Ricci Curvature of the Internet Topology

TL;DR

This paper investigates the discrete Ricci curvature of Internet topologies, revealing its distribution, correlations with network measures, and implications for understanding the network's geometric structure.

Contribution

It introduces the analysis of Ricci curvature in Internet topology, connecting local and global network properties and highlighting non-homogeneity in curvature distribution.

Findings

Ricci curvature distribution is broad, indicating non-homogeneity.

Ricci curvature correlates with node degree, clustering, and betweenness centrality.

The study links geometric curvature measures to network robustness and structure.

Abstract

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.