Mean-field nature of synchronization stability in networks with multiple interaction layers

The paper explicitly derives the mean-field equations (6)–(7) from the exact master-stability formulation (2) by introducing the mean-field spectral-overlap matrix Γ(α),MF with small rotation angle ε(α), and shows the associated complexity drop from O(N^2 M) to O(N M) per mode; these match the candidate solution’s parts (a)–(b) exactly . However, for part (c), the paper offers only numerical evidence that the sign of the largest Lyapunov exponent is “virtually always” assessed correctly and does not provide the analytic, small-ε robustness proof; the candidate solution supplies a correct argument via matrix-measure continuity bounds that fills this gap. Hence the paper is incomplete relative to (c), while the model’s proof is correct under mild, standard assumptions.