Fundamental structural constraint of random scale-free networks

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Random scale-free networks are fundamental models in network science.
• Understanding their structural constraints is crucial for predicting network properties.
• Previous studies explored graphicality but lacked rigor for general upper cutoffs.

Purpose of the Study:

• To rigorously determine the structural constraints on random scale-free networks.
• To establish the relationship between the degree exponent (γ) and the upper cutoff (k(c)).
• To investigate these constraints in the thermodynamic limit.

Main Methods:

• Employing the graphicality transitions framework.
• Developing a more rigorous application for general upper cutoff values (k(c)).
• Numerical verification using random and deterministic sampling of degree sequences.

Main Results:

• Established a critical upper bound for the degree cutoff: k(c) must be less than approximately N(1/γ) when γ < 2.
• Demonstrated that for degree exponents γ > 2, any upper cutoff is structurally permissible.
• Confirmed findings through extensive numerical simulations.

Conclusions:

• The degree exponent significantly dictates the permissible range of upper cutoffs in random scale-free networks.
• The findings provide a more precise understanding of network structural limitations.
• This work refines the graphicality criterion for broader applicability in network analysis.