Ricci curvature as a tool for analyzing networks and other complexes

In network science, concepts and quantities are often introduced in a rather ad hoc manner. I shall use analogies with another geometric discipline, Riemannian geometry, to develop network analysis tools in a systematic manner.

At the heart of Riemannian geometry are notions of curvature. In particular, in recent decades,fundamental aspects of Ricci curvature have been discovered and explored. In fact, there are two aspects of Ricci curvature that turn out to be useful in metric geometry. One is about the average dispersion of geodesics. The other is about comparing the distance between two points with the transportation cost between their local neighborhoods.

As I shall explain, both are useful for the theoretical investigation of networks and other complexes as well as for the analysis of empirical network data.