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Abstract: We develop a geometric framework to study the structure and function ofcomplex networks. We assume that hyperbolic geometry underlies these networks,and we show that with this assumption, heterogeneous degree distributions andstrong clustering in complex networks emerge naturally as simple reflections ofthe negative curvature and metric property of the underlying hyperbolicgeometry. Conversely, we show that if a network has some metric structure, andif the network degree distribution is heterogeneous, then the network has aneffective hyperbolic geometry underneath. We then establish a mapping betweenour geometric framework and statistical mechanics of complex networks. Thismapping interprets edges in a network as non-interacting fermions whoseenergies are hyperbolic distances between nodes, while the auxiliary fieldscoupled to edges are linear functions of these energies or distances. Thegeometric network ensemble subsumes the standard configuration model andclassical random graphs as two limiting cases with degenerate geometricstructures. Finally, we show that targeted transport processes without globaltopology knowledge, made possible by our geometric framework, are maximallyefficient, according to all efficiency measures, in networks with strongestheterogeneity and clustering, and that this efficiency is remarkably robustwith respect to even catastrophic disturbances and damages to the networkstructure.

Autor: Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, Marian Boguna

Fuente: https://arxiv.org/

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