  Pattern formation, arising from systems of autonomous reaction-diffusion
equations, on networks has become a common topic of study in the scientific
literature. In this work we focus primarily on directed networks. Although some
work prior has been done to understand how patterns arise on directed networks,
these works have restricted their attentions to networks for whom the Laplacian
matrix (corresponding to the network) is diagonalizable. Here, we address the
question "how does one detect pattern formation if the Laplacian matrix is not
diagonalizable?" To this end, we find it is useful to also address the related
problem of pattern formation arising from systems of reaction-diffusion
equations with non-local (global) reaction kinetics. These results are then
generalized to include non-autonomous systems as well as temporal networks,
i.e., networks whose topology is allowed to change in time.
