## 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.

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 |
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Article number | 109695 |

Number of pages | 7 |

Journal | Statistics and Probability Letters |

Volume | 193 |

Early online date | 19 Oct 2022 |

DOIs | |

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)