Key features of Turing systems are determined purely by network topology

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.