FIG 1. When the hubs may not be good spreaders. a, A schematic representation of a network under the k -shell decomposition. The two nodes of degree k = 8 (blue and yellow nodes) in this network are in different locations: one lies in the periphery, ( k$_{S}$ = 1) while the other hub is in the innermost core of the network, i.e. it has the largest k$_{S}$ ( k$_{S}$ = 3). 

b-d, The extent of the efficiency of the spreading process cannot be accurately predicted based on a measure of the immediate neighborhood of the node, such as the degree k . For the contact network of inpatients (CNI), we compare infections originating from single nodes having the same degree k = 96 (nodes A and B) or the same index k$_{S}$ = 63 (nodes A and C), with infection probability β = 0 . 035. In the corresponding plots, the colors indicate the probability that a node will be infected when spreading starts in the corresponding origin, as long as this probability is higher than 25%. The results are based on 10000 different realizations for each case. In the first case, where origin A has k$_{S}$ = 63, spreading reaches a much wider area more frequently, in contrast to origin B ( k$_{S}$ = 26), where the infection remains largely localized in the immediate neighborhood of B. Spreading is very similar between origins A and C, which have the same k$_{S}$ value, although the degree of C is much smaller than A. The importance of the network organization is also highlighted when we randomly rewire the network (preserving the same degree for all nodes). In this case the standard picture is recovered: the extent of spreading coincides and both hubs contribute equally largely to spreading (see SI-Section VI). 

FIG 2. The k -shell index predicts the outcome of spreading more reliably than the degree k or the betweenness centrality C$_{B}$ . The networks used are (top to bottom): email contacts ( β = 8%), CNI network ( β = 4%), the actors network ( β = 1%), and the Livejournal.com friendship network ( β = 1 . 5%). a, c, e, g Average infected size of the population M ( k$_{S}$, k ) when spreading originates in nodes with ( k$_{S}$, k ). b, d, f, h The infected size M ( k$_{S}$, C$_{B}$ ) when spreading originates in nodes of a given combination of k$_{S}$ and C$_{B}$ . In both cases, spreading is larger for nodes of higher k$_{S}$ , while nodes of a given k or C$_{B}$ value can result in either small or large spreading, depending on the value of k$_{S}$ . 

(There is an exception at large k$_{S}$ and small k of the livejournal database, which is due to artificial closed groups of virtual characters that connect with each other for the purpose of online gaming and do not correspond to regular users, as the rest of the database.) 

FIG 3. k -shell structure of the CNI network. a, The imprecision functions ϵ$_{k}$$_{S}$ ( p ), ϵ$_{k}$ ( p ), and ϵ$_{C}$$_{B}$ ( p ), for β = 4%. Even though both k -shell and k identification strategies 

