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Quasirandom geometric networks from low-discrepancy sequences.

Ernesto Estrada1

  • 1Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G11HX, United Kingdom.

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Summary
This summary is machine-generated.

Quasirandom geometric networks, built using low-discrepancy sequences, exhibit greater uniformity than random geometric networks. These quasirandom networks also show faster diffusion processes due to their structural homogeneity.

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Area of Science:

  • Network science
  • Computational geometry
  • Data analysis

Background:

  • Random geometric networks are widely studied but can exhibit heterogeneous node distributions.
  • Low-discrepancy sequences offer a deterministic approach to generating points with uniform distributions.
  • Understanding network properties based on point distribution is crucial for various applications.

Purpose of the Study:

  • To define and investigate quasirandom geometric networks using low-discrepancy sequences.
  • To compare the network-theoretic properties of quasirandom and random geometric networks.
  • To analyze diffusion processes on these networks and their dependence on structural properties.

Main Methods:

  • Generating d-dimensional networks where vertices are d-tuples from low-discrepancy sequences (Halton, Sobol, Niederreiter).
  • Connecting vertices within a specified connection radius.
  • Computationally investigating 11 network properties, degree distributions, and spectral densities for 2D networks.
  • Analyzing strategies (scrambling, skipping, leaping) for higher-dimensional network generation.
  • Simulating diffusive processes on both network types.

Main Results:

  • Quasirandom networks demonstrate superior vertex uniformity in the unit square compared to random geometric networks.
  • Up to dimension 20, scrambling, skipping, and leaping strategies maintain uniformity in higher-dimensional quasirandom networks.
  • Diffusion processes are significantly faster in quasirandom networks due to their structural homogeneity.
  • Random geometric networks show slower diffusion due to node clustering resulting from heterogeneous distributions.

Conclusions:

  • Quasirandom geometric networks offer a more uniform and predictable structure than random geometric networks.
  • Effective strategies exist for generating high-dimensional quasirandom networks with desired uniformity.
  • The structural homogeneity of quasirandom networks leads to more efficient diffusive processes.