  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-equivalent 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-equivalent 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.
