Arokiadoss's 5-page proof makes directed Laplacians diagonalizable for MSF

Aandrew Baggio Sahaya Arokiadoss has posted a compact 5-page paper on arXiv (2608.14439) that resolves a long-standing obstacle in network synchronization theory: the master stability function (MSF) traditionally requires a diagonalizable Laplacian to decompose variational equations into independent modes, but directed graphs often produce non-diagonalizable Laplacians. The author proves that every weakly connected digraph admits a strictly positive arc weighting that makes its weighted in-degree Laplacian diagonalizable—no topology changes required.

The construction is elegant and constructive. First, the author extracts a weakly connected spanning directed acyclic graph (DAG) with exactly one source vertex in each root strongly connected component. By choosing positive weights carefully, the weighted indegrees of all remaining vertices become pairwise distinct. Then, all other arcs of the original digraph receive a common, sufficiently small positive weight. This perturbation keeps the nonzero eigenvalues pairwise distinct, while the zero eigenvalue becomes semisimple with multiplicity equal to the number of root strongly connected components. The paper also provides a discriminant-based criterion to compute an admissible interval for that common arc weight, making the method practical for real networks.