  The interactions between the components of many real-world systems are best
modelled by networks with multiple layers. Different theories have been
proposed to explain how multilayered connections affect the linear stability of
synchronization in dynamical systems. However, the resulting equations are
computationally expensive, and therefore difficult, if not impossible, to solve
for large systems. To bridge this gap, we develop a mean-field theory of
synchronization for networks with multiple interaction layers. By assuming
quasi-identical layers, we obtain accurate assessments of synchronization
stability that are comparable with the exact results. In fact, the accuracy of
our theory remains high even for networks with very dissimilar layers, thus
posing a general question about the mean-field nature of synchronization
stability in multilayer networks. Moreover, the computational complexity of our
approach is only quadratic in the number of nodes, thereby allowing the study
of systems whose investigation was thus far precluded.
