A cosine rule-based discrete sectional curvature for graphs.

This article introduces and validates a new estimator for discrete sectional curvature applicable to graphs and other path metric spaces, focusing on random geometric graphs with low metric-distortion—graphs whose metric closely approximates that of underlying continuous manifolds with known constant sectional curvature. The estimator is based on a generalized cosine rule for right-angled triangles in spaces of constant curvature and is tested extensively on graphs constructed via a refined sprinkling algorithm ("hard annulus random geometric graphs") on spheres, hyperbolic planes, and Euclidean planes, demonstrating convergence of curvature estimates to continuum values as metric distortion decreases. Comparisons with existing discrete curvature notions, such as Wolfram–Ricci and mesoscopic Ollivier–Ricci curvatures, show improved convergence and robustness for the proposed estimator. Practical applications include estimating the Earth's radius from geographic data and analyzing sectional curvature distributions on self-similar fractals like the Sierpinski triangle, suggesting potential for broader use in network science, quantum gravity, and data science.