Title：Long-scale Ollivier-Ricci curvature of graphs

Speaker：Supanat Kamtue (Durham University, UK)

Time：2018年3月28日（周三）   下午  14:30-15:30

Room：东区管理科研楼  数学科学学院1518室

Abstract： Ricci curvature plays a very important role in the study of Riemannian manifolds, and it has then been adapted into various notions in more generalised metric spaces, in an attempt to explain some geometric properties of such spaces (e.g. Laplacian spectra, which is related to local connectivity). In 2009, Ollivier introduced one such curvature notion, based on optimal transport of probability measures. In the discrete setting of graphs, it gives rise to the Ollivier-Ricci curvature, which is defined between two vertices of a graph, with an additional parameter called 'idleness'.

Most previous work study this curvature under the condition that the two vertices are adjacent. We have generalised it to 'long scale' curvature where the adjacency restriction is removed. In this talk, I will present recent results which show that Ollivier-Ricci curvature as a function of idleness is concave and piecewise linear with at most 3 linear parts. The result can then be applied to compute the curvature in the Cartesian product of two regular graphs in terms of the curvature in its factors. I will also illustrate the benefits of studying this long scale curvature. This is a joint work with David Cushing.