Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

Abstract

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces.

Combinatorial quantum gravity: geometry from random bits

“…one of the distinguishing features of random graphs vs. random geometric graphs [8,9]. As a driver of the quantum phase transition I will consider the combinatorial Ollivier-Ricci curvature [10][11][12][13][14][15], which becomes the standard Ricci curvature scalar in the ordered phase. Note that this program is totally different from previous approaches to quantum gravity based on graph structures [16,17]: there is no need of auxiliary group variables and the action is a purely combinatorial version of the Einstein-Hilbert action.…”

“…Because of their random character, the Regge formulation of curvature is no more applicable, a purely combinatorial version of Ricci curvature is needed. Recently, exactly such a combinatorial Ricci curvature has been proposed by Ollivier [10,11] and further elaborated on in [12][13][14][15].…”

“…Fortunately, it becomes much simpler for bipartite graphs [15], which have no odd cycles. Since the Ollivier curvature of an edge depends only on the triangles, squares and pentagrams supported on that edge (a discrete form of locality) [12][13][14] and there are no triangles and pentagrams on graphs in the configuration space, one can use for all practical…”

Ricci-Ollivier Curvature of the Rooted Phylogenetic Subtree-Prune-Regraft Graph

“…This definition of curvature is invariant of p for small p [18] and can be used to avoid parity problems on graphs where the uniform random walk is periodic without choosing a specific laziness parameter (e.g. Ollivier often considered κ 1 2 (x, y) for this purpose).…”

“…We then computed curvatures for given pairs of trees directly, by using linear programming [18] to compute the minimal mass transport W 1 using the SAGE [35] front-end to the GLPK [1] solver; code can be found in [20] which grew from the code described in [18]. This would have required an enormous amount of computation to directly compute curvatures for the ((2n − 3)!!)…”