Vertices of high degree in the preferential attachment tree ∗

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number Dt(`) of vertices of each degree ` at each time t, focussing particularly on the case where ` is a growing function of t. We show that Dt(`) is concentrated around its mean, which is approximately 4t/`, for all ` ≤ (t/ log t)−1/3; this is best possible up to a logarithmic factor.