until there is no node left with k = 1 in the network. The removed nodes, along with the corresponding links, form a k -shell with index k$_{S}$ = 1. In a similar fashion, we iteratively remove the next k -shell, k$_{S}$ = 2, and continue removing higher k -shells until all nodes are removed. As a result, each node is associated with a unique k$_{S}$ index, and the network can be viewed as the union of all k -shells. The resulting classification of a node can be very different than when the degree k is used. To study the spreading process we apply the Susceptible-Infectious-Recovered (SIR) and Susceptible-Infectious-Susceptible (SIS) models. In the SIR model, all nodes are initially in susceptible state (S) except for one node in the infectious state (I). At each time step, the I nodes attempt to infect their susceptible neighbors with probability β and then enter the recovered state (R) where they become immunized and cannot be infected again. The SIS model aims to describe spreading processes that do not confer immunity on recovered individuals: infected individuals still try to infect their neighbors with probability β but they return to the susceptible state with probability λ (here we use λ = 0 . 8) and can be reinfected at subsequent time steps, while they remain infectious with probability 1 − λ . The betweenness centrality, C$_{B}$ ( i ), of a node i is defined as follows: Consider two nodes s and t and the set σ$_{st}$ of all possible shortest paths between these two nodes. If the subset of this set that contains the paths that pass through the node i is denoted by σ$_{st}$ ( i ), then the betweenness centrality of this node is given by: where the sum runs over all nodes s and t in the network. The imprecision function ϵ ( p ) quantifies the difference in the average spreading between the pN nodes (0 < p < 1) with highest k$_{S}$ , k , or C$_{B}$ from the average spreading of the pN most efficient spreaders ( N is the number of nodes in the network). Thus, it tests the merit of using k -shell, k and C$_{B}$ to identify the most efficient spreaders. For a given β value and 8 C$_{B}$ ( i ) = ∑ s = t σ$_{st}$ ( i ) σ$_{st}$ , (3) C. The imprecision function B. The spreading models