All scale-free networks are sparse
  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. 
Charo I. Del Genio, Thilo Gross and Kevin E. Bassler
physics.soc-ph cond-mat.stat-mech cs.SI