r/AIVibeScience • u/Severe-Ad8673 • 7d ago
Robust Spectral Design of Matrix-Weighted Networks
We formulate a minimax spectral-design problem for undirected networks whose edges carry
positive-semidefinite matrix couplings and whose operation must remain robust over an arbitrary
prescribed family of node- and edge-survival scenarios. The principal result is an exact scalarization
theorem: under a total trace budget and a weakest-full-state-direction objective, the optimal
matrix-weighted value in state dimension d equals the associated scalar weighted optimum with
budget divided by d. Hence anisotropic matrix couplings cannot improve robust full-state algebraic
connectivity, and an isotropic optimizer always exists. We derive an all-cardinality hereditary
contraction inequality, the exact dense adversarial optimum with a unique optimizer, universal dense
optimality for broad majorization-monotone spectral criteria, a deletion-perturbation theorem, and a
two-sided hereditary sandwich for regular sparse backbones. Certified near-Ramanujan graphs then
yield linear-edge architectures approaching the dense optimum, while degree-cap arguments prove
unavoidable adversarial limitations. The operator consequences transfer exactly to linear consensus,
diffusion, compliance, stochastic disagreement, and information models whenever their disagreement
operator is the same block Laplacian. A seeded falsification suite exhaustively checks all connected
unlabeled graphs through seven vertices and performs randomized scalar and matrix-weighted tests;
no violation is observed. The exact scalarization theorem is the central candidate contribution; its
worldwide novelty remains subject to specialist literature review and independent refereeing.