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2011 Scaled Gromov Four-Point Condition for Network Graph Curvature Computation
Edmond Jonckheere, Poonsuk Lohsoonthorn, Fariba Ariaei
Internet Math. 7(3): 137-177 (2011).

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.

Citation

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Edmond Jonckheere. Poonsuk Lohsoonthorn. Fariba Ariaei. "Scaled Gromov Four-Point Condition for Network Graph Curvature Computation." Internet Math. 7 (3) 137 - 177, 2011.

Information

Published: 2011
First available in Project Euclid: 13 October 2011

zbMATH: 07282357
MathSciNet: MR2837770

Rights: Copyright © 2011 A K Peters, Ltd.

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Vol.7 • No. 3 • 2011
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