  The location and nature of the percolation transition in random networks is a
subject of intense interest. Recently, a series of graph evolution processes
have been introduced that lead to discontinuous percolation transitions where
the addition of a single edge causes the size of the largest component to
exhibit a significant macroscopic jump in the thermodynamic limit. These
processes can have additional exotic behaviors, such as displaying a `Devil's
staircase' of discrete jumps in the supercritical regime. Here we investigate
whether the location of the largest jump coincides with the percolation
threshold for a range of processes, such as Erdos-Renyi percolation,
percolation via edge competition and via growth by overtaking. We find that the
largest jump asymptotically occurs at the percolation transition for
Erdos-Renyi and other processes exhibiting global continuity, including models
exhibiting an `explosive' transition. However, for percolation processes
exhibiting genuine discontinuities, the behavior is substantially richer. In
percolation models where the order parameter exhibits a staircase, the largest
discontinuity generically does not coincide with the percolation transition.
For the generalized Bohman-Frieze-Wormald model, it depends on the model
parameter. Distinct parameter regimes well in the supercritical regime feature
unstable discontinuous transitions, which is a novel and unexpected phenomenon
in percolation. We thus demonstrate that seemingly and genuinely discontinuous
percolation transitions can involve a rich behavior in supercriticality, a
regime that has been largely ignored in percolation.
