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Detecting hyperbolic geometry in networks: Why triangles are not enough.

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Geometric network models capture network properties using hyperbolic spaces. Standard statistics like clustering coefficients fail to detect this geometry, but a new statistic, weighted triangles, effectively identifies hyperbolic geometry in networks.

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Area of Science:

  • Network science
  • Complex systems
  • Data analysis

Background:

  • Geometric network models are widely used to represent real-world networks.
  • These models leverage the principle that similar nodes tend to connect, capturing properties like scale invariance and high clustering.
  • Many networks can be accurately modeled by embedding their graphs in hyperbolic spaces.

Purpose of the Study:

  • To address the limitations of current statistics in detecting hyperbolic geometry in networks.
  • To introduce a novel statistic for identifying geometry in network structures.
  • To demonstrate the efficacy of the proposed statistic in detecting hyperbolic geometry.

Main Methods:

  • Analysis of existing geometric network models and their limitations.
  • Development of a new statistic: weighted triangles.
  • Analytical derivations and validation using synthetic and real-world network data.

Main Results:

  • Standard network statistics like triangle counts and clustering coefficients are insufficient for detecting hyperbolic geometry.
  • Weighted triangles provide a more powerful and accurate measure for identifying hyperbolic geometry.
  • The effectiveness of weighted triangles is demonstrated through analytical proofs and empirical analysis.

Conclusions:

  • Hyperbolic geometry is a crucial underlying structure in many real-world networks.
  • Weighted triangles offer a superior method for detecting this geometry compared to traditional metrics.
  • This new statistic enhances our ability to understand and model complex network structures.