k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects

Abstract

We develop the theory of the $k$-core (bootstrap) percolation on uncorrelated random networks with arbitrary degree distributions. We show that the $k$-core percolation is an unusual, 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 transition. We describe the properties of the $k$-core, explain the meaning of the order parameter for the $k$-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 (corona is a subset of vertices in the $k$-core which have exactly $k$ neighbors in the $k$-core). 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 show 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 the $k$-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment.