We also highlight the location of the 25 largest hubs in the k -shell structure of the studied networks. Fig. 12 shows the results for the collaboration, actor, email, RL Internet, AS Internet, and Product space networks. High-degree nodes in most of the studied networks are scattered at different k -shells: the high- k nodes appear both in the periphery (starting as low as k$_{S}$ = 1) and in the network center (large k$_{S}$ value). In certain cases, such as in the actors network, the largest hubs are located in the highest k$_{S}$ layers. The relation of k$_{S}$ and k in the AS Internet and the product space is strongly monotonic, and there are very few nodes where k$_{S}$ is large or small compared to the degree k . This is a typical behavior for random networks, and the structure of these two networks is significantly close to their randomized counterparts. In these cases, choosing a node based on its degree or its k -shell index does not make a difference, since they practically lead to the same nodes. 1 10 k$_{S}$ 10 0 10 1 10 2 10 3 k 1 10 100 k$_{S}$ 10 0 10 1 10 2 k RL Internet Product space FIG. 13: Deviations from the average behavior highlight the importance of the k -shell structure. The average degree (red symbols) for a given k$_{S}$ index follows roughly a power-law dependence, as a function of k$_{S}$ . The deviation from this behavior can be significant, e.g. in RL internet, or negligible, as e.g. in the product space network. It is clear that the assortative behavior in a network can influence the extent to which hubs will appear in the periphery or in the core of a network. In principle, in a highly disassortative network we expect more hubs in the periphery, due to their tendency to connect to low-degree nodes. However, even in assortative networks it is possible that some hubs may still belong to low k -shells, so that the k$_{S}$ value will appropriately rank even these 12