Designing temporal networks that synchronize under resource constraints

The paper proves (at a proof-sketch level) that, under a commuting-Laplacian, slow-switching design with eigenvalue schedule splitting the nontrivial Laplacian spectrum into two oscillating groups, the maximum transverse Lyapunov exponent equals the averaged master stability function in the slow-switching limit and becomes strictly negative at σ = σc when Λ''(σc) < 0; it also explains why no static network can achieve strictly negative exponents at that budget. The candidate solution reproduces the same logic, adding a block/Grönwall estimate and explicit moment computations for the schedule in Eq. (5). The two arguments are substantially the same.