  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 birth points -- the
bootstrap percolation thresholds. We show that in networks with a finite mean
number z_2 of the second-nearest neighbors, the emergence of a k-core is a
hybrid phase transition. In contrast, if z_2 diverges, the networks contain an
infinite sequence of k-cores which are ultra-robust against random damage.
