Diameters in supercritical random graphs via first passage percolation

Jian Ding, Jeong Han Kim, Eyal Lubetzky, Yuval Peres

Research output: Contribution to journalArticle

Abstract

We study the diameter of C1, the largest component of the Erdos-Rényi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε/n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). uczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of 1000/7. We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of C1 is w.h.p. asymptotic to 2/3 D(ε,n), and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to 5/9D(ε,n).

Original languageEnglish (US)
Pages (from-to)729-751
Number of pages23
JournalCombinatorics Probability and Computing
Volume19
Issue number5-6
DOIs
StatePublished - Sep 2010

Fingerprint

First-passage Percolation
Random Graphs
Passage Time
Giant Component
Erdös
Upper and Lower Bounds
Flexibility
Entire
Cover
Valid
kernel
Imply
Estimate

ASJC Scopus subject areas

  • Applied Mathematics
  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Statistics and Probability

Cite this

Diameters in supercritical random graphs via first passage percolation. / Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval.

In: Combinatorics Probability and Computing, Vol. 19, No. 5-6, 09.2010, p. 729-751.

Research output: Contribution to journalArticle

Ding, Jian ; Kim, Jeong Han ; Lubetzky, Eyal ; Peres, Yuval. / Diameters in supercritical random graphs via first passage percolation. In: Combinatorics Probability and Computing. 2010 ; Vol. 19, No. 5-6. pp. 729-751.
@article{8e04f64064ba45648cd070e379e4bde1,
title = "Diameters in supercritical random graphs via first passage percolation",
abstract = "We study the diameter of C1, the largest component of the Erdos-R{\'e}nyi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε/n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). uczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of 1000/7. We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of C1 is w.h.p. asymptotic to 2/3 D(ε,n), and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to 5/9D(ε,n).",
author = "Jian Ding and Kim, {Jeong Han} and Eyal Lubetzky and Yuval Peres",
year = "2010",
month = "9",
doi = "10.1017/S0963548310000301",
language = "English (US)",
volume = "19",
pages = "729--751",
journal = "Combinatorics Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "5-6",

}

TY - JOUR

T1 - Diameters in supercritical random graphs via first passage percolation

AU - Ding, Jian

AU - Kim, Jeong Han

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2010/9

Y1 - 2010/9

N2 - We study the diameter of C1, the largest component of the Erdos-Rényi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε/n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). uczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of 1000/7. We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of C1 is w.h.p. asymptotic to 2/3 D(ε,n), and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to 5/9D(ε,n).

AB - We study the diameter of C1, the largest component of the Erdos-Rényi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε/n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). uczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of 1000/7. We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of C1 is w.h.p. asymptotic to 2/3 D(ε,n), and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to 5/9D(ε,n).

UR - http://www.scopus.com/inward/record.url?scp=78650479901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650479901&partnerID=8YFLogxK

U2 - 10.1017/S0963548310000301

DO - 10.1017/S0963548310000301

M3 - Article

VL - 19

SP - 729

EP - 751

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 5-6

ER -