Cluster analysis in multilayer networks using eigen vector centrality

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.