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](https://arxiv.org/abs/1111.5395).)
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[![enter image description here][1]][1]
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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](https://arxiv.org/abs/1105.3803).)

  [1]: https://i.sstatic.net/2xFHD.png