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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 

