Generic scale of the "scale-free" growing networks

Details Generic scale of the "scale-free" growing networks Generic scale of the "scale-free" growing networks

2000-11-07

arxiv.org/abs/cond-mat/0011115v1

Phys. Rev. E 63, 062101 (2001)

Physics

Condensed Matter

Statistical Mechanics

4 pages revtex, 3 figures

Scientific paper

10.1103/PhysRevE.63.062101

We show that the connectivity distributions $P(k,t)$ of scale-free growing networks ($t$ is the network size) have the generic scale -- the cut-off at $k_{cut} \sim t^\beta$. The scaling exponent $\beta$ is related to the exponent $\gamma$ of the connectivity distribution, $\beta=1/(\gamma-1)$. We propose the simplest model of scale-free growing networks and obtain the exact form of its connectivity distribution for any size of the network. We demonstrate that the trace of the initial conditions -- a hump at $k_h \sim k_{cut} \sim t^\beta$ -- may be found for any network size. We also show that there exists a natural boundary for the observation of the scale-free networks and explain why so few scale-free networks are observed in Nature.

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