  We study the effect of the connectivity pattern of complex networks on the
propagation dynamics of epidemics. The growth time scale of outbreaks is
inversely proportional to the network degree fluctuations, signaling that
epidemics spread almost instantaneously in networks with scale-free degree
distributions. This feature is associated with an epidemic propagation that
follows a precise hierarchical dynamics. Once the highly connected hubs are
reached, the infection pervades the network in a progressive cascade across
smaller degree classes. The present results are relevant for the development of
adaptive containment strategies.
