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Area of Science:

  • Network Science
  • Graph Theory
  • Complex Systems Analysis

Background:

  • Understanding local network geometry is crucial for global topology.
  • Current methods for analyzing complex real-world networks have limitations.
  • The 'degree difference' (DD) between connected vertices is an underexplored measure.

Purpose of the Study:

  • To analyze the 'degree difference' (DD) as a measure of local network geometry.
  • To explore the relationship between DD and global network assortativity.
  • To demonstrate DD's ability to reveal structural properties missed by other measures.

Main Methods:

  • Formal and conceptual analysis of DD and its connection to assortativity.
  • Evaluation of DD distribution across synthetic and real-world networks.
  • Comparison of DD with degree sequence and global assortativity for network classification.

Main Results:

  • DD is the fundamental unit of assortativity and reveals structural heterogeneity in mixing patterns.
  • DD distribution serves as a network signature, distinguishing networks undetectable by other metrics.
  • DD indicates the topological robustness of scale-free networks.

Conclusions:

  • DD is a simple, effective local measure for understanding complex network structures.
  • DD provides unique insights into network heterogeneity and robustness.
  • This measure enhances the characterization of real-world networks.