What Does Graph Construction Preserve? A Discrete Ricci Curvature Diagnostic for Point Cloud Graphs

Graph representation learning is a powerful general method for learning encodings for complex data, but the impact of graph construction methods on learning tasks remains understudied. Understanding how a graph construction method extracts particular data properties is important both for our understanding of what properties of the problem contribute to model performance, and to further improve downstream model performance. In this work we propose a diagnostic separation score—the Wasserstein-1 distance between discrete Ricci curvature distributions of labeled populations—to evaluate graph constructions. This diagnostic is training-free, and exposes some properties preserved by graph construction methods. We evaluate this diagnostic in the context of two application domains: quantum chemistry (the QM9 dataset) and collider physics (the JetSet dataset). On QM9, where bonds are known, separation based on the Ollivier-Ricci variant of discrete curvature tracks bond-recovery F1 across 19 constructions (Spearman $\rho =0.88$) without seeing the bonds and favors the $1.8$ \AA radius graph construction, whereas separation based on the Forman-Ricci variant does not. On JetSet, which have no ground truth, all construction methods separate the flavors, but once we control for the number of nodes in the graph, only the radius graph construction method keeps most of its separation over all nodes. In addition to the diagnostic, we also measure an incremental predictive value of curvature as a model input: on QM9, a model gains most on the $1.8$\AA graph teh diagnostic selects, whereas on jets curvature adds at most $0.02$ AUROC to a mean aggregating GNN. However, this gain is mostly matched by using a simpler feature, namely, the node degree.