  Real-world systems in epidemiology, social sciences, power transportation,
economics and engineering are often described as multilayer networks. Here we
first define and compute the symmetries of multilayer networks, and then study
the emergence of cluster synchronization in these networks. We distinguish
between independent layer symmetries which occur in one layer and are
independent of the other layers and dependent layer symmetries which involve
nodes in different layers. We study stability of the cluster synchronous
solution by decoupling the problem into a number of independent blocks and
assessing stability of each block through a Master Stability Function. We see
that blocks associated with dependent layer symmetries have a different
structure than the other blocks, which affects the stability of clusters
associated with these symmetries. Finally, we validate the theory in a fully
analog experiment in which seven electronic oscillators of three kinds are
connected with two kinds of coupling.
