k-Core organization of complex networks.
We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures--k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints--the bootstrap percolation thresholds. We show that in networks with a finite mean number zeta2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if zeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.
Departamento de Física da Universidade de Aveiro, 3810-193 Aveiro, Portugal and Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
S N
SN
Dorogovtsev
A V
AV
Goltsev
J F F
JF
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