Graph Neural Networks (GNNs) have emerged as formidable resources for processing graph-based information across diverse applications. While the expressive power of GNNs has traditionally been examined in the context of graph-level tasks, their potential for node-level tasks, such as node classification, where the goal is to interpolate missing node labels from the observed ones, remains relatively unexplored. In this study, we investigate the proficiency of GNNs for such classifications, which can also be cast as a function interpolation problem. Explicitly, we focus on ascertaining the optimal configuration of weights and layers required for a GNN to successfully interpolate a band-limited function over Euclidean cubes. Our findings highlight a pronounced efficiency in utilizing GNNs to generalize a bandlimited function within an $\varepsilon$-error margin. Remarkably, achieving this task necessitates only $O_d((\log\varepsilon^{-1})^d)$ weights and $O_d((\log\varepsilon^{-1})^d)$ training samples. We explore how this criterion stacks up against the explicit constructions of currently available Neural Networks (NNs) designed for similar tasks. Significantly, our result is obtained by drawing an innovative connection between the GNN structures and classical sampling theorems. In essence, our pioneering work marks a meaningful contribution to the research domain, advancing our understanding of the practical GNN applications.

翻译：图神经网络（GNN）已成为处理跨领域图信息的强大资源。尽管GNN的表达能力传统上在图级任务中得到检验，但它们在节点级任务（如节点分类，即从观测标签插值缺失节点标签）中的潜力仍相对未被探索。本研究探讨了GNN在此类分类任务中的效能，该任务也可被视作函数插值问题。具体而言，我们聚焦于确定GNN成功插值欧几里得立方体上全通带函数所需的最优权重与层数配置。我们的发现凸显了GNN在$\varepsilon$误差范围内泛化全通带函数的显著效率：完成该任务仅需$O_d((\log\varepsilon^{-1})^d)$个权重和$O_d((\log\varepsilon^{-1})^d)$个训练样本。我们进一步将该标准与当前用于类似任务的显式神经网络（NN）构造进行对比。值得注意的是，这一结果通过建立GNN结构与经典采样定理之间的创新联系而获得。本质上，我们的开创性工作为研究领域做出了重要贡献，深化了对GNN实际应用的理解。