Approaching the thermodynamic limit in equilibrated scale-free networks

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 the position of the cutoff in the degree distribution, k(cutoff), scales with N in a different way than predicted for N ->infinity; that is, subleading corrections to the scaling k(cutoff)similar to N-alpha are strong even for networks of order N similar to 10(9) nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution pi(k)similar to k(-3). On the other hand, the distribution of the maximal degree k(max) may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that k(max)similar to N-alpha' with an exponent alpha(')=min[alpha,1/(gamma-1)], where gamma is the exponent in the power law pi(k)similar to k(-gamma). We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.