  Bootstrap percolation is a simple but non-trivial model. It has applications
in many areas of science and has been explored on random networks for several
decades. In single layer (simplex) networks, it has been recently observed that
bootstrap percolation, which is defined as an incremental process, can be seen
as the opposite of pruning percolation, where nodes are removed according to a
connectivity rule. Here we propose models of both bootstrap and pruning
percolation for multiplex networks. We collectively refer to these two models
with the concept of "weak" percolation, to distinguish them from the somewhat
classical concept of ordinary ("strong") percolation. While the two models
coincide in simplex networks, we show that they decouple when considering
multiplexes, giving rise to a wealth of critical phenomena. Our bootstrap model
constitutes the simplest example of a contagion process on a multiplex network
and has potential applications in critical infrastructure recovery and
information security. Moreover, we show that our pruning percolation model may
provide a way to diagnose missing layers in a multiplex network. Finally, our
analytical approach allows us to calculate critical behavior and characterize
critical clusters.
