ceptible nodes become infected [18]. The quantity characterizing the role of nodes 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, which is reached asymptotically, ρ$_{i}$ becomes independent of t . The persistence ρ has been shown to be higher in hubs which are reinfected frequently due to the large number of their neighbors [7, 24, 25]. To uncover the role of k -shell layers in SIS spreading we use the joint persistence function ρ ( k$_{S}$, k ) ≡ ∑ i ∈ Υ( k$_{S}$,k ) ρ$_{i}$ N ( k$_{S}$, k ) . (6) 15 Here we present results for the virus persistence in the Actor, Collaboration, Email and RL Internet Networks. Similar to Fig. 4, we depict ρ ( k$_{S}$, k ) in both supercritical ( β > β$_{c}$ ) and subcritical ( β < β$_{c}$ ) regimes, where β$_{c}$ is the critical threshold. In the supercritical regime, ρ ( k, k$_{S}$ ) increases with both k and k$_{S}$ , with maximum values corresponding to hubs in the innermost layers (see Fig. 15). As depicted in Fig. 15, in the subcritical regime, viruses persist only in the highest k$_{S}$ layers, while the probability of finding an infected node in low k -shells is negligible. In order to determine in the above networks the actual epidemic threshold β$_{c}$ we study the behavior of SIS spreading over a wide range of β values. In order to highlight the role of k -shells in spreading, we organize several groups of nodes based on the k$_{S}$ layers of each network. Every such group comprises approximately 100 randomly chosen nodes with the corresponding k -shell indices. In order to achieve similar average degree in each of the groups, we pick nodes with uniform probability based on their degree. As shown in Fig. 16, virus persistence is consistently higher in the inner k -shells for all values of β . Moreover, we find substantially lower epidemic thresholds than in the random cases β$_{c}$ < β rand c in all considered networks except for the Email Contact network. The results of Figs. 15 and 16 suggest that the observed persistence of a virus is due to the dense sub-network formed by nodes in the innermost k -shell, which helps the virus to consistently survive locally in this area. Indeed, the innermost layers can be regarded as a small subgraph exclusively consisting of hubs. By definition, all nodes in this innermost k -shell will have degrees k ≥ k$_{S}$$_{max}$ . Therefore, as a simple approximation, one can regard the innermost core of a network as a regular graph consisting of nodes with the same degree k = k$_{S}$$_{max}$ . The mean-field solution of the SIS spreading in a regular graph can be found, for instance