Evolution of networks with preferential linking

Georgy Karev
NCBI, NIH

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     Last modified: April 25, 2006

Abstract
We suggest a modified preferential linking procedure of networks growing. Our first model is a model of a growing net with innovation and elimination of nodes and innovation of directed links; on the contrary with “classical” network models, the rates of innovation of links and nodes are not fixed, but depend on the net “request” or “capability” to attach new links according to the general principal of preferential linking. We study all possible stationary connectivity distributions of a wide class of these networks depending on the rates of innovation and elimination. The outcome of the first construction, the equilibrium network with a fixed number of nodes, is the initial object of the second model. We add the process of deletion of links and explore the final connectivity distributions depending on the rates of “birth” and “death” of links. We show that this distribution follow the power law if and only if these rates are asymptotically equal up to the second order. In the class of polynomial and rational models we can trace the evolution of the stable connectivity distribution (depending on the model parameters) from the Poisson one (as in random graphs) to the scale-free networks passing the intermediate cases of logarithmic and poly-logarithmic distributions.