On index expectation and curvature for networks

Oliver Knill

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arXiv · 1202.4514

On index expectation and curvature for networks

Oliver Knill

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Abstract

We prove that the expectation value of the index function i(x) over a probability space of injective function f on any finite simple graph G=(V,E) is equal to the curvature K(x) at the vertex x. This result complements and links Gauss-Bonnet sum K(x) = chi(G) and Poincare-Hopf sum i(x) = chi(G) which both hold for arbitrary finite simple graphs.

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Oliver Knill. 2012-02-21. On index expectation and curvature for networks. https://arxiv.org/abs/1202.4514

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