Analytical solution to the k-core pruning process

k-core decomposition is a widely-used method in ranking nodes or extracting important information of complex networks. It is a pruning process in which we recursively remove the vertices with degree less than k to obtain the core of a complex network. The simplicity and effectiveness of this approach has led to a variety of applications in many scientific fields, including bioinformatics, neurosciences, computer sciences, economics, and network sciences. However, the analytical theory of the k-core pruning process is still lacking. Here we find that in every pruning step of any given network, the Non-Backtracking Expansion Branch (NBEB) is directly related to the remaining k-core. Using this NBEB method, we obtain the analytical results of the k-core pruning process and its detailed critical behaviour.