Universal scaling of distances in complex networks

Area of Science:

• Network Science
• Graph Theory
• Complex Systems Analysis

Background:

• Real-world networks exhibit diverse structures and properties.
• Understanding distance metrics in networks is crucial for analyzing information flow and connectivity.
• Previous studies have explored network properties but lacked a universal scaling law for distances.

Purpose of the Study:

• To identify and describe universal scaling laws for distances between network vertices.
• To propose a theoretical framework explaining the observed scaling phenomenon.
• To investigate the relationship between scaling parameters and network characteristics like node degree and clustering.

Main Methods:

• Analysis of distance scaling in various network models (Erdos-Rényi, Barabási-Albert) and real-world networks (collaboration, biological, Internet, transport).
• Derivation of a mathematical relationship for mean distance based on node degrees: (l(ij)) = A - B log(k(i)k(j)).
• Development of a simple theory to explain the origin of this scaling.

Main Results:

• Observed universal scaling of distances across a wide range of network types.
• Established a mean distance formula dependent on the product of node degrees.
• Demonstrated that scaling parameters (A, B) are determined by nearest neighbor node degrees and clustering coefficients.

Conclusions:

• A universal scaling law for network distances has been identified, applicable across diverse systems.
• The proposed theory provides a simple explanation for this fundamental network property.
• Node degree and clustering are key factors influencing inter-node distances in complex networks.