The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph

Michael J G Wilsher; Carl P Dettmann; A J Ganesh

doi:10.1016/j.spl.2022.109695

The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph

Michael J G Wilsher, Carl P Dettmann , A J Ganesh^*

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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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.

Original language	English
Article number	109695
Number of pages	7
Journal	Statistics and Probability Letters
Volume	193
Early online date	19 Oct 2022
DOIs	https://doi.org/10.1016/j.spl.2022.109695
Publication status	Published - Feb 2023

Bibliographical note

Funding Information:
MW was supported by the EPSRC Centre for Doctoral Training in Communications, UK ( EP/I028153/1 and EP/L016656/1 ).

Publisher Copyright:
© 2022 The Author(s)

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https://arxiv.org/abs/2202.08044

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title = "The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph",

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 scalingregime, 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.",

author = "Wilsher, \{Michael J G\} and Dettmann, \{Carl P\} and Ganesh, \{A J\}",

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AU - Wilsher, Michael J G

AU - Dettmann , Carl P

AU - Ganesh, A J

PY - 2023/2

Y1 - 2023/2

N2 - 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 scalingregime, 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.

AB - 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 scalingregime, 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.

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DO - 10.1016/j.spl.2022.109695

M3 - Article (Academic Journal)

SN - 0167-7152

VL - 193

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

M1 - 109695

ER -

The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph

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