All scale-free networks are sparse

Phys Rev Lett. 2011 Oct 21;107(17):178701. doi: 10.1103/PhysRevLett.107.178701. Epub 2011 Oct 17.


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.