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Central loops in random planar graphs.

Benjamin Lion1, Marc Barthelemy1

  • 1Institut de Physique Théorique, CEA, CNRS-URA 2306, F-91191 Gif-sur-Yvette, France.

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Summary
This summary is machine-generated.

Central loops emerge in random planar graphs when loop link weights are low. A toy model shows loop centrality depends on branch number and size, revealing key factors for complex network organization.

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Area of Science:

  • Network Science
  • Graph Theory
  • Complex Systems

Background:

  • Characterizing random planar graph structure is vital for various applications.
  • Local graph properties are insufficient; path-related measures, like betweenness centrality (BC), reveal organizational patterns.
  • Nodes with high BC can form nontrivial structures, including central loops.

Purpose of the Study:

  • To investigate the emergence of nontrivial patterns, specifically central loops, in random planar graphs.
  • To propose and analyze a simplified model for understanding the conditions leading to central loop formation.
  • To explore the impact of randomness on loop centrality within these structures.

Main Methods:

  • Empirical analysis of betweenness centrality in different random planar graphs.
  • Development and analysis of a toy model comprising a star network with superimposed loops.
  • Estimation of BC for the central node and loop nodes within the toy model.

Main Results:

  • A toy model demonstrates that a superimposed loop can become more central than the origin node under specific conditions (w < w_c).
  • The transition threshold scales as w_c ~ n/N_b, where n is branch size and N_b is the number of branches.
  • An optimal loop position is found to scale as ℓ_opt ~ N_b*w/4, indicating dependence on network parameters.

Conclusions:

  • The number and spatial extension of radial branches are critical factors controlling the existence of central loops.
  • The proposed toy model provides insights into the organization of random structures and the effect of randomness on loop centrality.
  • Understanding these structural properties is essential for characterizing complex random networks.