Nearest-neighbor graphs on random point sets and their applications to sensor networks

We study the graph NN(d,k) constructed on a Poisson point process in d dimensions by connecting each point to the k points nearest to it. We show that if k geq 188, NN(2,k) has an infinite connected component and this infinite component has an infinite subset of points with the property that the distance along the edges of the graphs between any pair of these points is at most a constant factor larger than their Euclidean distance. Finally we discuss the relevance of our results to the study of wireless sensor networks.