Abstract
Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.
Highlights
Introduction and statement of resultsRicci curvature is a fundamental notion in the study of Riemannian manifolds
We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness
We study the long-scale curvature for the Cartesian product of two regular graphs
Summary
Ricci curvature is a fundamental notion in the study of Riemannian manifolds. This notion has been generalized in various ways from the smooth setting of manifolds to more general metric spaces. In [1] the authors investigate the Ollivier Ricci idleness function p → κp(x, y), which takes the idleness parameter p ∈ [ , ] as a variable and gives the value of curvature between the xed two adjacent vertices x and y (or equivalently, the curvature given on an edge of the graph joining x and y). End this sentence with a full-stop, after the intervals expression p → κp(x, y) is concave and piecewise linear on [ , ] with at most 3 linear parts, and it is linear on the intervals.
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