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We develop the theory of the k-core (bootstrap) percolation on uncorrelatedrandom networks with arbitrary degree distributions. We show that the k-corepercolation is an unusual, hybrid phase transition with a jump emergence of thek-core as at a first order phase transition but also with a criticalsingularity as at a continuous transition. We describe the properties of thek-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 aso-called ``corona'' of the k-core plays a crucial role (corona is a subset ofvertices in the k-core which have exactly k neighbors in the k-core). It turnsout that the k-core percolation threshold is at the same time the percolationthreshold of finite corona clusters. The mean separation of vertices in coronaclusters plays the role of the correlation length and diverges at the criticalpoint. We show that a random removal of even one vertex from the k-core mayresult in the collapse of a vast region of the k-core around the removedvertex. The mean size of this region diverges at the critical point. We find anexact mapping of the k-core percolation to a model of cooperative relaxation.This model undergoes critical relaxation with a divergent rate at some criticalmoment.Comment: 11 pages, 8 figure

physical review. e, statistical, nonlinear and soft matter physics

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects

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