Diagonalizable Directed Laplacians by Positive Arc-Weight Design for Master Stability Analysis

The standard master stability function (MSF) formulation has traditionally relied on a diagonalizable network Laplacian, since diagonalizability allows the variational equations to be decomposed into independent equations. Directed Laplacians, however, need not be diagonalizable. We show that every weakly connected digraph admits a strictly positive arc weighting for which its weighted in-degree Laplacian is diagonalizable. Our construction first extracts a weakly connected spanning directed acyclic graph having exactly one source vertex in each root strongly connected component and assigns positive weights so that the weighted indegrees of all remaining vertices are pairwise distinct. The remaining arcs of the original digraph are then assigned a common sufficiently small positive weight. The nonzero eigenvalues remain pairwise distinct under this perturbation, while the zero eigenvalue is semisimple, with multiplicity equal to the number of root strongly connected components. Consequently, the resulting Laplacian admits a complete set of eigenvectors and restores the fully decoupled form of the MSF variational equations. We further give a discriminant-based criterion for computing an admissible interval of the common arc weight.