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Levy geometric graphs.

S Plaszczynski1, G Nakamura1, C Deroulers1

  • 1Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France and Université Paris-Cité, IJCLab, 91405 Orsay, France.

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Summary
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We introduce Levy graphs, built from random walks, where connectivity depends on scale, not size. These graphs exhibit scale-invariant clusters, useful for detecting Levy processes and analyzing networks.

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Area of Science:

  • Complex Systems
  • Network Science
  • Statistical Physics

Background:

  • Random walks are fundamental stochastic processes.
  • Geometric graphs connect points within a specified distance.
  • Levy flights exhibit power-law increments with infinite variance.

Purpose of the Study:

  • Introduce a novel family of graphs, termed Levy graphs.
  • Analyze the structural properties of these Levy graphs.
  • Explore potential applications in network analysis.

Main Methods:

  • Constructing geometric graphs from random walk trajectories.
  • Analyzing graph properties such as node degree and component distribution.
  • Utilizing dimensional arguments for theoretical insights.

Main Results:

  • Node degree follows a scale-dependent gamma distribution with exponential decay.
  • Number of clusters scales inversely with the connection distance.
  • Component size distribution is scale-invariant, indicating self-similarity.
  • Levy graphs do not exhibit a giant component or percolation phase transition.

Conclusions:

  • Levy graphs possess unique properties derived from underlying Levy processes.
  • The scale-invariant nature allows testing for Levy process origins in graphs.
  • Potential applications include community detection and analysis of collective behaviors in networks.