On the Expressive Power of Geometric Graph Neural Networks

Geometric GNNs are an emerging class of GNNs for spatially embedded graphs across science and engineering, e.g. SchNet for molecules, Tensor Field Networks for materials, GemNet for electrocatalysts, MACE for molecular dynamics, and E(n)-Equivariant Graph ConvNet for macromolecules.

How powerful are geometric GNNs? How do key design choices influence expressivity and how to build maximally powerful ones?

Check out this recent paper from Chaitanya K. Joshi, Cristian Bodnar, Simon V. Mathis, Taco Cohen, and Pietro Liò for more:

📄 PDF: http://arxiv.org/abs/2301.09308

💻 Code: http://github.com/chaitjo/geometric-gnn-dojo

❓Research gap: Standard theoretical tools for GNNs, such as the Weisfeiler-Leman graph isomorphism test, are inapplicable for geometric graphs. This is due to additional physical symmetries (roto-translation) that need to be accounted for.

💡Key idea: notion of geometric graph isomorphism + new geometric WL framework --> upper bound on geometric GNN expressivity.

The Geometric WL framework formalises the role of depth, invariance vs. equivariance, body ordering in geometric GNNs.
- Invariant GNNs cannot tell apart one-hop identical geometric graphs, fail to compute global properties.
- Equivariant GNNs distinguish more graphs; how? Depth propagates local geometry beyond one-hop.

What about practical implications? Synthetic experiments highlight challenges in building maximally powerful geom. GNNs:
- Oversquashing of geometric information with increased depth.
- Utility of higher order order spherical tensors over cartesian vectors.

P.S. Are you new to Geometric GNNs, GDL, PyTorch Geometric, etc.? Want to understand how theory/equations connect to real code?

Try this Geometric GNN 101 notebook before diving in:
https://github.com/chaitjo/geometric-gnn-dojo/blob/main/geometric_gnn_101.ipynb