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Phys Rev Lett. 2006 Feb 3;96(4):040601. Epub 2006 Feb 2.

k-Core organization of complex networks.

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Departamento de FĂ­sica da Universidade de Aveiro, 3810-193 Aveiro, Portugal and Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.


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.

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