All Scale-free networks are sparse

We study the graphicality of power-law distributed degree sequences, showing that the fraction of graphical sequences undergoes two sharp transitions at the values 0 and 2 of the power-law exponent. We characterize these transitions as first-order, and provide an analytic explanation of their nature. Further numerical calculations, based on extreme value arguments, verify this treatment, and introduce a method to determine transition points for any given degree distribution. Our results reveal a fundamental reason why scale-free networks with no constraints on minimum and maximum degree must be sparse for positive power-law exponents, and dense otherwise.