Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a unified framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.

翻译：设计具有高表达能力的图神经网络（GNN）是图学习领域的一个基础课题。迄今为止，GNN的表达能力主要通过Weisfeiler-Lehman（WL）层级来评估。然而，这种表达能力度量方法存在显著局限：它本质上是粗糙的、定性的，且可能无法很好地反映实际需求（例如，编码子结构的能力）。在本文中，我们引入了一个统一框架，用于定量研究GNN架构的表达能力，从而克服上述所有局限。具体来说，我们识别出一个基本的表达能力度量——同态表达能力，它量化了GNN模型在同态意义下计数图的能力。同态表达能力提供了一套完整且实用的评估工具：其完整性使得不同GNN模型之间可以直接进行表达能力比较，而其实用性则有助于理解GNN的具体能力，例如子图计数。通过对四类典型GNN的案例研究，我们推导出了它们在不变性和等变性设定下同态表达能力的简单、统一且优雅的描述。我们的研究结果为一系列先前的工作提供了新颖见解，统一了该领域不同子方向的格局，并解决了若干开放性问题。在实证方面，我们在合成任务和真实世界任务上进行了大量实验，验证了我们的理论，结果表明GNN模型的实际性能与所提出的度量指标高度吻合。