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On index expectation and curvature for networks (1202.4514v1)
Published 21 Feb 2012 in math.DG, cs.DM, and math.GN
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.