On Bollobás‐Riordan random pairing model of preferential attachment graph

Abstract

AbstractBollobás‐Riordan random pairing model of a preferential attachment graph is studied. Let {Wj}j ≤ mn + 1 be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first vertices are jointly, and uniformly, asymptotic to , and that with high probability (whp) the smallest of these degrees is , at least. Next we bound the probability that there exists a pair of large vertex sets without connecting edges, and apply the bound to several special cases. We propose to measure an influence of a vertex v by the size of a maximal recursive tree (max‐tree) rooted at v. We show that whp the set of the first vertices does not contain a max‐tree, and the largest max‐tree has size of order n. We prove that, for m > 1, . We show that the distribution of scaled size of a generic max‐tree in converges to a mixture of two beta distributions.