Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.

本文利用 Ollivier 和 Lin 的离散 Ricci 曲率分析了互联网的曲率，发现 Ricci 曲率的分布广泛，表明网络拓扑结构是不均匀的，与节点度和聚类系数等本地量和介数中心性和网络连通性等全局量以及地理距离等辅助属性具有有趣的联系，这些观察结果丰富了复杂网络理论中的几何结构。

互联网拓扑的里奇曲率

Ricci Curvature of the Internet Topology

The description of the Internet topology is an important open problem,
recently tackled with the introduction of scale-free networks. In this paper we
focus on the topological and dynamical properties of real Internet maps in a
three years time interval. We study higher order correlation functions as well
as the dynamics of several quantities. We find that the Internet is
characterized by non-trivial correlations among nodes and different dynamical
regimes. We point out the importance of node hierarchy and aging in the
Internet structure and growth. Our results provide hints towards the realistic
modeling of the Internet evolution.

本文旨在研究真实互联网地图的拓扑和动力学性质，探讨节点层次结构和老化在互联网结构和增长中的重要性，并提供了互联网演化的现实建模线索。