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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).


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.


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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.


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

zbMATH: 07282357
MathSciNet: MR2837770

Rights: Copyright © 2011 A K Peters, Ltd.


Vol.7 • No. 3 • 2011
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