Connectivity of 1d random geometric graphs

TL;DR

This paper investigates the structure of 1d random geometric graphs by counting k-hop paths, revealing a novel connection to lattice path combinatorics and providing exact probability distributions.

Contribution

It introduces a new combinatorial approach linking 1d RGG path counts to lattice paths, with explicit generating functions and distributions.

Findings

Derived the probability generating function for k-hop paths.

Established a bijection between path counts and lattice path volumes.

Linked spatial random graphs to lattice path combinatorics.

Abstract

A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the space is the 1d interval $[0,1]$. We show how the $k$-hop path count between two vertices at Euclidean distance $|x-y|$ is in bijection with the volume enclosed by a uniformly random $d$-dimensional lattice path joining the corners of a $(k-1)$-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of this $k$-hop path count as a sum over lattice paths, incorporating the idea of restricted integer partitions with limited number of parts. We therefore demonstrate and describe an important link between spatial random graphs, and lattice path combinatorics, where the $d$-dimensional lattice paths correspond to spatial permutations of the geometric points on the line.