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We investigate three aspects of the importance of nodes with respect toSusceptible-Infectious-Removed (SIR) disease dynamics: influence maximization(the expected outbreak size given a set of seed nodes), the effect ofvaccination (how much deleting nodes would reduce the expected outbreak size)and sentinel surveillance (how early an outbreak could be detected with sensorsat a set of nodes). We calculate the exact expressions of these quantities, asfunctions of the SIR parameters, for all connected graphs of three to sevennodes. We obtain the smallest graphs where the optimal node sets are notoverlapping. We find that: node separation is more important than centralityfor more than one active node, that vaccination and influence maximization arethe most different aspects of importance, and that the three aspects are moresimilar when the infection rate is low.

Three faces of node importance in network epidemiology: Exact results for small graphs