Following up on Ryan Budney's comment, here is Forman's definition of the combinatorial Ricci curvature of an edge (where {vertex, edge, face} = {0-cell, 1-cell, 2-cell}):

---

![RicciCurvature][1]

---

In words: The curvature of an edge $e$ of the manifold $\mathbb{M}$ is the number of vertices that are in the boundary of $e$, plus the number of faces whose boundaries include $e$, minus the number of edges that share *only* a vertex (and not a face) with $e$, minus the number of edges that share a face *but not a vertex* with $e$.

Note that the curvature of an edge $e$ under this definition is only dependent upon the 2-skeleton of $\mathbb{M}$. For a cube, Ric$(e)=2+2-0-2=2$:
 

The above definition is taken from Paul McCormick's thesis, "Combinatorial Curvature of Cellular Complexes" (PDF download), which in turn relies on Forman's paper, "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature" (Springer link).