  We hereby develop the theory of Turing instability for relativistic
reaction-diffusion systems defined on complex networks. Extending the framework
introduced by Cattaneo in the 40's, we remove the unphysical assumption of
infinite propagation velocity holding for reaction-diffusion systems, reducing
thus the gap between theory and experiments. We analytically prove that Turing
instability emerges for a much broader set of conditions, e.g., once the
activator diffuses faster than the inhibitor, overcoming thus the classical
Turing instability framework. Analytical results are compared to direct
simulations made on the FitzHugh-Nagumo model, extended to the relativistic
reaction-diffusion framework with a complex network as substrate for the
dynamics. We found stationary patterns, oscillatory ones, as well as a new
interesting class of solutions with a stationary-like transient regime followed
by an oscillatory one.
