  In recent years, many variants of percolation have been used to study network
structure and the behavior of processes spreading on networks. These include
bond percolation, site percolation, $k$-core percolation, bootstrap
percolation, the generalized epidemic process, and the Watts Threshold Model
(WTM). We show that --- except for bond percolation --- each of these processes
arises as a special case of the WTM and bond percolation arises from a small
modification. In fact "heterogeneous $k$-core percolation", a corresponding
"heterogeneous bootstrap percolation" model, and the generalized epidemic
process are completely equivalent to one another and the WTM. We further show
that a natural generalization of the WTM in which individuals "transmit" or
"send a message" to their neighbors with some probability less than $1$ can be
reformulated in terms of the WTM, and so this apparent generalization is in
fact not more general. Finally, we show that in bond percolation, finding the
set of nodes in the component containing a given node is equivalent to finding
the set of nodes activated if that node is initially activated and the node
thresholds are chosen from the appropriate distribution. A consequence of these
results is that mathematical techniques developed for the WTM apply to these
other models as well, and techniques that were developed for some particular
case may in fact apply much more generally.
