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

Abstract

We consider a class of dynamic random graphs known as preferential attachment models, where the probability that a new vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, the tail of the limiting distribution may behave like a power law or a stretched exponential. Using Stein's method we provide rates of convergence to zero of the total variation distance between the finite distribution and its limit. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour.

A central limit theorem for the number of isolated vertices in a preferential attachment random graph

“…Lemma 1.1 (Lemma 3.1 in [BDO19]). For the preferential attachment model defined above and f (k) ≤ γk + 1 for all k and some γ ∈ (0, 1), we have, for all…”

“…where as above " ≈ " means that equality holds up to a factor c ∈ (0, ∞), independent of n. Note that by Theorem 1.6 in [BDO19], the outdegree is asymptotically Poisson distributed with finite expectation. Since the outdegree of every vertex is at least one with positive probability, this in particular implies that p j,0 = P(outdegree(j) = 0) ≥ p 0 for all j ∈ N, where p 0 is a constant strictly larger than 0.…”

“…In [DM13] the authors look at component sizes in this model and give an abstract criterion for the existence of a giant component for general concave attachment functions f , which becomes explicit when restricting to linear functions. Developing Stein's method for this class of limiting distributions, the authors in [BDO19] give error bounds in the total variation distance between the indegree distribution and the corresponding limit for that very same model. They also prove rates convergence of the outdegree distribution towards a Poisson limit.…”

On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

“…However, the reader can refer to the book by van der Hofstad [2017] and to Newman et al [2006] and Newman [2003], which would give an excellent overview. For generalizations of the basic preferential-attachment model of Barabási and Albert see for instance Bollobás et al [2001], Krapivsky et al [2000], Krapivsky and Redner [2001], Athreya [2007], Dereich and Mörters [2009], Dereich and Mörters [2013], Betken et al [2019], Collevecchio et al [2013], and the book by van der Hofstad [2017].…”