  In this work we investigate a bootstrap percolation process on random graphs
generated by a random graph model which combines preferential attachment and
edge insertion between previously existing vertices. The probabilities of
adding either a new vertex or a new connection between previously added
vertices are time dependent and given by a function $f$ called the edge-step
function. We show that under integrability conditions over the edge-step
function the graphs are highly susceptible to the spread of infections, which
requires only $3$ steps to infect a positive fraction of the whole graph. To
prove this result, we rely on a quantitative lower bound for the maximum degree
that might be of independent interest.
