Project Euclid

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.