Graph Neural Networks (GNNs) have achieved much success on graph-structured data. In light of this, there have been increasing interests in studying their expressive power. One line of work studies the capability of GNNs to approximate permutation-invariant functions on graphs, and another focuses on the their power as tests for graph isomorphism. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the expressive power of GNNs that incorporates both of these viewpoints using the language of sigma-algebra, through which we compare the expressive power of different types of GNNs together with other graph isomorphism tests. In particular, we prove that the second-order Invariant Graph Network fails to distinguish non-isomorphic regular graphs with the same degree. Then, we extend it to a new architecture, Ring-GNN, which succeeds in distinguishing these graphs and achieves good performances on real-world datasets.

翻译：图神经网络（GNN）在图结构数据上取得了显著成功。为此，对其表达能力的研究日益受到关注。一方面，相关研究探讨了GNN逼近图上置换不变函数的能力；另一方面，另一些工作聚焦于其作为图同构测试工具的潜力。本研究将这两种视角联系起来，并证明了它们之间的等价性。我们进一步利用σ-代数语言构建了一个融合上述两种观点的GNN表达能力分析框架，借此比较了不同类型GNN及其他图同构测试的表达能力。特别地，我们证明了二阶不变图网络无法区分具有相同度数的非同构正则图。随后，我们将其扩展为一种新型架构Ring-GNN，该架构不仅能成功区分这些图，还在真实世界数据集上取得了优异性能。