Size-dependent degree distribution of a scale-free growing network

We propose the simplest model of scale-free growing networks and obtain the exact form of its degree distribution for any size of the network (degree is a number of connections of a node). We demonstrate that a trace of initial conditions - a hump near cutoff of the degree distribution at k(cut) approximately t(beta)--may be found for any network size. Here beta=1/(gamma-1), where gamma is the exponent of the degree distribution of the network. These size effects implement a natural boundary for the observation of the scale-free networks.