  $k$-core decomposition is widely used to identify the center of a large
network, it is a pruning process in which the nodes with degrees less than $k$
are recursively removed. Although the simplicity and effectiveness of this
method facilitate its implementation on broad applications across many
scientific fields, it produces few analytical results. We here simplify the
existing theoretical framework to a simple iterative relationship and obtain
the exact analytical solutions of the $k$-core pruning process on large
uncorrelated networks. From these solutions we obtain such statistical
properties as the degree distribution and the size of the remaining subgraph in
each of the pruning steps. Our theoretical results resolve the long-lasting
puzzle of the $k$-core pruning dynamics and provide an intuitive description of
the dynamic process.
