Hyperbolic Geometry of Complex Networks

TL;DR

This paper proposes a hyperbolic geometric framework for understanding complex networks, explaining their heterogeneity and clustering, and demonstrating how this geometry enables efficient transport processes even under damage.

Contribution

It introduces a novel hyperbolic geometric model for complex networks, linking geometry with network properties and dynamics, and showing its implications for network efficiency and robustness.

Findings

Heterogeneous degree distributions and clustering emerge naturally from hyperbolic geometry.

Networks with hyperbolic structure enable maximally efficient transport without global topology knowledge.

The geometric framework unifies various network models and explains robustness against damages.

Abstract

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.