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 journalArticle (Academic Journal)peer-review

1 Citation (Scopus)

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 languageEnglish
Article number109695
Number of pages7
JournalStatistics and Probability Letters
Volume193
Early online date19 Oct 2022
DOIs
Publication statusPublished - 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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