Curvature and temperature of complex networks

TL;DR

This paper demonstrates that the heterogeneity and clustering in complex networks like the Internet can be explained by hyperbolic geometry, with implications for improved routing strategies based on local information.

Contribution

It introduces a hyperbolic geometric model linking network topology features to curvature and temperature, validated by Internet embedding.

Findings

Heterogeneous degree distributions arise from hyperbolic space expansion.

Clustering correlates with temperature in the model.

Internet embedding aligns with the hyperbolic model.

Abstract

We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the Internet into the hyperbolic plane, and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of Internet routing.