### 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 language | English |
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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

Norros, I., & Reittu, H. (2006). On a conditionally Poissonian graph process.

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