Convergence of existing discrete scalar curvature definitions to manifold scalar curvature

Determine whether the discrete scalar curvature at a node, defined either as the sum or as the mean of the incident edge curvatures using Forman-Ricci curvature (Sandhu et al., 2015) or Ollivier-Ricci curvature (Sreejith et al., 2017), converges to the scalar curvature of the underlying Riemannian manifold.

Background

The paper reviews prior attempts to define discrete scalar curvature on graphs by aggregating edge-based discrete Ricci curvature at nodes. Sandhu et al. used Forman-Ricci curvature, while Sreejith et al. used Ollivier-Ricci curvature, taking either sums or means over incident edges to define a node-level scalar curvature.

While these definitions are natural analogues inspired by the relationship between Ricci and scalar curvature in Riemannian geometry, the authors note that it is not established whether such aggregations recover the manifold’s scalar curvature in any sampling limit. This motivates the present work, which introduces a new weighted aggregation (scalar Ollivier-Ricci curvature) and proves its convergence to the manifold scalar curvature under random geometric graph sampling. The open question remains for the earlier sum/mean-based definitions.