  We consider bootstrap percolation on uncorrelated complex networks. We obtain
the phase diagram for this process with respect to two parameters: $f$, the
fraction of vertices initially activated, and $p$, the fraction of undamaged
vertices in the graph. We observe two transitions: the giant active component
appears continuously at a first threshold. There may also be a second,
discontinuous, hybrid transition at a higher threshold. Avalanches of
activations increase in size as this second critical point is approached,
finally diverging at this threshold. We describe the existence of a special
critical point at which this second transition first appears. In networks with
degree distributions whose second moment diverges (but whose first moment does
not), we find a qualitatively different behavior. In this case the giant active
component appears for any $f>0$ and $p>0$, and the discontinuous transition is
absent. This means that the giant active component is robust to damage, and
also is very easily activated. We also formulate a generalized bootstrap
process in which each vertex can have an arbitrary threshold.
