We study the graph constructed on a Poisson point process in @math dimensions by connecting each point to the @math points nearest to it. This graph a.s. has an infinite cluster if @math where @math , known as the critical value, depends only on the dimension @math . This paper presents an improved upper bound of 188 on the value of @math . We also show that if @math the infinite cluster of @math has an infinite subset of points with the property that the distance along the edges of the graphs between these points is at most a constant multiplicative factor larger than their Euclidean distance. Finally we discuss in detail the relevance of our results to the study of multi-hop wireless sensor networks.
The "nearest-neighbor" relation, or more generally the "k-nearest-neighbors" relation, defined for a set of points in a metric space, has found many uses in computational geometry and clustering analysis, yet surprisingly little is known about some of its basic properties. In this paper we consider some natural questions that are motivated by geometric embedding problems. We derive bounds on the relationship between size and depth for the components of a nearest-neighbor graph and prove some probabilistic properties of the k-nearest-neighbors graph for a random set of points.