We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures—-cores. The -core is the largest subgraph where vertices have at least interconnections. We find the structure of -cores, their sizes, and their birthpoints—the bootstrap percolation thresholds. We show that in networks with a finite mean number of the second-nearest neighbors, the emergence of a -core is a hybrid phase transition. In contrast, if diverges, the networks contain an infinite sequence of -cores which are ultrarobust against random damage.
