General formulation of long-range degree correlations in complex networks

Area of Science:

• Network Science
• Statistical Physics
• Data Analysis

Background:

• Understanding node interdependencies is crucial in complex networks.
• Nearest-neighbor degree correlations are well-studied, but long-range correlations remain less understood.
• Finite-size effects can obscure intrinsic network properties.

Purpose of the Study:

• To develop a general framework for analyzing long-range degree correlations in complex networks.
• To introduce probability distributions that fully describe these correlations based on node degrees and path lengths.
• To differentiate intrinsic long-range correlations from those arising from finite-size effects.

Main Methods:

• Introduction of one joint and four conditional probability distributions.
• Derivation of general relations among these distributions.
• Analytical evaluation of distributions for random networks using a mean-field approximation.

Main Results:

• A comprehensive method to quantify long-range degree correlations is presented.
• The framework clarifies the relationship between long-range and nearest-neighbor correlations.
• Distributions are identified as meaningful only in finite-size networks, distinguishing them from nearest-neighbor correlations.
• A baseline for intrinsic long-range correlations is established using random network models.

Conclusions:

• The proposed framework effectively analyzes long-range degree correlations in complex networks.
• The methods allow for the identification of intrinsic long-range correlations, independent of network size.
• The approach is validated through application to real-world complex networks.