Fundamental structural constraint of random scale-free networks.

We study the structural constraint of random scale-free networks that determines possible combinations of the degree exponent γ and the upper cutoff k(c) in the thermodynamic limit. We employ the framework of graphicality transitions proposed by Del Genio and co-workers [Phys. Rev. Lett. 107, 178701 (2011)], while making it more rigorous and applicable to general values of k(c). Using the graphicality criterion, we show that the upper cutoff must be lower than k(c)∼N(1/γ) for γ<2, whereas any upper cutoff is allowed for γ>2. This result is also numerically verified by both the random and deterministic sampling of degree sequences.