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

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.