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.

本文旨在加深对大型图上图神经网络（GNNs）的理论理解，特别是着重于它们的表达能力。通过对等变GNNs生成的函数空间进行更完整和直观的描述，我们强调了输入节点特征的作用，并研究了节点位置编码（PEs）的影响，这是最近一项在实践中表现出卓越成果的研究。通过对大型随机图的PEs的研究，我们将先前已知的普适性结果扩展到了更为普遍的模型，同时提出了一些规范化技巧，这在合成和实际数据上得到了正面的验证。

图神经网络能在随机图上计算哪些函数？位置编码的作用