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Graph neural networks (GNN) are very popular methods in machine learning andhave been applied very successfully to the prediction of the properties ofmolecules and materials. First-order GNNs are well known to be incomplete,i.e., there exist graphs that are distinct but appear identical when seenthrough the lens of the GNN. More complicated schemes have thus been designedto increase their resolving power. Applications to molecules (and moregenerally, point clouds), however, add a geometric dimension to the problem.The most straightforward and prevalent approach to construct graphrepresentation for molecules regards atoms as vertices in a graph and draws abond between each pair of atoms within a chosen cutoff. Bonds can be decoratedwith the distance between atoms, and the resulting "distance graph NNs" (dGNN)have empirically demonstrated excellent resolving power and are widely used inchemical ML, with all known indistinguishable configurations being resolved inthe fully-connected limit, which is equivalent to infinite or sufficientlylarge cutoff. Here we present a counterexample that proves that dGNNs are notcomplete even for the restricted case of fully-connected graphs induced by 3Datom clouds. We construct pairs of distinct point clouds whose associatedgraphs are, for any cutoff radius, equivalent based on a first-orderWeisfeiler-Lehman test. This class of degenerate structures includeschemically-plausible configurations, both for isolated structures and forinfinite structures that are periodic in 1, 2, and 3 dimensions. The existenceof indistinguishable configurations sets an ultimate limit to the expressivepower of some of the well-established GNN architectures for atomistic machinelearning. Models that explicitly use angular or directional information in thedescription of atomic environments can resolve this class of degeneracies.

Incompleteness of graph neural networks for points clouds in three dimensions