A Simple Differential Geometry for Networks and its Generalizations

TL;DR

This paper introduces new, intuitive, and computationally efficient curvature measures for networks based on classical differential geometry concepts, applicable to various types of networks and higher-dimensional structures.

Contribution

It proposes novel definitions of sectional, Ricci, and scalar curvature for networks, inspired by classical curvature notions, and simplifies their computation for large-scale applications.

Findings

New curvature measures are more intuitive and easier to compute.

Applicable to weighted, unweighted, directed, and undirected networks.

Provides flexible curvature definitions based on a discrete Gauss-Bonnet theorem.

Abstract

Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, directed or undirected networks, are far more intuitive and easier to compute, than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible. We also propose even simpler and more intuitive substitutes of the Haantjes curvature, that allow for even faster and easier computations in large-scale networks.