FIG. 10: The infected percentage is always higher in higher k -shells, independently of the infection probability β . Nodes are grouped according to their k -shell and we calculate the average infected percentage for each group as a function of β . The solid lines correspond to the grand average over all nodes acting as spreading origins. The networks that were analyzed are: a, the email network, b, the CNI, c, the adult IMDB actors network, and d, the cond-mat collaboration network. The nodes in higher k -shells are consistently reaching a larger fraction of the network. Our main interest is in the β range where we are above the critical point, 〈 M 〉 > 0, but the average infection reaches a finite but small fraction, in the range of 1-20%. When the average spreading is even larger, nodes of lower k -shells can become efficient too, because in this case there is a high probability to reach the ‘core’ of the network, and this would enable the spreading to extend over an even larger part of the network. For β values in this ‘intermediate’ range, the distribution P ( M ) of the infected percentage M is composed by two well-defined peaks (Fig. 11). The first is at M = 0 and corresponds to those instances where the infection dies within the first few infection steps. The second peak is at a finite fraction M , and it seems to be at the same point for all origins. However, the intensity of each peak strongly differs, depending on the k$_{S}$ value of the origin. For 9 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 M 1 <= k$_{S}$ < 4 4 <= k$_{S}$ < 8 8 <= k$_{S}$ < 12 12 <= k$_{S}$ < 16 16 <= k$_{S}$ < 20 20 <= k$_{S}$ < 24 Average a 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.4 0.6 M c 0 0.02 0.04 0.06 0.08 0.2 0.4 0.6 0.8 M 1 <= k$_{S}$ < 10 10 <= k$_{S}$ < 20 20 <= k$_{S}$ < 30 30 <= k$_{S}$ < 40 40 <= k$_{S}$ < 50 50 <= k$_{S}$ < 60 60 <= k$_{S}$ < 70 Average b 0 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 M 1 <= k$_{S}$ < 5 5 <= k$_{S}$ < 10 10 <= k$_{S}$ < 15 15 <= k$_{S}$ < 20 20 <= k$_{S}$ < 25 Average d