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Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij})

Part of: Operations research and management science Limit theorems Graph theory

Published online by Cambridge University Press:  04 January 2016

Kaisheng Lin  and
Gesine Reinert
Kaisheng Lin*
Affiliation:
University of Oxford
Gesine Reinert*
Affiliation:
University of Oxford
*
∗ Email address: k.lin@wolfson.oxon.org
∗∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. Email address: reinert@stats.ox.ac.uk
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Abstract

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In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Keywords

Stein's methodsize-biased couplingvertex degreeinhomogeneous random graphpower law

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 05C80: Random graphs 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 139 - 165
Copyright
© Applied Probability Trust 

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