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For a family of linear preferential attachment graphs, we provide rates ofconvergence for the total variation distance between the degree of a randomlychosen vertex and an appropriate power law distribution as the number ofvertices tends to infinity. Our proof uses a new formulation of Stein's methodfor the negative binomial distribution, which stems from a distributionaltransformation that has the negative binomial distributions as the only fixedpoints.

Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution