论文标题
复杂网络的简单微分几何形状
A Simple Differential Geometry for Complex Networks
论文作者
论文摘要
我们介绍了针对网络及其较高维度的截面，RICCI和标量曲率的新定义，这些定义来自两个经典的曲率概念，用于一般度量空间中的曲线，即Menger Curvature和Haantjes曲率。这些曲率适用于未加权或加权，无向或有向网络，并且比其他网络曲线更直观，更易于计算。特别是，基于Haantjes定义为测量曲率的解释的提议曲率使我们能够给出经典的局部高斯河网定理的网络类似物。此外，我们为Haantjes曲率提出了更简单，更直观的代理，该代理可以在大型网络中更快，更轻松地计算。此外，我们还研究了所提出的RICCI曲线的嵌入性能。最后，我们还研究了本文中介绍的曲线的行为，这些曲线具有更既定的RICCI曲率和其他广泛使用的网络测量的概念。
We introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks, and are 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 allow us to give a network analogue of the classical local Gauss-Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behaviour, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely-used network measures.
