Network Rewiring in the r-K Plane.

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Scale-free networks are ubiquitous in nature and technology.
• Network correlations, such as assortativity and nearest neighbor degree, significantly impact network properties.
• Controlling these correlations in network models is crucial for understanding and designing complex systems.

Purpose of the Study:

• To introduce a new rewiring algorithm for generating correlated scale-free networks within the configuration model.
• To enable simultaneous tuning of the Newman assortativity coefficient (r) and the average nearest neighbor degree (K).
• To explore the limits of network assortativity, including previously unconsidered cases with small minimum degrees.

Main Methods:

• A novel rewiring algorithm based on the Metropolis acceptance probability, incorporating a variable temperature (T).
• Calculation of local variations in assortativity (Δr) and nearest neighbor degree (ΔK) at each rewiring step.
• Analysis of rewiring trajectories in the r-K plane and monitoring of network properties like giant component size and entropy.

Main Results:

• A general relation between Δr and ΔK was derived, connecting two distinct topological measures.
• The algorithm successfully generates networks across a wide range of assortativity coefficients (-1 ≤ r ≤ 1) and nearest neighbor degrees (K ≥ 〈k〉).
• The average number of second neighbors (z̄₂,B) was proven constant for Markovian networks, independent of correlations.

Conclusions:

• The developed rewiring algorithm provides a powerful tool for constructing and analyzing correlated scale-free networks.
• The study establishes a fundamental link between network assortativity and nearest neighbor degree.
• The findings offer new insights into network connectivity and the behavior of complex systems with varying degrees of correlation.