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We introduce the heterogeneous k-core, which generalizes the k-core, and contrast it with bootstrap percolation. Vertices have a threshold ri that may be different at each vertex. If a vertex has fewer than ri neighbors it is pruned from the network. The heterogeneous k-core is the subgraph remaining after no further vertices can be pruned. If the thresholds ri are 1 with probability ƒ, or k ⩾ 3 with probability 1 - ƒ, the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation, which is an activation process. 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 the network structure has a crucial effect on these processes, with the giant heterogeneous k-core appearing immediately at a finite value for any ƒ > 0 when the degree distribution tends to a power law P(q) ∼ q-γ with γ < 3. [ABSTRACT FROM AUTHOR]