On index expectation and curvature for networks

Oliver Knill

On index expectation and curvature for networks

Differential Geometry 2012-02-22 v1 Discrete Mathematics General Topology

Authors: Oliver Knill

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.

Keywords

index expectation curvature finite simple graphs Gauss-Bonnet theorem Poincare-Hopf theorem Euler characteristic

Cite

@article{arxiv.1202.4514,
  title  = {On index expectation and curvature for networks},
  author = {Oliver Knill},
  journal= {arXiv preprint arXiv:1202.4514},
  year   = {2012}
}