  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.
