On Nearest-Neighbor Graphs

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.