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This study introduces the Graphlet Correlation Distance with 11 orbits (GCD11) for analyzing small, dense graphs. A new statistical test using GCD11 challenges standard assumptions in fisheries, highlighting challenges in comparing real-world small graphs to models.

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Area of Science:

  • Graph theory
  • Network analysis
  • Statistical modeling

Background:

  • Graph models are standard for representing relationships between entities.
  • Existing methods like Graphlet Correlation Distance (GCD) are well-established for large graphs.
  • Small graphs with high connection densities are less explored but relevant in sociology, ecology, and fisheries.

Purpose of the Study:

  • To investigate the distinguishability of common graph models (Erdős-Rényi, Scale-Free, Small-World, Geometric) using a specific GCD measure (GCD11).
  • To develop and apply a randomization statistical test based on GCD11 for comparing empirical graphs against null models.
  • To analyze pairwise proximity in a fishing vessel case study.

Main Methods:

  • Numerical experiments to evaluate GCD11's ability to distinguish graph models based on density and order.
  • Development of a randomization statistical test utilizing GCD11.
  • Application of the statistical test to empirical data from a fishing fleet.

Main Results:

  • Identified conditions under which different graph models can be distinguished by GCD11.
  • The statistical test ruled out the assumption of independent pairing within the studied fishing fleet.
  • Demonstrated the challenges in identifying similarities between real-world small graphs and theoretical graph models.

Conclusions:

  • GCD11 provides a method for distinguishing between various small graph models.
  • The developed statistical test offers a novel approach for analyzing empirical network data.
  • Findings from the fishing case study have implications for understanding vessel interactions and challenging existing assumptions in fisheries science.