k-core architecture and k-core percolation on complex networks

We analytically describe the architecture of uncorrelated complex networks with arbitrary degree distributions as a set of successively enclosed substructures-k-cores. 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 neighbours, the emergence of a k-core is a unique hybrid phase transition with a jump emergence of the k-core as at a first order phase transition but also with a critical singularity as at a continuous phase transition. In contrast, if z(2) diverges, the networks contain an infinite sequence of k-cores which are ultra-robust against random damage. We explain the meaning of the order parameter for k-core percolation, and reveal the origin of the specific critical phenomena. We demonstrate that a so-called "corona" of the k-core plays a crucial role. It turns out that the k-core percolation threshold is at the same time the percolation threshold of finite corona clusters. The mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point. We demonstrate that a random removal of even one vertex from the k-core may result in the collapse of a vast region of the k-core around the removed vertex. The mean size of this region diverges at the critical point. We find an exact mapping of k-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment. (c) 2006 Elsevier B.V. All rights reserved.