  The concept of symmetry in multilayer networks and its use for cluster
analysis of the network have recently been reported in [Rossa et al., Nat.
Commun. 11, 1 (2020)]. It has been shown that clusters in a multilayer network
can be determined by finding the symmetry group of the multilayer from the
symmetry groups of the independent individual layers. However, finding symmetry
group elements of large complex networks consisting of several layers can often
be difficult. Here we present a new mathematical framework involving the
structure of the eigen vector centrality (EVC) of the adjacency matrix for
cluster analysis in multilayer networks. The framework is based on an
analytical result showing a direct correspondence between the EVC elements and
the clusters of the network. Using this result, cluster analysis is performed
successfully for several multilayer networks and in each case the results are
found to be consistent with that obtained using the symmetry group analysis
method. Finally, cluster synchronization in multilayer networks on
Sakaguchi-Kuramoto (SK) model is investigated under the proposed framework.
Stability analysis of the cluster synchronization states are also done using
master stability function approach.
