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Poisson Approximation of the Number of Cliques in Random Intersection Graphs

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  14 July 2016

Katarzyna Rybarczyk  and
Dudley Stark
Katarzyna Rybarczyk*
Affiliation:
Adam Mickiewicz University
Dudley Stark*
Affiliation:
Queen Mary, University of London
*
∗Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland.
∗∗Postal address: School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK. Email address: [email protected]
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Abstract

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A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects Wv ⊆ W such that w ∈ Wv with probability p, independently for all v ∈ V and all w ∈ W. Given two vertices v1, v2 ∈ V, we set v1 ∼ v2 if and only if Wv1 ∩ Wv2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

Keywords

Stein's methodPoisson approximationrandom intersection graph

MSC classification

Secondary: 60F05: Central limit and other weak theorems 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 826 - 840
Copyright
Copyright © Applied Probability Trust 2010 

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