Fundamental Structural Constraint of Random Scale-Free Networks
This work addresses a fundamental problem in network theory for researchers studying scale-free networks, providing incremental improvements by making existing graphicality criteria more rigorous and applicable.
The study tackled the structural constraints of random scale-free networks by determining possible combinations of the degree exponent γ and upper cutoff kc in the thermodynamic limit, showing that for γ < 2, the upper cutoff must be lower than kc N^(1/γ), while for γ > 2, any upper cutoff is allowed, with numerical verification provided.
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.