sponds to the index of the k -shell this node belongs to. The k$_{S}$ index provides a different type of information on a node than that provided by the degree k . By definition, a given layer with index k$_{S}$ can be occupied with nodes of degree k ≥ k$_{S}$ . In the case of random model networks, such as the configurational model, there is a strong correlation between k and the k$_{S}$ index of a node and, therefore, both quantities provide the same type of informa- tion. Thus, the low-degree nodes are generally in the periphery, and the high-degree nodes are generally in the innermost k -shells. In real networks, however, this relation is often not true. In real networks hubs may have very different k$_{S}$ values and can be located both in the periphery (yellow node in Fig. 5) or in the core (blue node in Fig. 5) of the network. 0 5 10 15 20 25 k$_{S}$ (original) 0 5 10 15 20 25 k S  (after removal) 0 20 40 60 k$_{S}$ (original) 0 20 40 60 k S  (after removal) 0 25 50 75 100 125 150 175 200 k$_{S}$ (original) 0 25 50 75 100 125 150 175 200 k S  (after removal) 0 20 40 60 80 100 k$_{S}$ (original) 0 20 40 60 80 100 k S  (after removal) Email CNI Livejournal Adult IMDB FIG. 6: Robustness of k$_{S}$ under incomplete network information . We randomly remove 10% of the network links and 50% of the network links (results shown in black and red symbols, respectively). The relative ranking of the nodes remains invariant under both removals, for all the networks studied: Email, Hospital, Adult IMDB, and Livejournal.com. The assignment of a k$_{S}$ index to a node is also quite robust. We have randomly removed 10% and 50% of the links in the networks that we study, simulating thus incomplete infor- 4