TITLE: Curvature of a finite metric space
QUESTION [18 upvotes]: I am sorry to ask a very vague question, but:

What are good ways to define the curvature of a finite metric space?

The best way I can think of is: the curvature of a finite metric space $M$
is the infimum of the real $k$ such that there is a geodesic metric space $X$ which is $Cat(k)$ and $M$ embeds isometrically in $X$. However, I don't know how to compute this curvature form the distance matrice... Is there a way? Moreover
is this notion defined somewhere, and has its properties being studied? 
I would also be open to any other sensible definition, or reference discussing this question. Thanks.
PS: I ask this question for experimental purposes. I have a metric set of data I would like to define and compute the curvature of.

REPLY [5 votes]: If something more analogous to Ricci curvature than to sectional curvature would interest you, then there has been some work done on the Ricci curvature of discrete spaces. I don't know much about it, but you can find some articles on Yann Olivier's website, and Jürgen Jost and Christian Leonard gave a minicourse on it at the IHS last spring. These might be some starting points for you.