Universal network characteristics, such as the scale-free degree distribution and the small world phenomena, are the bread and butter of network science. But how do we translate such topological findings into an understanding of the system's dynamic behavior: for instance, how does the small world structure impact the patterns of flow in the system? Or how does the presence of hubs affect the distribution of influence? In essence, whether it's communicable diseases, genetic regulation, or the spread of failures in an infrastructure network, these questions touch on the patterns of information spread in the network. It all begins with a local perturbation, such as a sudden disease outbreak or a local power failure, which then propagates to impact all other nodes. The challenge is that the resulting spatio-temporal propagation patterns are diverse and unpredictable -- indeed, a zoo of spreading patterns -- that seem to be only loosely connected to the network topology. We show that we can tame this zoo by exposing a systematic translation of topological elements into their dynamic outcome, allowing us to navigate the network, and, most importantly, to expose a deep universality behind the seemingly diverse dynamics.