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On a conditionally Poissonian graph process

Part of: Graph theory

Published online by Cambridge University Press:  01 July 2016

Ilkka Norros  and
Hannu Reittu
Ilkka Norros*
Affiliation:
VTT Technical Research Centre of Finland
Hannu Reittu*
Affiliation:
VTT Technical Research Centre of Finland
*
∗ Postal address: VTT Technical Research Centre of Finland, PO Box 10000, FIN-02044 VTT, Finland.
∗ Postal address: VTT Technical Research Centre of Finland, PO Box 10000, FIN-02044 VTT, Finland.
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Abstract

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Random (pseudo)graphs GN with the following structure are studied: first, independent and identically distributed capacities Λi 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(Λi Λj / ∑k=1N Λk). The main result of the paper is that when P(Λ1 > x) ≥ x−τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, GN 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.

Keywords

Random graphbranching processgiant componentpower law

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C12: Distance in graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 59 - 75
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Presented at the ICMS Workshop on Spatial Stochastic Modelling with Applications to Communications Networks (Edinburgh, June 2004).

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