  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 $a(t)$.
