Reduced Arc-Weight Heterogeneity for Master Stability Analysis of Directed Networks

Master stability analysis can be applied to directed networks even when their Laplacian matrices are not diagonalizable. Nevertheless, a diagonalizable Laplacian is easier to analyse because the perturbations governing synchronization separate into independent modes. This paper asks how many different positive coupling strengths are needed to obtain this simplification without adding or removing any directed connections. We first give an elementary construction showing that suitable positive weights exist for every weakly connected directed network and explain its relationship to stronger genericity results in the literature. We then introduce a measure of the smallest number of distinct coupling strengths required for diagonalizability. If a common coupling strength already produces a diagonalizable Laplacian, only one value is needed. For directed acyclic networks, we give a constructive upper bound based on the length of the longest directed path from a source. This bound is exact for rooted outward directed trees. Consequently, an outward star requires only one coupling value, whereas a directed path requires a different value on every arc. These results reduce coupling heterogeneity while retaining the simpler modal form of master stability analysis. Diagonalizability is treated as an analytical convenience, not as a sufficient condition for synchronization.