Generalization of core percolation on complex networks

We introduce a $k$-leaf removal algorithm as a generalization of the so-called leaf removal algorithm. In this pruning algorithm, vertices of degree smaller than $k$, together with their first nearest neighbors and all incident edges are progressively removed from a random network. As the result of this pruning the network is reduced to a subgraph which we call the Generalized $k$-core ($Gk$-core). Performing this pruning for the sequence of natural numbers $k$, we decompose the network into a hierarchy of progressively nested $Gk$-cores. We present an analytical framework for description of $Gk$-core percolation for undirected uncorrelated networks with arbitrary degree distributions (configuration model). To confirm our results, we also derive rate equations for the $k$-leaf removal algorithm which enable us to obtain the structural characteristics of the $Gk$-cores in another way. Also we apply our algorithm to a number of real-world networks and perform the $Gk$-core decomposition for them.