On the Expressive Power of Sparse Geometric MPNNs

TL;DR

This paper investigates the expressive capabilities of sparse geometric message-passing neural networks, demonstrating their ability to distinguish non-isomorphic graphs under certain conditions and introducing a new architecture, EGENNET, that achieves these guarantees.

Contribution

It provides theoretical guarantees for separating geometric graphs with sparse, rotation-equivariant message-passing networks and introduces the EGENNET architecture for practical applications.

Findings

Rotation-equivariant networks can distinguish non-isomorphic graphs when connected.

Invariant features guarantee separation for globally rigid graphs.

EGENNET performs well on synthetic and chemical graph benchmarks.

Abstract

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.