On the relation between graph distance and Euclidean distance in random geometric graphs

• Published in: Advances in applied probability Vol. 48; no. 3; pp. 848 - 864
• Main Authors: Díaz, J., Mitsche, D., Perarnau, G., Pérez-Giménez, X.
• Format: Journal Article Publication
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2016; Applied Probability Trust
• Subjects: 60 Probability theory and stochastic processes; 60D05 Geometric probability, stochastic geometry, random sets; Asymptotic properties; Classificació AMS; Demutualization; diameter; Estimates; Euclidean distance; Euclidean geometry; graph distance; Graph theory; Graphs; Lower bounds; Matemàtiques i estadística; Mathematics; Probabilitat; Probability; Random geometric graph; State regulation; Symbols; Upper bounds; Àrees temàtiques de la UPC; Random geometric graph; diameter; Secondary 68R10; Primary 05C80; Euclidean distance; graph distance; Graph distance; Diameter 2010 Mathematics Subject Classification: Primary 05C80 Secondary 68R10; Random geometric graphs
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2016.31

Given any two vertices u, v of a random geometric graph G(n, r), denote by d E (u, v) their Euclidean distance and by d E (u, v) their graph distance. The problem of finding upper bounds on d G (u, v) conditional on d E (u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on d E (u, v) conditional on d E (u, v).