  How does one generalize differential geometric constructs such as curvature
of a manifold to the discrete world of graphs and other combinatorial
structures? This problem carries significant importance for analyzing models of
discrete spacetime in quantum gravity; inferring network geometry in network
science; and manifold learning in data science. The key contribution of this
paper is to introduce and validate a new estimator of discrete sectional
curvature for random graphs with low metric-distortion. The latter are
constructed via a specific graph sprinkling method on different manifolds with
constant sectional curvature. We define a notion of metric distortion, which
quantifies how well the graph metric approximates the metric of the underlying
manifold. We show how graph sprinkling algorithms can be refined to produce
hard annulus random geometric graphs with minimal metric distortion. We
construct random geometric graphs for spheres, hyperbolic and euclidean planes;
upon which we validate our curvature estimator. Numerical analysis reveals that
the error of the estimated curvature diminishes as the mean metric distortion
goes to zero, thus demonstrating convergence of the estimate. We also perform
comparisons to other existing discrete curvature measures. Finally, we
demonstrate two practical applications: (i) estimation of the earth's radius
using geographical data; and (ii) sectional curvature distributions of
self-similar fractals.
