We aim to deepen the theoretical understanding of Graph Neural Networks (GNNs) on large graphs, with a focus on their expressive power. Existing analyses relate this notion to the graph isomorphism problem, which is mostly relevant for graphs of small sizes, or studied graph classification or regression tasks, while prediction tasks on nodes are far more relevant on large graphs. Recently, several works showed that, on very general random graphs models, GNNs converge to certains functions as the number of nodes grows. In this paper, we provide a more complete and intuitive description of the function space generated by equivariant GNNs for node-tasks, through general notions of convergence that encompass several previous examples. We emphasize the role of input node features, and study the impact of node Positional Encodings (PEs), a recent line of work that has been shown to yield state-of-the-art results in practice. Through the study of several examples of PEs on large random graphs, we extend previously known universality results to significantly more general models. Our theoretical results hint at some normalization tricks, which is shown numerically to have a positive impact on GNN generalization on synthetic and real data. Our proofs contain new concentration inequalities of independent interest.

翻译：我们旨在加深对图神经网络（GNN）在大规模图上理论层面的理解，重点关注其表达能力。现有分析通常将这一概念与图同构问题相联系，这主要适用于小规模图，或研究图分类与回归任务，而节点预测任务在大规模图上更具实际意义。近期多项研究表明，在非常一般的随机图模型上，随着节点数量增长，GNN会收敛到某些特定函数。本文通过涵盖先前多个例子的广义收敛概念，对等变GNN在节点任务上生成的函数空间给出了更完整且直观的描述。我们重点阐述了输入节点特征的作用，并研究了节点位置编码（PE）的影响——这一研究方向在实际应用中已取得最先进成果。通过分析大规模随机图上多种PE的实例，我们将先前已知的普适性结论推广至更为一般的模型。理论结果揭示了某些归一化技巧的有效性，数值实验表明这些技巧能显著提升GNN在合成数据与真实数据上的泛化能力。本文的证明过程还包含了若干具有独立价值的新型浓度不等式。