III. PROBABILITY AND TIME OF INFECTION IV. THE IMPRECISION FUNCTIONS 5 mation. When we measure the new k$_{S}$ value for the same nodes in the resulting networks (Fig. 6) we find that their relative ranking remains the same. We recover a practically linear dependence on the k$_{S}$ values of the original and the incomplete networks, showing that this measure would work equally well for predicting the spreading efficiency of nodes in a network with missing information. We have demonstrated that the location of a node, as described through the k$_{S}$ index, is important for the extent of spreading M$_{i}$ when this node is the spreading origin. Here, we show that nodes with high k$_{S}$ are more probable to be infected during an epidemic outbreak and are infected earlier than nodes with low k$_{S}$ , when spreading starts at a random node. We introduce the quantity E$_{i}$ , as the probability that a node i is going to be infected during an epidemic outbreak originating at a random location, and T$_{i}$ , as the average time before node i is infected during the same process. As shown in Figs. 7a-d all three quantities that characterize the role of a node in an epidemics process, M$_{i}$ , E$_{i}$ and T$_{i}$ are strongly correlated. The nodes that are infected by a given node i form a cluster of size M$_{i}$ , and they are statistically the nodes that can reach i when they act as origins themselves. Thus, the probability E$_{i}$ to reach this node in general is directly proportional to the size M$_{i}$ , as shown in the plots. The average time T$_{i}$ to reach a node is inversely proportional to its spreading efficiency M$_{i}$ , which emphasizes the fact that these nodes are easily reachable from different network locations. In conclusion, the nodes with the largest k$_{S}$ values consistently a) are infecting larger parts of the network, b) are infected more frequently, and c) are infected earlier, than nodes with smaller k$_{S}$ values. We quantify the spreading efficiency of an individual origin i through the infected number of nodes M$_{i}$ . In order to compare the different methods, we rank all network nodes according to their spreading efficiency, independently of their other properties, and we consider a fraction p of the most efficient spreaders ( p ∈ [0 , 1]). We designate this set by Υ$_{eff}$( p ). Similarly, we define Υ$_{k}$$_{S}$( p ) as the set of individuals with highest k -shell values. In order to