  We discuss how various models of scale-free complex networks approach their
limiting properties when the size N of the network grows. We focus mainly on
equilibrated networks and their finite-size degree distributions. Our results
show that subleading corrections to the scaling of the position of the cutoff
are strong even for networks of order 10^9 nodes. We observe also a logarithmic
correction to the scaling for degenerated graphs when the degree distribution
follows a power law k^(-3). We study also the distribution of the maximal
degree and show that it may have a different scaling than the cutoff and,
moreover, it approaches the thermodynamic limit much faster. We present also
some results on the cutoff function and the distribution of the maximal degree
in equilibrated networks.
