On the One dimensional Poisson Random Geometric Graph

TL;DR

This paper analyzes a one-dimensional Poisson random geometric graph, deriving explicit formulas for the distribution of its connected components based on a Poisson process on a bounded interval.

Contribution

It provides an explicit distribution formula for the number of connected components in a 1D Poisson random geometric graph, using Laplace transform inversion techniques.

Findings

Explicit distribution of connected components derived

Laplace transform inversion used for analysis

Results applicable to 1D Poisson processes

Abstract

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.