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arXiv · 0912.4812

Joint Vertex Degrees in an Inhomogeneous Random Graph Model

K. Lin·G. Reinert
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Abstract

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 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.

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K. Lin, G. Reinert. 2009-12-24. Joint Vertex Degrees in an Inhomogeneous Random Graph Model. https://arxiv.org/abs/0912.4812

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