了先前结果。每个基元素对应一个特定多重图，其在图数据输入上的计算对应一个张量缩并问题。其次，我们提出利用张量缩并序列评估GNN表达性的算法工具，并计算了主流GNN的表达能力。最后，受理论启发，我们通过添加多项式特征或额外操作/聚合方案，增强了常见GNN架构的表达性。这些增强型GNN在多个图学习基准实验中的表现达到了最先进水平。