On the Expressive Power of Geometric Graph Neural Networks
This work addresses a foundational gap in understanding geometric GNNs for researchers in machine learning and scientific domains like chemistry and physics, though it is incremental in extending existing theoretical frameworks to geometric settings.
The paper tackles the problem of characterizing the expressive power of geometric Graph Neural Networks (GNNs) for graphs embedded in Euclidean space, such as biomolecules and materials, by proposing a geometric version of the Weisfeiler-Leman test (GWL) that respects physical symmetries, and it shows that equivariant layers and higher-order tensors enable maximally powerful geometric GNNs.
The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at \url{https://github.com/chaitjo/geometric-gnn-dojo}