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assess the merit of using k -shell decomposition to identify the most efficient SIR spreaders one needs to compare the two sets Υ$_{eff}$( p ) and Υ$_{k}$$_{S}$( p ). In order to consider individual M$_{i}$ values, we calculate the average M$_{eff}$ ( p ) and M$_{k}$$_{S}$ ( p ) values for the sets Υ$_{eff}$( p ) and Υ$_{k}$$_{S}$( p ) respectively: M$_{k}$$_{S}$ ( p ) ≡ ∑ i ∈ Υ$_{k}$$_{S}$ ( p $_{)}$M$_{i}$ /Np and M$_{eff}$ ( p ) ≡ ∑ i ∈ Υ$_{eff}$ ( p $_{)}$M$_{i}$ /Np , where Np is 

