The power of choice combined with preferential attachment ∗

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible neighbors are sampled from the existing vertices with probability proportional to degree. Of these possibilities, the new vertex attaches to the vertex from the sample that has the highest degree. The maximal degree in this model has linear or near-linear behavior. This behavior contrasts sharply with the behavior in the same choice model with uniform attachment as well as the preferential attachment model without choice. The proof is based on showing the tree has a persistent hub by comparison with the standard preferential attachment model, as well as martingale and stochastic approximation arguments.