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The theory of pattern formation in reaction-diffusion systems is extended tothe case of a directed network. Due to the structure of the network Laplacianof the scrutinised system, the dispersion relation has both real and imaginaryparts, at variance with the conventional case for a symmetric network. It isfound that the homogeneous fixed point can become unstable due to the topologyof the network, resulting in a new class of instabilities which cannot beinduced on undirected graphs. Results from a linear stability analysis allowthe instability region to be analytically traced. Numerical simulations showthat the instability can lead to travelling waves, or quasi-stationarypatterns, depending on the characteristics of the underlying graph. The resultspresented here could impact on the diverse range of disciplines where directednetworks are found, such as neuroscience, computer networks and trafficsystems.

Topology-driven instabilities: the theory of pattern formation on directed networks