  We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and
contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which
may be different at each vertex. If a vertex has less than $k_i$ neighbors it
is pruned from the network. The heterogeneous-$k$-core is the sub-graph
remaining after no further vertices can be pruned. If the thresholds $k_i$ are
$1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process
forms one branch of an activation-pruning process which demonstrates
hysteresis. The other branch is formed by ordinary bootstrap percolation. We
show that there are two types of transitions in this heterogeneous-$k$-core
process: the giant heterogeneous-$k$-core may appear with a continuous
transition and there may be a second, discontinuous, hybrid transition. We
compare critical phenomena, critical clusters and avalanches at the
heterogeneous-$k$-core and bootstrap percolation transitions. We also show that
network structure has a crucial effect on these processes, with the giant
heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$
when the degree distribution tends to a power law $P(q) \sim q^{-\gamma}$ with
$\gamma < 3$.
