Joint Vertex Degrees in an Inhomogeneous Random Graph Model

TL;DR

This paper investigates the dependence structure of vertex degree counts in inhomogeneous random graphs, showing that joint counts can be approximated as independent only for large degrees, with implications for understanding apparent power-law behaviors.

Contribution

It introduces new approximation techniques for joint degree counts in inhomogeneous graphs using Stein's method and a novel size-biased coupling.

Findings

Joint degree counts can be approximated as independent for large degrees.

Inhomogeneous graphs can exhibit pseudo-power-law behavior without true power-law degree distributions.

New bounds on distributional distance are established using Stein's method.

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.