Fundamental Structural Constraint of Scale-Free Networks

We study the structural constraint of scale-free networks that determines possible combinations of the degree exponent $\gamma$ and the upper cutoff $k_c$ in the thermodynamic limit. In order to obtain the fundamental constraint that is independent of the mechanism for network generation, we employ the framework of graphicality transition proposed by [Del Genio {\em et al.}, Phys. Rev. Lett. {\bf 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 $k_c \sim N^{1/\gamma}$ for $\gamma < 2$, whereas any upper cutoff is allowed for $\gamma > 2$. This result is also numerically verified by both random and deterministic sampling of degree sequences.