  We analyze both continuous and discrete-time Ollivier-Ricci curvatures of
locally-finite weighted graphs $\G$ equipped with a given distance "$\dist$"
(w.r.t. which $\G$ 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 continuous-time
Ollivier-Ricci. these properties were previously known only for very specific
distances and very specific random walks. As an application of Lipschitz
continuity, we obtain existence and uniqueness of generalized continuous-time
Ollivier-Ricci curvature flows. Along the way, we obtain -- by optimizing
McMullen's upper bounds -- a sharp upper bound estimate on the number of
vertices of a convex polytope in terms of number of its facets and the ambient
dimension, which might be of independent interest in convex geometry. The said
upper bound allows us to bound the number of polynomial pieces of the
discrete-time Ollivier-Ricci curvature as a function of time in the
time-polynomial random walk. The limit-free formulation we establish allows us
to define an operator theoretic Ollivier-Ricci curvature which is a non-linear
concave functional on suitable operator spaces.
