On the distribution of typical shortest-path lengths in connected random geometric graphs

Abstract

Stationary point processes in ?2 with two different types of points, say H and L, are considered where the points are located on the edge set G of a random geometric graph, which is assumed to be stationary and connected. Examples include the classical Poisson---Voronoi tessellation with bounded and convex cells, aggregate Voronoi tessellations induced by two (or more) independent Poisson processes whose cells can be nonconvex, and so-called β-skeletons being subgraphs of Poisson---Delaunay triangulations. The length of the shortest path along G from a point of type H to its closest neighbor of type L is investigated. Two different meanings of "closeness" are considered: either with respect to the Euclidean distance (e-closeness) or in a graph-theoretic sense, i.e., along the edges of G (g-closeness). For both scenarios, comparability and monotonicity properties of the corresponding typical shortest-path lengths C e? and C g? are analyzed. Furthermore, extending the results which have recently been derived for C e?, we show that the distribution of C g? converges to simple parametric limit distributions if the edge set G becomes unboundedly sparse or dense, i.e., a scaling factor ? converges to zero and infinity, respectively.