  We investigate three aspects of the importance of nodes with respect to
Susceptible-Infectious-Removed (SIR) disease dynamics: influence maximization
(the expected outbreak size given a set of seed nodes), the effect of
vaccination (how much deleting nodes would reduce the expected outbreak size)
and sentinel surveillance (how early an outbreak could be detected with sensors
at a set of nodes). We calculate the exact expressions of these quantities, as
functions of the SIR parameters, for all connected graphs of three to seven
nodes. We obtain the smallest graphs where the optimal node sets are not
overlapping. We find that: node separation is more important than centrality
for more than one active node, that vaccination and influence maximization are
the most different aspects of importance, and that the three aspects are more
similar when the infection rate is low.
