### Abstract

_{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 language | English |
---|---|

Pages (from-to) | 59-75 |

Number of pages | 17 |

Journal | Advances in Applied Probability |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 |

MoE publication type | A1 Journal article-refereed |

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### Cite this

*Advances in Applied Probability*,

*38*(1), 59-75. https://doi.org/10.1239/aap/1143936140

}

*Advances in Applied Probability*, vol. 38, no. 1, pp. 59-75. https://doi.org/10.1239/aap/1143936140

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

Research output: Contribution to journal › Article › Scientific › peer-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 -