On index expectation and curvature for networks

TL;DR

This paper establishes a mathematical link between the expected index function and curvature in finite simple graphs, connecting classical topological invariants through probabilistic and geometric perspectives.

Contribution

It proves that the expected value of the index function equals the curvature at each vertex, bridging Gauss-Bonnet and Poincare-Hopf theorems in graph theory.

Findings

Expected index equals curvature at each vertex

Links Gauss-Bonnet sum and Poincare-Hopf sum for graphs

Provides a probabilistic interpretation of geometric invariants

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.