Connectivity in One-Dimensional Soft Random Geometric Graphs

TL;DR

This paper analyzes the connectivity of one-dimensional soft random geometric graphs, deriving bounds and thresholds for connectivity, and contrasting their properties with hard RGGs.

Contribution

It provides new analytical bounds and thresholds for connectivity in 1D soft RGGs, highlighting differences from hard RGGs.

Findings

Bounds on full connectivity probability

Analytic expressions for isolated nodes

Negligible probability of uncrossed gaps at critical scaling

Abstract

In this paper, we study the connectivity of a one-dimensional soft random geometric graph (RGG). The graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on the distance between them. We derive bounds on the probability that the graph is fully connected by analysing key modes of disconnection. In particular, analytic expressions are given for the mean and variance of the number of isolated nodes, and a sharp threshold established for their occurrence. Bounds are also derived for uncrossed gaps, and it is shown analytically that uncrossed gaps have negligible probability in the scaling at which isolated nodes appear. This is in stark contrast to the hard RGG in which uncrossed gaps are the most important factor when considering network connectivity.