We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.
Figure 1: Graphicality of the degree maximizing sequence (\(N = 10^6\)) given by Eq. 11 for scale-free distributions vs.
exponent \(\gamma\). The graphicality transitions points are correctly identified.