  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.
