Graphicality of random scale-free networks with general degree cutoffs

We study graphicality of random scale-free networks with arbitrary degree cutoffs in the thermodynamic limit, which refers to realizability of degree sequences randomly generated with the degree exponent $\gamma$ and the upper degree cutoff $k_c$ as the number of nodes $N$ goes to infinity. While a recent study\footnote{C. I. Del Genio, T. Gross, and K. E. Bassler, Phys. Rev. Lett. {\bf 107}, 178701 (2011).} found that only degree sequences with $\gamma > 2$ or $\gamma < 0$ are graphical if $k_c = N-1$ using the graphicality criterion proved by Erd\"os and Gallai,\footnote{P. Erd\"os and T. Gallai, Matematikai lapok {\bf 11}, 264 (1960).} we generalize the study to different upper cutoffs. To ensure graphicality of degree sequences, it is found 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 is also numerically verified, using both random and deterministic sampling of degree sequences. Our result can be interpreted as giving a fundamental constraint on the structure of random scale-free networks.