  A bootstrap percolation process on a graph $G$ is an "infection" process
which evolves in rounds. Initially, there is a subset of infected nodes and in
each subsequent round each uninfected node which has at least $r$ infected
neighbours becomes infected and remains so forever. The parameter $r\geq 2$ is
fixed. Such processes have been used as models for the spread of ideas or
trends within a network of individuals.
  We analyse bootstrap percolation process in the case where the underlying
graph is an inhomogeneous random graph, which exhibits a power-law degree
distribution, and initially there are $a(n)$ randomly infected nodes. The main
focus of this paper is the number of vertices that will have been infected by
the end of the process. The main result of this work is that if the degree
sequence of the random graph follows a power law with exponent $\beta$, where
$2 < \beta < 3$, then a sublinear number of initially infected vertices is
enough to spread the infection over a linear fraction of the nodes of the
random graph, with high probability.
  More specifically, we determine explicitly a critical function $a_c(n)$ such
that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number
of vertices of the underlying random graph, if $a(n) \ll a_c(n)$, then the
process does not evolve at all, with high probability as $n$ grows, whereas if
$a(n)\gg a_c(n)$, then there is a constant $\eps>0$ such that, with high
probability, the final set of infected vertices has size at least $\eps n$. It
turns out that when the maximum degree is $o(n^{1/(\beta -1)})$, then $a_c(n)$
depends also on $r$. But when the maximum degree is $\Theta (n^{1/(\beta
-1)})$, then $a_c (n)=n^{\beta -2 \over \beta -1}$.
