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The distribution of the number of isolated nodes in the 1-Dimensional soft random geometric graph

Author

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  • Wilsher, Michael
  • Dettmann, Carl P.
  • Ganesh, Ayalvadi J.

Abstract

We study the number of isolated nodes in a soft random geometric graph whose vertices constitute a Poisson process on the torus of length L (the line segment [0,L] with periodic boundary conditions), and where an edge is present between two nodes with a probability which depends on the distance between them. Edges between distinct pairs of nodes are mutually independent. In a suitable scaling regime, we show that the number of isolated nodes converges in total variation to a Poisson random variable. The result implies an upper bound on the probability that the random graph is connected.

Suggested Citation

  • Wilsher, Michael & Dettmann, Carl P. & Ganesh, Ayalvadi J., 2023. "The distribution of the number of isolated nodes in the 1-Dimensional soft random geometric graph," Statistics & Probability Letters, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:stapro:v:193:y:2023:i:c:s0167715222002085
    DOI: 10.1016/j.spl.2022.109695
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