Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions

A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n1/d]d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega (\frac{\log n}{r^{d-1}} )$\end{document}, their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n1/d/r) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight.