Somewhat related, but my old PhD supervisor had a very interesting paper on extending some ideas from Riemannian geometry to graphs. http://arxiv.org/abs/1011.5

Somewhat related, but my old PhD supervisor had a very interesting paper on extending some ideas from Riemannian geometry to graphs. http://arxiv.org/abs/1011.5898 http://arxiv.org/abs/1011.5898