  The theory of pattern formation in reaction-diffusion systems is extended to
the case of a directed network. Due to the structure of the network Laplacian
of the scrutinised system, the dispersion relation has both real and imaginary
parts, at variance with the conventional case for a symmetric network. It is
found that the homogeneous fixed point can become unstable due to the topology
of the network, resulting in a new class of instabilities which cannot be
induced on undirected graphs. Results from a linear stability analysis allow
the instability region to be analytically traced. Numerical simulations show
that the instability can lead to travelling waves, or quasi-stationary
patterns, depending on the characteristics of the underlying graph. The results
presented here could impact on the diverse range of disciplines where directed
networks are found, such as neuroscience, computer networks and traffic
systems.
