Master Stability Function

Network synchronization stability — Pecora-Carroll framework, 1998

Master Stability Function (MSF): For a network of identical oscillators coupled through Laplacian L with coupling σ, synchronization is stable iff all transverse modes αk = σ·λk fall in the stable region of the MSF.
ẋi = F(xi) − σ Σj Lij H(xj)
The MSF Λmax(α) is the largest Lyapunov exponent of the variational equation. Synchronization requires Λmax(σλk) < 0 for all k ≥ 2. Each dot on the MSF plot shows a network eigenvalue — green = stable, red = unstable.