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Inhomogeneous random graphs, isolated vertices, and Poisson approximation

Part of: Graph theory Limit theorems

Published online by Cambridge University Press:  28 March 2018

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of Bath
*
* Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: m.d.penrose@bath.ac.uk
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Abstract

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

Keywords

Inhomogeneous random graphrandom connection modelstochastic block modellatent variable modelrandom geometric graphPoisson approximationStein's methodU-statistic

MSC classification

Primary: 05C80: Random graphs
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 112 - 136
Copyright
Copyright © Applied Probability Trust 2018 

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