Connectivity threshold for Bluetooth graphs

TL;DR

This paper analyzes the connectivity thresholds of Bluetooth graphs modeled as irrigation subgraphs of random geometric graphs, revealing how parameters like radius and edges per node influence network connectivity.

Contribution

It provides a precise characterization of the connectivity threshold in Bluetooth graphs near the critical radius for the underlying geometric graph.

Findings

Connectivity cannot be achieved with high probability outside certain parameter ranges.

The connectivity threshold in the number of edges per node is characterized for radii near the critical value.

The results help understand the conditions for reliable connectivity in wireless ad hoc networks.

Abstract

We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius $r$ that determines the "visible neighbors" of each node and the number of edges $c$ that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters $r, c$ and completely characterize the connectivity threshold (in $c$) for values of $r$ close the critical value for connectivity in the underlying random geometric graph.