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Random rectangular graphs.

Ernesto Estrada1, Matthew Sheerin1

  • 1Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom.

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Summary
This summary is machine-generated.

This study introduces random rectangular graphs (RRGs), a generalization of random geometric graphs (RGGs). RRGs reveal how graph properties like connectivity and clustering depend on rectangle dimensions and connection radius.

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Area of Science:

  • Graph theory
  • Network science
  • Computational geometry

Background:

  • Random geometric graphs (RGGs) model networks on a unit square.
  • Understanding network topology is crucial in various scientific fields.

Purpose of the Study:

  • Generalize the RGG model to random rectangular graphs (RRGs).
  • Analyze topological properties of RRGs as a function of rectangle dimensions and connection radius.
  • Investigate the impact of geometric constraints on network structure.

Main Methods:

  • Developed a mathematical model for RRGs on a unit area rectangle.
  • Derived analytical expressions for key graph metrics (average degree, connectivity, etc.).
  • Validated theoretical findings through computational simulations.

Main Results:

  • Most RRG properties (degree, connectivity, path length) depend monotonically on connection radius and rectangle side lengths.
  • The clustering coefficient exhibits a non-monotonic behavior, peaking at slight rectangular elongations.
  • The model accurately captures RGGs (square case) and 1D RGGs (highly elongated rectangles).

Conclusions:

  • RRGs offer a flexible framework for studying network topology under varying geometric conditions.
  • The shape of the underlying space significantly influences network properties.
  • The study provides valuable insights into the relationship between geometry and network structure.