Graph Neural Networks (GNNs) are a broad class of connectionist models for graph processing. Recent studies have shown that GNNs can approximate any function on graphs, modulo the equivalence relation on graphs defined by the Weisfeiler--Lehman (WL) test. However, these results suffer from some limitations, both because they were derived using the Stone--Weierstrass theorem -- which is existential in nature, -- and because they assume that the target function to be approximated must be continuous. Furthermore, all current results are dedicated to graph classification/regression tasks, where the GNN must produce a single output for the whole graph, while also node classification/regression problems, in which an output is returned for each node, are very common. In this paper, we propose an alternative way to demonstrate the approximation capability of GNNs that overcomes these limitations. Indeed, we show that GNNs are universal approximators in probability for node classification/regression tasks, as they can approximate any measurable function that satisfies the 1--WL equivalence on nodes. The proposed theoretical framework allows the approximation of generic discontinuous target functions and also suggests the GNN architecture that can reach a desired approximation. In addition, we provide a bound on the number of the GNN layers required to achieve the desired degree of approximation, namely $2r-1$, where $r$ is the maximum number of nodes for the graphs in the domain.

翻译：图神经网络（GNN）是用于图处理的一类广泛连接主义模型。近期研究表明，GNN能够近似图上的任意函数，其近似能力受限于Weisfeiler-Lehman（WL）测试定义的图等价关系。然而，这些结果存在若干局限性：一方面，它们基于Stone-Weierstrass定理推导——该定理本质上是存在性的；另一方面，它们假设待逼近的目标函数必须连续。此外，现有结论均针对图分类/回归任务（即GNN需为整张图输出单一结果），而节点分类/回归问题（需为每个节点单独返回输出）亦十分常见。本文提出一种替代性方法证明GNN的近似能力，从而突破上述局限。具体而言，我们证明GNN是节点分类/回归任务的概率通用逼近器：它们能近似任意满足节点1-WL等价性的可测函数。所提出的理论框架不仅允许逼近一般的不连续目标函数，还能指明达到所需近似精度应采用的GNN架构。此外，我们给出了实现目标近似程度所需的GNN层数上界，即$2r-1$，其中$r$为定义域图中节点的最大数量。