Transition from small to large world in growing networksS. N. Dorogovtsev1, 2, P. L. Krapivsky3 and J. F. F. Mendes1
1 Departamento de Física da Universidade de Aveiro - 3810-193 Aveiro, Portugal
2 A. F. Ioffe Physico-Technical Institute - 194021 St. Petersburg, Russia
3 Department of Physics and Center for Molecular Cybernetics, Boston University - Boston, MA 02215, USA
received 19 September 2007; accepted in final form 26 November 2007; published February 2008
published online 31 December 2007
We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite or finite dimensional depending on whether the attachment probability decays slower or faster than (age)-1. The network becomes one dimensional when the attachment probability decays faster than (age)-2. We describe structural characteristics of these phases and transitions between them.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
89.75.Fb - Structures and organization in complex systems.
89.75.Hc - Networks and genealogical trees.
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