The ability of graph neural networks (GNNs) to count homomorphisms has recently been proposed as a practical and fine-grained measure of their expressive power. Although several existing works have investigated the homomorphism counting power of certain GNN families, a simple and unified framework for analyzing the problem is absent. In this paper, we first propose \emph{generalized folklore Weisfeiler-Leman (GFWL)} algorithms as a flexible design basis for expressive GNNs, and then provide a theoretical framework to algorithmically determine the homomorphism counting power of an arbitrary class of GNN within the GFWL design space. As the considered design space is large enough to accommodate almost all known powerful GNNs, our result greatly extends all existing works, and may find its application in the automation of GNN model design.

翻译：图神经网络（GNN）的同态计数能力最近被提出作为衡量其表达能力的一种实用且细粒度的指标。尽管已有若干研究工作探讨了特定GNN家族的同态计数能力，但尚缺乏一个简单统一的分析框架。本文首先提出**广义民俗Weisfeiler-Leman（GFWL）**算法作为表达能力强的GNN的灵活设计基础，随后构建了一个理论框架，用于在算法层面确定GFWL设计空间中任意一类GNN的同态计数能力。由于所考虑的设计空间足以容纳几乎所有已知的高性能GNN，我们的结果极大地扩展了现有研究，并有望在图神经网络模型设计的自动化中得到应用。