There has been very interesting work on defining curvature on a discrete graph. For example:
Knill, Oliver. "A graph theoretical Gauss-Bonnet-Chern theorem." arXiv:1111.5395 (2011). (Abstract.)
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Another discrete curvature is the Ollivier-Ricci curvature:
Bauer, Frank, Jürgen Jost, and Shiping Liu. "Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator." arXiv:1105.3803 (2011). (Abstract.)