Positive Arc-Weight Design Makes Every Directed Laplacian Diagonalizable
For directed networks, the Laplacian need not be diagonalizable, so the standard master-stability variational equations cannot in general be fully decoupled into independent eigenmodes. We prove that this obstruction can always be removed by coupling-strength design: every weakly connected digraph admits a strictly positive weighting of its existing arcs for which the weighted in-degree Laplacian is diagonalizable. The construction uses a spanning directed acyclic subgraph with one source in each root strongly connected component, assigns distinct positive weighted indegrees to its non-source vertices, and then restores all remaining arcs with a common sufficiently small positive weight. The zero eigenvalue remains semisimple and all nonzero eigenvalues remain simple. We also give a discriminant criterion that computes an admissible interval of restoring weights. Thus, any fixed weakly connected directed topology can be positively weighted so that master-stability perturbations admit a complete modal decomposition.