Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A phase transition in the evolution of bootstrap percolation processes
  on preferential attachment graphs   The theme of this paper is the analysis of bootstrap percolation processes on
random graphs generated by preferential attachment. This is a class of
infection processes where vertices have two states: they are either infected or
susceptible. At each round every susceptible vertex which has at least $r\geq
2$ infected neighbours becomes infected and remains so forever. Assume that
initially $a(t)$ vertices are randomly infected, where $t$ is the total number
of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average
degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg
a_c(t)$, complete infection occurs with high probability as $t \rightarrow
\infty$, but when $a(t) \ll a_c (t)$, then with high probability the process
evolves only for a bounded number of rounds and the final set of infected
vertices is asymptotically equal to $a(t)$.

8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga