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

Michael Wilsher; Carl Dettmann; Ayalvadi Ganesh

doi:10.1016/j.spl.2022.109695

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arXiv · 2202.08044

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

Michael Wilsher·Carl Dettmann·Ayalvadi Ganesh

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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.

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Michael Wilsher, Carl Dettmann, Ayalvadi Ganesh. 2022-02-16. The Distribution of the Number of Isolated Nodes in the 1-Dimensional Soft Random Geometric Graph. https://doi.org/10.1016/j.spl.2022.109695

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