一种简单的网络Differential Geometry及其推广
A Simple Differential Geometry for Networks and its Generalizations
摘要：
基于一般度量空间中曲线曲率的两个经典概念，即Menger曲率和Haantjes曲率，我们引入了网络及其高维对应的截面曲率、Ricci曲率和数量曲率的新定义.这些适用于加权和非加权、有向或无向网络的新类型的曲率比其他网络曲率更直观、更容易计算。特别是，建议的曲率的基础上的解释Haantjes定义为测地曲率，并通过拟合离散高斯-邦纳定理，是相当灵活的。我们还提出了更简单，更直观的Haantjes曲率的替代品，允许在大规模网络中更快，更容易的计算。
英文摘要：
Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, directed or undirected networks, are far more intuitive and easier to compute, than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible. We also propose even simpler and more intuitive substitutes of the Haantjes curvature, that allow for even faster and easier computations in large-scale networks.
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