Hyperbolic geometry of complex networks

作     者：Dmitri Krioukov Fragkiskos Papadopoulos Maksim Kitsak Amin Vahdat Marián Boguñá 

作者机构：Cooperative Association for Internet Data Analysis (CAIDA) University of California–San Diego (UCSD) La Jolla California 92093 USA Department of Electrical and Computer Engineering University of Cyprus Kallipoleos 75 Nicosia 1678 Cyprus Department of Computer Science and Engineering University of California–San Diego (UCSD) La Jolla California 92093 USA Departament de Física Fonamental Universitat de Barcelona Martí i Franquès 1 08028 Barcelona Spain 

出 版 物：《Physical Review E》 (物理学评论E辑：统计、非线性和软体物理学)

年 卷 期：2010年第82卷第3期

页      面：036106-036106页

核心收录：

学科分类：07[理学] 070203[理学-原子与分子物理] 0702[理学-物理学] 

基　　金：National Science Foundation, NSF Directorate for Computer and Information Science and Engineering, CISE, (0434996, 0722070, 0964236) 

主　　题：Complex networks 

摘      要：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.