Ordinal Preferential Attachment: A Self-Organizing Principle Generating Dense Scale-Free Networks

Area of Science:

• Network Science
• Complex Systems Analysis
• Statistical Physics

Background:

• Real-world complex systems are often modeled as scale-free networks with power-law degree distributions and growing average degrees.
• Generating dense scale-free networks has been challenging, often requiring external cutoffs for the scale-free regime.
• Understanding the self-organizing processes that lead to dense scale-free networks is crucial for network modeling.

Purpose of the Study:

• To propose a novel growing network model capable of producing dense scale-free networks with dynamically generated cutoffs.
• To investigate a weak preferential attachment mechanism based on node degree order relations.
• To analytically and numerically study the emergent properties of these networks.

Main Methods:

• Development of a new growing network model with a link formation rule based on relative node degrees.
• Analytical derivation of network properties including degree distribution, degree correlation, and local clustering coefficient.
• Comparison of analytical predictions with results from numerical simulations.

Main Results:

• The proposed model successfully generates scale-free networks with arbitrary scaling exponents greater than 1.
• Networks produced by the model exhibit density when scaling exponents are 2 or less.
• Analytical calculations for network properties show strong agreement with numerical simulations.
• The model demonstrates that both sparse and dense scale-free networks can emerge from the same self-organizing process.

Conclusions:

• A new, self-organizing growing network model can generate dense scale-free networks without external cutoffs.
• The model's preferential attachment rule, based on degree order, is key to producing tunable network densities.
• This work provides a unified framework for understanding the emergence of both sparse and dense scale-free networks.