  The heterogeneous structure implies that a very few nodes may play the
critical role in maintaining structural and functional properties of a
large-scale network. Identifying these vital nodes is one of the most important
tasks in network science, which allow us to better conduct successful social
advertisements, immunize a network against epidemics, discover drug target
candidates and essential proteins, and prevent cascading breakdowns in power
grids, financial markets and ecological systems. Inspired by the nested nature
of real networks, we propose a decomposition method where at each step the
nodes with the lowest degree are pruned. We have strictly proved that this
so-called lowest degree decomposition (LDD) is a subdivision of the famous
k-core decomposition. Extensive numerical analyses on epidemic spreading,
synchronization and nonlinear mutualistic dynamics show that the LDD can more
accurately find out the most influential spreaders, the most efficient
controllers and the most vulnerable species than k-core decomposition and other
well-known indices. The present method only makes use of local topological
information, and thus has high potential to become a powerful tool for network
analysis.
