Discrete Ollivier-Ricci curvature
Zohreh Fathi, Sajjad Lakzian

TL;DR
This paper extends the theory of Ollivier-Ricci curvature to both continuous and discrete settings on weighted graphs, establishing new properties, existence criteria, and applications including curvature flows and bounds in convex geometry.
Contribution
It generalizes the definitions of Ollivier-Ricci curvature to broader classes of random walks and distances, providing new properties and a limit-free formulation.
Findings
Established Lipschitz continuity and concavity of curvatures.
Provided existence and uniqueness results for curvature flows.
Derived a sharp upper bound on vertices of convex polytopes.
Abstract
We analyze both continuous and discrete-time Ollivier-Ricci curvatures of locally-finite weighted graphs equipped with a given distance "" (w.r.t. which is metrically complete) and for general random walks. We show the continuous-time Ollivier-Ricci curvature is well-defined for a large class of Markovian and non-Markovian random walks and provide a criterion for existence of continuous-time Ollivier-Ricci curvature; the said results generalize the previous rather limited constructions in the literature. In addition, important properties of both discrete-time and continuous-time Ollivier-Ricci curvatures are obtained including -- to name a few -- Lipschitz continuity, concavity properties, piece-wise regularity (piece-wise linearity in the case of linear walks) for the discrete-time Ollivier-Ricci as well as Lipschitz continuity and limit-free formulation for the…
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Taxonomy
TopicsPoint processes and geometric inequalities · Geometric Analysis and Curvature Flows · Markov Chains and Monte Carlo Methods
