the network and, in particular, they need not be connected with each other. Clearly, the step-like features in the plot of highest k$_{S}$ nodes (red solid curve in Fig. 3d) suggest that the infected percentage remains constant as long as the infected nodes belong in the same shell. 

Including just one node from a different shell results in a significantly increased spreading. 

This result suggests that a better spreading strategy using multiple n spreaders is to choose either the highest k or k$_{S}$ nodes with the requirement that no two of the n spreaders are directly linked to each other. This scheme then provides the largest infected area of the network as shown in Fig. 3d. 

Many contagious infections, including most sexually transmitted infections [23], do not confer full immunity after infection as assumed in the SIR model, and therefore are suitably described by the SIS epidemic model, where an infectious node returns to the susceptible state with probability λ . In an SIS epidemic the number of infectious nodes eventually reaches a dynamic equilibrium “endemic” state where as many infectious individuals become susceptible as susceptible nodes become infectious [18]. In contrast to SIR, in the initial state of our SIS simulations 20% of the network nodes are already infected. The spreading efficiency of a given node i in SIS spreading is the persistence, ρ$_{i}$ ( t ), defined as the probability that node i is infected at time t [7]. In an endemic SIS state, ρ$_{i}$ ( t → ∞ ) becomes independent of t (see SI-Section VII). Previous studies have shown that the largest persistence ρ$_{i}$ ( t → ∞ ) is found in the network hubs which are re-infected frequently due to the large number of neighbors [7, 24, 25]. However, we find that this result holds only in randomized network structures. In the real network topologies studied here, we find that viruses persist mainly in high k$_{S}$ layers instead, irrespectively of the degree of the nodes in the core. 

In the case of random networks, it is found that viruses propagate to the entire network above an epidemic threshold given by β > β rand c ≡ λ 〈 k 〉 / 〈 k $^{2}$〉 [24, 26]. In real networks, such as the CNI network, the threshold β$_{c}$ is different from β rand c . Furthermore, in real networks, we find that viruses can survive locally even when β < β$_{c}$ , but only within the high k$_{S}$ layers of the network, while virus persistence in peripheral k$_{S}$ layers is negligible (Fig. 4a- c). Since the k -shell structure depends on the network assortativity the lower threshold is in agreement with the observation that high positive assortativity [27] may decrease the epidemic threshold. 

The importance of high k$_{S}$ nodes in SIS spreading is confirmed when we analyze the asymptotic probability that nodes of given ( k$_{S}$, k ) values will be infected. This probability 

