## Internet Mathematics

- Internet Math.
- Volume 7, Number 3 (2011), 137-177.

### Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Edmond Jonckheere, Poonsuk Lohsoonthorn, and Fariba Ariaei

#### Abstract

In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the thin triangle condition (TTC), to the computationally simplified, but less intuitive, four-point condition (FPC). The original motivation was that for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta measuring the thinness of a triangle scaled by its diameter must be below a certain threshold all across the graph. Here we develop various ways of scaling the 4-point delta, and develop upper bounds for the scaled 4-point delta in various spaces. A significant theoretical advantage of the TTC over the FPC is that the latter allows for a Gromov-like characterization of Ptolemaic spaces. As a major network application, it is shown that scale-free networks tend to be scaled Gromov hyperbolic, while small-world networks are rather scaled positively curved.

#### Article information

**Source**

Internet Math., Volume 7, Number 3 (2011), 137-177.

**Dates**

First available in Project Euclid: 13 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1318514901

**Mathematical Reviews number (MathSciNet)**

MR2837770

#### Citation

Jonckheere, Edmond; Lohsoonthorn, Poonsuk; Ariaei, Fariba. Scaled Gromov Four-Point Condition for Network Graph Curvature Computation. Internet Math. 7 (2011), no. 3, 137--177. https://projecteuclid.org/euclid.im/1318514901