## Advances in Applied Probability

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

### Joint vertex degrees in the inhomogeneous random graph model *G*(*n*, {*p*_{ij}})

Kaisheng Lin and Gesine Reinert

#### 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 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

**Source**

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

**Dates**

First available in Project Euclid: 8 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1331216648

**Digital Object Identifier**

doi:10.1239/aap/1331216648

**Mathematical Reviews number (MathSciNet)**

MR2951550

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aap/1331216648