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

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 attent...

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Vydáno v:Advances in applied probability Ročník 48; číslo 3; s. 848 - 864
Hlavní autoři: Díaz, J., Mitsche, D., Perarnau, G., Pérez-Giménez, X.
Médium: Journal Article Publikace
Jazyk:angličtina
Vydáno: Cambridge, UK Cambridge University Press 01.09.2016
Applied Probability Trust
Témata:
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
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Popis
Shrnutí: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).
Bibliografie:ObjectType-Article-1
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ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2016.31