Degree asymptotics with rates for preferential attachment random graphs

Abstract

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein's method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.Comment: Published in at http://dx.doi.org/10.1214/12-AAP868 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fluctuations in a general preferential attachment model via Stein's method

“…Moreover, for the model with random outdegree [10], we can also obtain rate of convergence results for the outdegree when compared with a Poisson distribution. For linear f , a result analogous to ours was shown in [12,15,19]. In the thesis [12] various versions of the preferential attachment model (including random and fixed outdegree) were studied using couplings and Stein's method.…”

“…However, in her work only the former was used successfully to derive rates of convergence, which in the best case are of the form (1). For the linear model with fixed outdegree and where connections are made with probability proportional to the degree plus a constant , there is an alternative proof of (1) first in [15] (for = 0) and then in [19]. These proofs use Stein's method, but crucially rely on the fact that the limiting distribution can be represented as a mixture of geometric (resp.…”

“…Other questions that have been studied include error bounds for the degree of a fixed vertex. Peköz, Röllin, and Ross [15] apply Stein's method successfully to prove a rate of convergence for the Kolmogorov distance between the indegree distribution of any fixed vertex and a power law distribution by comparing it to a mixed negative binomial distribution, whereas in [16] they provide rates of convergence for the total variation distance between a random vertex in a uniform attachment graph and the geometric distribution. In [17] the authors prove a rate of convergence in the multidimensional case for the joint degree distribution.…”

Spectrum of Complex Networks

“…Lastly, we will be using a result from [21] which gives us explicit rates of convergence from the degrees to their limiting distribution. To describe this limiting distribution, we first define…”

A Stein characterisation of the generalized hyperbolic distribution

“…Such Stein equations were uncommon in the literature, although recently [21,37,39] have obtained second order Stein equations. In fact, n-th order Stein equations have recently been obtained for the product of n independent beta, gamma and central normal random variables [24,25] and general linear combinations of gamma random variables [1].…”