On Bollob\'as-Riordan random pairing model of preferential attachment graph

Bollob'as-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\{W_j{\}}_{j\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first ${\nu }_m^n:=n^{\frac{m}{m+2}-ϵ}$ vertices are jointly, and uniformly, asymptotic to ${2(mn)^{1/2}\bigl(W^{1/2}{mj}-W^{1/2}{m(j-1)}\bigr)}_{j\in [t]}$,andthatwithhighprobability(whp)thesmallestofthesedegreesis$n^{\frac{\epsilon(m+2)}{2m}}$, at least. In contrast, the degrees of vertices below the top by any fraction of $n$ are whp of $O(\log n)$ order. Next we bound the probability that there exists a pair of large vertex sets with no edges joining them, 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 $v$. The set of the first ${\nu }_m^n$ vertices is shown whp not to contain a max-tree of any size. Whp the largest recursive tree has size of order $n$. We prove that, for $m>1$, $\Bbb P(G_m^n$\text{ is connected}$)\ge 1- O\bigl((\log n)^{-(m-1)/3+o(1)}\bigr)$. We show that the distribution of the scaled size of a generic max-tree in $G_1^n$ converges to a mixture of two beta distributions.