On a conditionally Poissonian graph process

Ilkka Norros, Hannu Reittu

Research output: Contribution to journalArticleScientificpeer-review

86 Citations (Scopus)

Abstract

Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.
Original languageEnglish
Pages (from-to)59-75
Number of pages17
JournalAdvances in Applied Probability
Volume38
Issue number1
DOIs
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed

Fingerprint

Giant Component
Poisson distribution
Graph in graph theory
Identically distributed
Connectivity
Architecture

Cite this

@article{a8bb2dd4771f4f8ba47155ecd57d2f13,
title = "On a conditionally Poissonian graph process",
abstract = "Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.",
author = "Ilkka Norros and Hannu Reittu",
year = "2006",
doi = "10.1239/aap/1143936140",
language = "English",
volume = "38",
pages = "59--75",
journal = "Advances in Applied Probability",
issn = "0001-8678",
publisher = "Cambridge University Press",
number = "1",

}

On a conditionally Poissonian graph process. / Norros, Ilkka; Reittu, Hannu.

In: Advances in Applied Probability, Vol. 38, No. 1, 2006, p. 59-75.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - On a conditionally Poissonian graph process

AU - Norros, Ilkka

AU - Reittu, Hannu

PY - 2006

Y1 - 2006

N2 - Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.

AB - Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.

U2 - 10.1239/aap/1143936140

DO - 10.1239/aap/1143936140

M3 - Article

VL - 38

SP - 59

EP - 75

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -