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is quantified by the persistence function 

<img src='content_image/364197.jpg'>

as a function of ( k$_{S}$, k ) at different β values (Fig. 4a and b). High k$_{S}$ layers in networks might be closely related to the concept of a core group in Sexually Transmitted Infections re- search [23]. The core groups are defined as subgroups in the general population characterized by high partner turnover rate and extensive intergroup interaction [23]. 

Similar to the core group, the dense sub-network formed by nodes in the innermost k - shells helps the virus to consistently survive locally in the inner-core area and infect other nodes adjacent to the area. These k -shells preserve the existence of a virus, in contrast to e.g. isolated hubs in the periphery. Note that a virus cannot survive in the degree-preserving randomized version of the CNI network, due to the absence of high k -shells. 

The importance of the inner-core nodes in spreading is not influenced by the infection probability values, β . In both models, SIS and SIR, we find that the persistence ρ or the average infected fraction M , respectively, is systematically larger for nodes in inner k -shells compared to nodes in outer shells, over the entire β range that we studied (Fig. 4c,d). Thus, the k -shell measure is a robust indicator for the spreading efficiency of a node. 

Finding the most accurate ranking of individual nodes for spreading in a population can influence the success of dissemination strategies. When spreading starts from a single node, the k$_{S}$ value is enough for this ranking, while in the case of many simultaneous origins, spreading is greatly enhanced when we additionally repel the spreaders with large degree or k$_{S}$ . In the case of infections that do not confer immunity on recovered individuals, the core of the network in the large k$_{S}$ layers forms a reservoir where infection can survive locally. 

## I. METHODS 

## A. The k -shell decomposition 

Nodes are assigned to k -shells according to their remaining degree, which is obtained by successive pruning of nodes with degree smaller than the k$_{S}$ value of the current layer. We start by removing all nodes with degree k = 1. After removing all the nodes with k = 1, some nodes may be left with one link, so we continue pruning the system iteratively 

