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

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

Statistics & Probability Letters, 2023, vol. 193, issue C

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.

Keywords: Random graphs; Connectivity; Vehicular networks (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spl.2022.109695

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