Graph machine learning architectures are typically tailored to specific tasks on specific datasets, which hinders their broader applicability. This has led to a new quest in graph machine learning: how to build graph foundation models capable of generalizing across arbitrary graphs and features? In this work, we present a recipe for designing graph foundation models for node-level tasks from first principles. The key ingredient underpinning our study is a systematic investigation of the symmetries that a graph foundation model must respect. In a nutshell, we argue that label permutation-equivariance alongside feature permutation-invariance are necessary in addition to the common node permutation-equivariance on each local neighborhood of the graph. To this end, we first characterize the space of linear transformations that are equivariant to permutations of nodes and labels, and invariant to permutations of features. We then prove that the resulting network is a universal approximator on multisets that respect the aforementioned symmetries. Our recipe uses such layers on the multiset of features induced by the local neighborhood of the graph to obtain a class of graph foundation models for node property prediction. We validate our approach through extensive experiments on 29 real-world node classification datasets, demonstrating both strong zero-shot empirical performance and consistent improvement as the number of training graphs increases.

翻译：图机器学习架构通常针对特定数据集上的特定任务进行定制，这限制了其更广泛的应用。这引发了图机器学习领域的新探索：如何构建能够泛化至任意图结构和特征的图基础模型？本研究从第一性原理出发，提出了一种面向节点级任务的图基础模型设计方法。本研究的核心基础是对图基础模型必须遵循的对称性进行系统性探究。简而言之，我们认为除了图中每个局部邻域常见的节点置换等变性外，标签置换等变性与特征置换不变性同样不可或缺。为此，我们首先刻画了对节点与标签置换具有等变性、对特征置换具有不变性的线性变换空间。随后，我们证明了所得网络在遵循上述对称性的多重集上具有通用逼近能力。我们的方法通过在图的局部邻域诱导生成的特征多重集上应用此类层，构建了用于节点属性预测的图基础模型类别。我们在29个真实世界节点分类数据集上进行了广泛实验验证，结果表明该方法不仅具备强大的零样本实证性能，而且随着训练图数量的增加表现出持续的性能提升。