  We present a thorough inspection of the dynamical behavior of epidemic
phenomena in populations with complex and heterogeneous connectivity patterns.
We show that the growth of the epidemic prevalence is virtually instantaneous
in all networks characterized by diverging degree fluctuations, independently
of the structure of the connectivity correlation functions characterizing the
population network. By means of analytical and numerical results, we show that
the outbreak time evolution follows a precise hierarchical dynamics. Once
reached the most highly connected hubs, the infection pervades the network in a
progressive cascade across smaller degree classes. Finally, we show the
influence of the initial conditions and the relevance of statistical results in
single case studies concerning heterogeneous networks. The emerging theoretical
framework appears of general interest in view of the recently observed
abundance of natural networks with complex topological features and might
provide useful insights for the development of adaptive strategies aimed at
epidemic containment.
