related to the degree of the node, k , where the spreading have started. Spreading may be very different even when it starts from hubs of similar degree as comparatively shown in Figs. 1b and c. Instead, the location of the spreading origin given by its k$_{S}$ index predicts more accurately the size of the infected population. For instance, Figs. 1b and 1d show that nodes in the same k$_{S}$ layer produce similar spreading areas even if they have different k (by definition, in a given layer there could be many nodes with k ≥ k$_{S}$ ). 

The above example suggests that the position of the node relative to the organization of the network determines its spreading influence more than a local property of a node, like the degree k . To quantify the influence of a given node i in an SIR spreading process we study the average size of the population M$_{i}$ infected in an epidemic originating at node i with a given ( k$_{S}$, k ). The infected population is averaged over all the origins with the same ( k$_{S}$, k ) values: 

where Υ( k$_{S}$, k ) is the union of all N ( k$_{S}$, k ) nodes with ( k$_{S}$, k ) values. 

The analysis of M ( k$_{S}$, k ) in the studied social networks reveals three general results (see Fig. 2): (a) For a fixed degree, there is a wide spread of M ( k$_{S}$, k ) values. In particular, there are many hubs located in the periphery of the network (large k , low k$_{S}$ ) that are poor spreaders. (b) For a fixed k$_{S}$ , M ( k$_{S}$, k ) is approximately independent of the degree of the nodes. This result is revealed in the vertically layered structure of M ( k$_{S}$, k ) suggesting that infected nodes located in the same k -shell produce similar epidemic outbreaks M ( k$_{S}$, k ) independent of the value of k of the infection origin. (c) The most efficient spreaders are located in the inner-core of the network (large k$_{S}$ region) fairly independently of their degree. These results indicate that the k -shell index of a node is a better predictor of spreading influence. When an outbreak starts in the core of the network (large k$_{S}$ ) there exist many pathways through which a virus can infect the rest of the network; this result is valid regardless of the node degree. The existence of these pathways implies that during a typical epidemic outbreak from a random origin, nodes located in high k$_{S}$ layers are more likely to be infected and they will be infected earlier than other nodes (see SI-Section III). The neighborhood of these nodes makes them more efficient in sustaining an infection at the early stages, allowing thus the epidemics to reach a critical mass that will allow it to fully develop. Similar results on the efficiency of high- k$_{S}$ nodes are obtained from the analysis 

