On the equivalence between graph isomorphism testing and function approximation with GNNs

TL;DR

This paper establishes the theoretical equivalence between GNNs' ability to approximate permutation-invariant functions and their power to test graph isomorphism, introducing a new architecture that improves graph distinction capabilities.

Contribution

It proves the equivalence between GNNs' expressive power in function approximation and graph isomorphism testing, and introduces Ring-GNN to distinguish challenging regular graphs.

Findings

Second-order Invariant Graph Network cannot distinguish certain regular graphs

Ring-GNN successfully distinguishes non-isomorphic regular graphs

Ring-GNN performs well on real-world datasets

Abstract

Graph Neural Networks (GNNs) have achieved much success on graph-structured data. In light of this, there have been increasing interests in studying their expressive power. One line of work studies the capability of GNNs to approximate permutation-invariant functions on graphs, and another focuses on the their power as tests for graph isomorphism. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the expressive power of GNNs that incorporates both of these viewpoints using the language of sigma-algebra, through which we compare the expressive power of different types of GNNs together with other graph isomorphism tests. In particular, we prove that the second-order Invariant Graph Network fails to distinguish non-isomorphic regular graphs with the same degree. Then, we extend it to a new architecture, Ring-GNN, which succeeds in distinguishing these graphs and achieves good performances on real-world datasets.