Heterogeneousk-core versus bootstrap percolation on complex networks (original) (raw)
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 heterogeneousk-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-kcore 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.
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