  Global transport and communication networks enable information, ideas and
infectious diseases now to spread at speeds far beyond what has historically
been possible. To effectively monitor, design, or intervene in such
epidemic-like processes, there is a need to predict the speed of a particular
contagion in a particular network, and to distinguish between nodes that are
more likely to become infected sooner or later during an outbreak. Here, we
study these quantities using a message-passing approach to derive simple and
effective predictions which are validated against epidemic simulations on a
variety of real-world networks with good agreement. In addition to
individualized predictions for different nodes, we find an overall sudden
transition from low density to almost full network saturation as the contagion
develops in time. Our theory is developed and explained in the setting of
simple contagions on tree-like networks, but we are also able to show how the
method extends remarkably well to complex contagions and highly clustered
networks.
