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

TL;DR

This paper analyzes the distribution of isolated nodes in a 1D soft random geometric graph with Poisson distributed vertices on a torus, showing convergence to a Poisson distribution and implications for connectivity.

Contribution

It provides a new probabilistic analysis of isolated nodes in 1D soft RGGs, establishing convergence to a Poisson distribution under certain scaling regimes.

Findings

Number of isolated nodes converges to a Poisson distribution.

Derived an upper bound on the probability of graph connectivity.

Established probabilistic behavior of isolated nodes in 1D soft RGGs.

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.