  Identifying the most influential spreaders is one of outstanding problems in
physics of complex systems. So far, many approaches have attempted to rank the
influence of nodes but there is still the lack of accuracy to single out
influential spreaders. Here, we directly tackle the problem of finding
important spreaders by solving analytically the expected size of epidemic
outbreaks when spreading originates from a single seed. We derive and validate
a theory for calculating the size of epidemic outbreaks with a single seed
using a message-passing approach. In addition, we find that the probability to
occur epidemic outbreaks is highly dependent on the location of the seed but
the size of epidemic outbreaks once it occurs is insensitive to the seed. We
also show that our approach can be successfully adapted into weighted networks.
