Improving Local Connectivity in Sparse and Rigid Graph Constructions

Nemo Delingat, Justin M. Baker, Andrea L. Bertozzi

Keywords: complete invariant, geometric gnn, geometric deep learning, graph construction, fundamental forms, graph neural networks

Abstract: Geometric graph neural networks (GNNs) depend critically on the construction of the underlying graph structure for molecular modeling, protein structure prediction, and 3D shape analysis. Recent rigidity-theory-inspired graph constructions aim to enhance the expressivity of geometric GNNs–separating geometric graphs up to isometries, while remaining sparse. However, their theoretical and empirical behavior remains poorly understood, and—crucially—they often fail to preserve local structure, introducing distortions that hinder learning of neighborhood-scale geometry. We introduce the Fundamental Forms Graph Construction (FFGC), a construction method that generates sparse, invariant, and architecture-agnostic graphs by pairing genus-0 projections with curvature-aware geometry and iterative optimization for reconciling global separation with local fidelity.

Submission Number: 14