k-core organization in complex networks

Abstract

We analyse the complex network architectures based on the k-core notion, where the k-core is the maximal subgraph in a network, whose vertices all have internal degree at least k. We explain the nature of the unusual “hybrid” phase transition of the emergence of a giant k-core in a network, which combines a jump in the order parameter and a critical singularity, and relate this transition to critical phenomena in other systems. Finally, we indicate generic features and differences between the k-core problem and bootstrap percolation.