网络图曲率计算的缩放Gromov四点条件

Q3 Mathematics Internet Mathematics Pub Date : 2011-08-30 DOI:10.1080/15427951.2011.601233
E. Jonckheere, P. Lohsoonthorn, F. Ariaei
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引用次数: 23

摘要

摘要本文将最初为薄三角形条件(TTC)而发展的尺度Gromov双曲图的概念推广到计算简化但不太直观的四点条件(FPC)。最初的动机是,对于一个大而有限的网络图来说,要享受负弯曲黎曼流形的一些典型性质,测量三角形的厚度的delta必须低于整个图的某个阈值。本文给出了4点函数的各种缩放方法,并给出了缩放后的4点函数在不同空间中的上界。TTC相对于FPC的一个重要的理论优势是,后者允许对托勒密空间进行格罗莫夫式的表征。作为一种主要的网络应用,无标度网络倾向于缩放的Gromov双曲,而小世界网络则倾向于缩放的正曲线。
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Scaled Gromov Four-Point Condition for Network Graph Curvature Computation
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.
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Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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