On the Sum of Ricci-Curvatures for Weighted Graphs

Abstract

In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph.A weighted graph G = (V, E, d) is an undirected graph G = (V, E) associated with a distance function d : E → [0, ∞). By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between u and v is exactly d(u, v) for any edge uv. Now consider a random walk whose transitive probability from an vertex u to its neighbor v (a jump move along the edge uv) is proportional to wuv := F (d(u, v))/d(u, v) for some given function F (•). We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of κ(x, y) using a so called * -coupling functions. The total curvature K(G) is defined to be the sum of Ricci curvatures over all edges of G. We proved the following theorems: ifBoth equalities hold if and only if d is a constant function plus the girth is at least 6.In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least 6, where the graph Ricci curvature is defined geometrically in terms of optimal transport.

Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs

“…We will also exhibit an upper bound on the action, for arbitrary graphs. Together with the lower bound on the action worked out in [15], these two bounds restrict the range of the action for arbitrary finite graphs. In Section 5 we will give some more general solutions to the equations of motion, in the case where no boundary condition is imposed.…”

“…Other papers which have explored related ideas are [11][12][13][14][15] (see also the survey article [16]).…”

Graphs with nonnegative Ricci curvature and maximum degree at most 3

“…For fixed vertices u, v ∈ E, it was shown in [BCL + 18] that the function ε → κ ε (u, v) is concave and piecewise linear, so the limitation exists and κ is well defined. There are also equivalent limit-free definitions via coupling function or Laplacian on graphs; see [BHLY20,MW19]. When we say Ricci curvature or curvature in the following, we always mean the Lin-Lu-Yau-Ollivier Ricci curvature defined above.…”