  We present a generalized method for calculating the k-shell structure of
weighted networks. The method takes into account both the weight and the degree
of a network, in such a way that in the absence of weights we resume the shell
structure obtained by the classic k-shell decomposition. In the presence of
weights, we show that the method is able to partition the network in a more
refined way, without the need of any arbitrary threshold on the weight values.
Furthermore, by simulating spreading processes using the
susceptible-infectious-recovered model in four different weighted real-world
networks, we show that the weighted k-shell decomposition method ranks the
nodes more accurately, by placing nodes with higher spreading potential into
shells closer to the core. In addition, we demonstrate our new method on a real
economic network and show that the core calculated using the weighted k-shell
method is more meaningful from an economic perspective when compared with the
unweighted one.
