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 $\gamma$ 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. {\bf 107}, 178701 (2011)], while making it more rigorous and applicable to general values of kc. Using the graphicality criterion, we show that the upper cutoff must be lower than $k_c{N}^{1\mathrm{/}\gamma }$ for $\gamma <2$, whereas any upper cutoff is allowed for $\gamma >2$. This result is also numerically verified by both the random and deterministic sampling of degree sequences.