On Bollobas-Riordan random pairing model of preferential attachment graph

Bollobas-Riordan random pairing model of a preferential attachment graph Gmn is studied. Let {W-j}(j <= mn + 1) be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first nu mn colon equals nmm+2-epsilon vertices are jointly, and uniformly, asymptotic to {2(mn)1/2(Wmj1/2-Wm(j-1)1/2)}j is an element of[nu mn], and that with high probability (whp) the smallest of these degrees is n epsilon(m+2)2m, 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 nu mn vertices does not contain a max-tree, and the largest max-tree has size of order n. We prove that, for m > 1, DOUBLE-STRUCK CAPITAL P(Gmnis connected)>= 1-O((logn)-(m-1)/3+o(1)). We show that the distribution of scaled size of a generic max-tree in G1n converges to a mixture of two beta distributions.