March 10, 2020
Answered by: Jonathan Gorard
Why do you get a Euclidean/Riemannian metric, as opposed to a taxicab metric, induced on your hypergraphs?
If one measures distances by considering lengths of single geodesics between pairs of points in a hypergraph, using (for instance) some variant of Dijkstra’s algorithm, then evidently the induced metric will be discrete, and akin to a generalized taxicab metric. However, our derivation of the Einstein field equations involves first defining the Ollivier-Ricci curvature, which works by considering finite “bundles” of such geodesics (where, for two points in the hypergraph, one constructs a pair of geodesic balls surrounding those points by considering collections of random walks, and then one determines the geodesic distances between corresponding points on those balls using the Wasserstein transportation metric between the balls, considered as probability measures). This has the consequence of effectively “softening” the natural combinatorial metric on the hypergraph into an appropriately discretized version of a Riemannian metric; the full details are given in our formal discussion of general relativity.