Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Q3 Mathematics
E. Jonckheere, P. Lohsoonthorn, F. Ariaei
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引用次数: 23

Abstract

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.
网络图曲率计算的缩放Gromov四点条件
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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