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 presence of hubs affect the propagation of information? In essence, whether it is communicable diseases, genetic regulation or the spread of failures in an infrastructure 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 spatiotemporal 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. Exposing a deep universality behind seemingly diverse dynamics.