On index expectation and curvature for networks

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.

References in corpus (2)

• A graph theoretical Gauss-Bonnet-Chern Theorem
• A graph theoretical Poincare-Hopf Theorem

Cited by in corpus (30)

• A Survey on Centrality Metrics and Their Implications in Network Resilience
• Complex network view of evolving manifolds
• Ollivier-Ricci curvature convergence in random geometric graphs
• The McKean-Singer Formula in Graph Theory
• The Dirac operator of a graph
• An index formula for simple graphs
• The Lusternik-Schnirelmann theorem for graphs
• Gauss-Bonnet for multi-linear valuations
• A notion of graph homeomorphism
• Coloring graphs using topology
• Curvature from Graph Colorings
• An integrable evolution equation in geometry
• Classical mathematical structures within topological graph theory
• The amazing world of simplicial complexes
• The strong ring of simplicial complexes
• Isospectral deformations of the Dirac operator
• A Sard theorem for graph theory
• Dehn-Sommerville from Gauss-Bonnet
• A Reeb sphere theorem in graph theory
• Poincare Hopf for vector fields on graphs
• The Kuenneth formula for graphs
• Integral geometric Hopf conjectures
• On Helmholtz free energy for finite abstract simplicial complexes
• If Archimedes would have known functions
• A Brouwer fixed point theorem for graph endomorphisms
• On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes
• Constant index expectation curvature for graphs or Riemannian manifolds
• More on Poincare-Hopf and Gauss-Bonnet
• The Curvature of Graph Products
• Graph complements of circular graphs