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Connectivity in one-dimensional soft random geometric graphs.

Michael Wilsher1, Carl P Dettmann1, Ayalvadi Ganesh1

  • 1School of Mathematics, University of Bristol, Woodland Road, Bristol, BS8 1UG, United Kingdom.

Physical Review. E
|January 20, 2021
PubMed
Summary
This summary is machine-generated.

This study examines connectivity in one-dimensional soft random geometric graphs (RGGs). Unlike hard RGGs, isolated nodes, not gaps, are the primary factor determining connectivity in soft RGGs.

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Area of Science:

  • Mathematics
  • Computer Science
  • Network Theory

Background:

  • Random geometric graphs (RGGs) are crucial models for network analysis.
  • Understanding connectivity in soft RGGs is essential for various applications.
  • Previous research often focused on hard RGGs, leaving soft RGGs less explored.

Purpose of the Study:

  • To analyze the connectivity of one-dimensional soft random geometric graphs (RGGs).
  • To identify and quantify key factors influencing disconnection in these networks.
  • To establish a contrast with the connectivity properties of hard RGGs.

Main Methods:

  • Developing analytic expressions for the mean and variance of isolated nodes.
  • Deriving bounds on the probability of graph disconnection.
  • Analyzing the occurrence and impact of uncrossed gaps.

Main Results:

  • Isolated nodes are identified as the critical factor for disconnection in soft RGGs.
  • A sharp threshold for the occurrence of isolated nodes is established.
  • Uncrossed gaps are shown to have negligible probability in the relevant scaling.

Conclusions:

  • The connectivity of soft RGGs is dominated by isolated nodes, differing significantly from hard RGGs.
  • The findings provide a theoretical foundation for understanding network formation in soft RGG models.
  • This research offers insights into network robustness and design in one-dimensional systems.