A new perspective on the approximation capability of GNNs

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 nodes defined by the Weisfeiler - Lehman 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. In this paper, we propose an alternative way to demonstrate the approximation capability of GNNs that overcomes these limitations. In particular, some new results are proved, which allow to: (1) define GNN architectures capable of obtaining a given approximation; (2) show that the Weisfeiler-Lehman test converges in r+1 steps, where r is the diameter of the graph; (3) derive a formal relationship between the Weisfeiler-Lehman test and the unfolding trees, that is trees that can be built by visiting the graph starting from a given node. These results provide a more comprehensive understanding of the approximation power of GNNs, definitely showing that the 1-WL test and the unfolding tree concepts can be used interchangeably to study the their expressiveness.