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Graph Neural Networks (GNNs) have emerged as formidable resources forprocessing graph-based information across diverse applications. While theexpressive power of GNNs has traditionally been examined in the context ofgraph-level tasks, their potential for node-level tasks, such as nodeclassification, where the goal is to interpolate missing node labels from theobserved ones, remains relatively unexplored. In this study, we investigate theproficiency of GNNs for such classifications, which can also be cast as afunction interpolation problem. Explicitly, we focus on ascertaining theoptimal configuration of weights and layers required for a GNN to successfullyinterpolate a band-limited function over Euclidean cubes. Our findingshighlight a pronounced efficiency in utilizing GNNs to generalize a bandlimitedfunction within an $\varepsilon$-error margin. Remarkably, achieving this tasknecessitates only $O_d((\log\varepsilon^{-1})^d)$ weights and$O_d((\log\varepsilon^{-1})^d)$ training samples. We explore how this criterionstacks up against the explicit constructions of currently available NeuralNetworks (NNs) designed for similar tasks. Significantly, our result isobtained by drawing an innovative connection between the GNN structures andclassical sampling theorems. In essence, our pioneering work marks a meaningfulcontribution to the research domain, advancing our understanding of thepractical GNN applications.

Theoretical guarantees for the advantage of GNNs over NNs in generalizing bandlimited functions on Euclidean cubes