Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\nu_m^n:=n^{\frac{m}{m+2}-\epsilon}$ vertices are jointly, and uniformly,
asymptotic to $\{2(mn)^{1/2}\bigl(W^{1/2}_{mj}-W^{1/2}_{m(j-1)}\bigr)\}_{j\in
[t]}$, and that with high probability (whp) the smallest of these degrees is
$n^{\frac{\epsilon(m+2)}{2m}}$, 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.

14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg