  Bootstrap percolation is a well-known activation process in a graph, in which
a node becomes active when it has at least $r$ active neighbors. Such process,
originally studied on regular structures, has been recently investigated also
in the context of random graphs, where it can serve as a simple model for a
wide variety of cascades, such as the spreading of ideas, trends, viral
contents, etc. over large social networks. In particular, it has been shown
that in $G(n,p)$ the final active set can exhibit a phase transition for a
sub-linear number of seeds. In this paper, we propose a unique framework to
study similar sub-linear phase transitions for a much broader class of graph
models and epidemic processes. Specifically, we consider i) a generalized
version of bootstrap percolation in $G(n,p)$ with random activation thresholds
and random node-to-node influences; ii) different random graph models,
including graphs with given degree sequence and graphs with community structure
(block model). The common thread of our work is to show the surprising
sensitivity of the critical seed set size to extreme values of distributions,
which makes some systems dramatically vulnerable to large-scale outbreaks. We
validate our results running simulation on both synthetic and real graphs.
