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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)
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