  Turing's theory of pattern formation is a universal model for
self-organization, applicable to many systems in physics, chemistry and
biology. Essential properties of a Turing system, such as the conditions for
the existence of patterns and the mechanisms of pattern selection are well
understood in small networks. However, a general set of rules governing how
network topology determines fundamental system properties and constraints has
not be found. Here we provide a first general theory of Turing network
topology, which proves why three key features of a Turing system are directly
determined by the topology: the type of restrictions that apply to the
diffusion rates, the robustness of the system, and the phase relations of the
molecular species.
