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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. {\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 $k_c N^{1/γ}$ 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.