  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 every uninfected node which has at least r infected neighbours
becomes infected and remains so forever. The parameter r > 1 is fixed.
  We consider this process in the case where the underlying graph is an
inhomogeneous random graph whose kernel is of rank 1. Assuming that initially
every vertex is infected independently with probability p > 0, we provide a law
of large numbers for the number of vertices that will have been infected by the
end of the process.
  We also focus on a special case of such random graphs which exhibit a
power-law degree distribution with exponent in (2,3). The first two authors
have shown the existence of a critical function a_c(n) such that a_c(n)=o(n)
with the following property. Let n be the number of vertices of the underlying
random graph and let a(n) be the number of the vertices that are initially
infected. Assume that a set of a(n) vertices is chosen randomly and becomes
externally infected. If a(n) << a_c(n), then the process does not evolve at
all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high
probability the final set of infected vertices is linear.
  Using the techniques of the previous theorem, we give the precise asymptotic
fraction of vertices which will be eventually infected when a(n) >> a_c (n) but
a(n) = o(n). Note that this corresponds to the case where p approaches 0.
