Fundamental Structural Constraint of Random Scale-Free Networks

TL;DR

This paper establishes a fundamental structural constraint on the degree cutoff in random scale-free networks, revealing how the cutoff scales with network size depending on the degree exponent.

Contribution

It provides a more rigorous and general framework for the graphicality transition in scale-free networks, clarifying the relationship between degree exponent and cutoff constraints.

Findings

Upper cutoff must be below $k_c N^{1/ ext{gamma}}$ for $ ext{gamma} < 2$

Any upper cutoff is permissible for $ ext{gamma} > 2$

Results confirmed by numerical sampling of degree sequences

Abstract

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 $k_c N^{1/\gamma}$ 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.