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Degree asymptotics with rates for preferential attachment random graphs. (English) Zbl 1271.60019
The authors study the distribution of the degree of a fixed vertex in two preferential attachment models $$G_n$$. Let $$W_{n,i}$$ be the degree of vertex $$i$$ in $$G_n$$. They show the optimal rates of convergence in the Kolmogrov metric of $$W_{n,i}/(\operatorname{E}W^2_{n,i})^{1/2}$$ to the asymptotic distribution of its distribution limit $$K_s$$ as $$n$$ goes to infinity. The distribution function $$K_s$$ is defined by its density as $k_s(x)=\Gamma(s)\sqrt{\frac{2}{s\pi}}\exp\left(\frac{-x^2}{2s}\right)U\left(s-1,0.5,\frac{x^2}{2s}\right)$ for $$x>0$$ and $$s\geq1/2$$. Here, $$\Gamma$$ and $$U$$ represent the gamma function and the confluent hypergeometric function of the second kind, respectively. The main approach is a development of Stein’s method for the distribution $$K_s$$ using fixed points of distributional transformations.

##### MSC:
 60C05 Combinatorial probability 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
preferential attachment; Stein’s method
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##### References:
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