Advances in Applied Probability

Joint vertex degrees in the inhomogeneous random graph model G(n, {pij})

Kaisheng Lin and Gesine Reinert

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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.

Article information

Adv. in Appl. Probab., Volume 44, Number 1 (2012), 139-165.

First available in Project Euclid: 8 March 2012

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Mathematical Reviews number (MathSciNet)

Primary: 60F05: Central limit and other weak theorems
Secondary: 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

Stein's method size-biased coupling vertex degree inhomogeneous random graph power law


Lin, Kaisheng; Reinert, Gesine. Joint vertex degrees in the inhomogeneous random graph model G ( n , { p ij }). Adv. in Appl. Probab. 44 (2012), no. 1, 139--165. doi:10.1239/aap/1331216648.

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