Hyperbolic geometry of complex networks
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 noninteracting 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.
 Publication:

Physical Review E
 Pub Date:
 September 2010
 DOI:
 10.1103/PhysRevE.82.036106
 arXiv:
 arXiv:1006.5169
 Bibcode:
 2010PhRvE..82c6106K
 Keywords:

 89.75.Hc;
 02.40.k;
 67.85.Lm;
 89.75.Fb;
 Networks and genealogical trees;
 Geometry differential geometry and topology;
 Degenerate Fermi gases;
 Structures and organization in complex systems;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Computer Science  Networking and Internet Architecture;
 Physics  Physics and Society
 EPrint:
 Phys. Rev. E 82, 036106 (2010)