  Bootstrap percolation is a prominent framework for studying the spreading of
activity on a graph. We begin with an initial set of active vertices. The
process then proceeds in rounds, and further vertices become active as soon as
they have a certain number of active neighbors. A recurring feature in
bootstrap percolation theory is an `all-or-nothing' phenomenon: either the size
of the starting set is so small that the process stops very soon, or it
percolates (almost) completely.
  Motivated by several important phenomena observed in various types of
real-world networks we propose in this work a variant of bootstrap percolation
that exhibits a vastly different behavior. Our graphs have two types of
vertices: some of them obstruct the diffusion, while the others facilitate it.
We study the effect of this setting by analyzing the process on
Erd\H{o}s-R\'enyi random graphs. Our main findings are two-fold. First we show
that the presence of vertices hindering the diffusion does not result in a
stable behavior: tiny changes in the size of the starting set can dramatically
influence the size of the final active set. In particular, the process is
non-monotone: a larger starting set can result in a smaller final set. In the
second part of the paper we show that this phenomenom arises from the
round-based approach: if we move to a continuous time model in which every edge
draws its transmission time randomly, then we gain stability, and the process
stops with an active set that contains a non-trivial constant fraction of all
vertices. Moreover, we show that in the continuous time model percolation
occurs significantly faster compared to the classical round-based model. Our
findings are in line with empirical observations and demonstrate the importance
of introducing various types of vertex behaviors in the mathematical model.
