Positive Arc-Weight Design Makes Every Directed Laplacian Diagonalizable

What the math gives to ML

The paper gives a constructive spectral-design mechanism for directed graphs: every weakly connected digraph can be assigned strictly positive arc weights so that its in-degree Laplacian is diagonalizable, with a semisimple zero eigenvalue and simple nonzero eigenvalues. The construction selects a spanning directed acyclic subgraph, assigns distinct positive weighted indegrees to non-source vertices, and restores omitted arcs with a common sufficiently small positive weight. This transfers directly to graph neural ODEs, directed recurrent message passing, and graph state-space models, where defective Laplacians create Jordan-chain transients that cannot be separated into independent modes. The key engineering tests are eigenvalue-collision boundaries, eigenvector conditioning, modal reconstruction error, and long-horizon transient growth.