  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.
