We introduce the heterogeneous-k-core, which generalizes the k-core, and
contrast it with bootstrap percolation. Vertices have a threshold ki​ which
may be different at each vertex. If a vertex has less than ki​ 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 ki​ are
1 with probability f or k≥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)∼q−γ with
γ<3.Comment: 10 pages, 4 figure