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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| Published in: | Advances in applied probability Vol. 48; no. 3; pp. 848 - 864 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article Publication |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
01.09.2016
Applied Probability Trust |
| Subjects: | |
| ISSN: | 0001-8678, 1475-6064 |
| Online Access: | Get full text |
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| Abstract | 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). |
|---|---|
| AbstractList | 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=[...]([radical]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). Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(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=¿(vlogn) (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 dE(u, v) conditional on dE(u, v). Peer Reviewed 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(√log n ) (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 ). 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). Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(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 ... (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 dE(u, v) conditional on dE(u, v). (ProQuest: ... denotes formulae/symbols omitted.) 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_G(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 = \omega\sqrt(log n)) (i.e. 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_G(u, v) conditional on d_E(u, v). Given any two vertices u, υ of a random geometric graph g(n, r), denote by dE(u, υ) their Euclidean distance and by dG(u, υ) their graph distance. The problem of finding upper bounds on dG(u, υ) conditional on dE(u, υ) 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 $r = \omega \left( {\sqrt {\log n} } \right)$ (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 dG(u, υ) conditional on dE(u, υ). |
| Author | Perarnau, G. Mitsche, D. Díaz, J. Pérez-Giménez, X. |
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| CitedBy_id | crossref_primary_10_1016_j_ejc_2022_103616 crossref_primary_10_1016_j_amc_2018_09_038 crossref_primary_10_1109_TPDS_2019_2933839 crossref_primary_10_1109_TWC_2018_2808290 crossref_primary_10_1016_j_ejc_2023_103842 crossref_primary_10_1007_s00373_017_1768_5 crossref_primary_10_1002_aaai_12210 crossref_primary_10_1038_s41598_020_67421_8 crossref_primary_10_1103_PhysRevResearch_3_013211 crossref_primary_10_1007_s00454_023_00507_y crossref_primary_10_1109_TGRS_2025_3552629 crossref_primary_10_1002_rsa_20922 crossref_primary_10_1177_0278364918802957 crossref_primary_10_1214_24_AAP2052 |
| Cites_doi | 10.1017/CBO9781139004114.009 10.1007/s00453-006-0172-y 10.1137/0109045 10.1093/acprof:oso/9780198506263.001.0001 10.1214/aoap/1034625335 10.1007/s00454-012-9482-9 10.1137/1.9781611973075.114 10.1007/978-3-642-25591-5_21 |
| ContentType | Journal Article Publication |
| Contributor | Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions Universitat Politècnica de Catalunya. ALBCOM - Algorismia, Bioinformàtica, Complexitat i Mètodes Formals Universitat Politècnica de Catalunya. Departament de Ciències de la Computació |
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| Copyright | Copyright © Applied Probability Trust 2016 Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/es Distributed under a Creative Commons Attribution 4.0 International License |
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| Keywords | 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 |
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| References | S0001867816000318_ref1 Goel (S0001867816000318_ref5) 2004 S0001867816000318_ref10 S0001867816000318_ref4 S0001867816000318_ref3 S0001867816000318_ref2 S0001867816000318_ref9 Muthukrishnan (S0001867816000318_ref7) 2005 Penrose (S0001867816000318_ref8) 1997; 7 S0001867816000318_ref6 |
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| Snippet | 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... Given any two vertices u, υ of a random geometric graph g(n, r), denote by dE(u, υ) their Euclidean distance and by dG(u, υ) their graph distance. The problem... 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... 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... Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem... 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_G(u, v) their graph distance. The... |
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| SubjectTerms | 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 |
| Title | On the relation between graph distance and Euclidean distance in random geometric graphs |
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