Internet Mathematics

Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Edmond Jonckheere, Poonsuk Lohsoonthorn, and Fariba Ariaei

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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

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

First available in Project Euclid: 13 October 2011

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Jonckheere, Edmond; Lohsoonthorn, Poonsuk; Ariaei, Fariba. Scaled Gromov Four-Point Condition for Network Graph Curvature Computation. Internet Math. 7 (2011), no. 3, 137--177.

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