Approximate conditional distributions of distances between nodes in a two-dimensional sensor network

TL;DR

This paper investigates the conditional distributions of graph and Euclidean distances between sensor nodes in a 2D network, providing analytical and simulation results that could improve sensor localization methods.

Contribution

It introduces the first analysis of conditional distributions of sensor distances in a 2D network, combining analytical and simulation approaches.

Findings

Derived conditional probability distributions for sensor distances

Provided analytical solutions and simulation validation

Opened new avenues for sensor localization improvements

Abstract

When we represent a network of sensors in Euclidean space by a graph, there are two distances between any two nodes that we may consider. One of them is the Euclidean distance. The other is the distance between the two nodes in the graph, defined to be the number of edges on a shortest path between them. In this paper, we consider a network of sensors placed uniformly at random in a two-dimensional region and study two conditional distributions related to these distances. The first is the probability distribution of distances in the graph, conditioned on Euclidean distances; the other is the probability density function associated with Euclidean distances, conditioned on distances in the graph. We study these distributions both analytically (when feasible) and by means of simulations. To the best of our knowledge, our results constitute the first of their kind and open up the possibility of discovering improved solutions to certain sensor-network problems, as for example sensor localization.