  Complex networks such as the sexual partnership web or the Internet often
show a high degree of redundancy and heterogeneity in their connectivity
properties. This peculiar connectivity provides an ideal environment for the
spreading of infective agents. Here we show that the random uniform
immunization of individuals does not lead to the eradication of infections in
all complex networks. Namely, networks with scale-free properties do not
acquire global immunity from major epidemic outbreaks even in the presence of
unrealistically high densities of randomly immunized individuals. The absence
of any critical immunization threshold is due to the unbounded connectivity
fluctuations of scale-free networks. Successful immunization strategies can be
developed only by taking into account the inhomogeneous connectivity properties
of scale-free networks. In particular, targeted immunization schemes, based on
the nodes' connectivity hierarchy, sharply lower the network's vulnerability to
epidemic attacks.
