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Cluster synchronization in networks of coupled oscillators is the subject ofbroad interest from the scientific community, with applications ranging fromneural to social and animal networks and technological systems. Most of thesenetworks are directed, with flows of information or energy that propagateunidirectionally from given nodes to other nodes. Nevertheless, most of thework on cluster synchronization has focused on undirected networks. Here wecharacterize cluster synchronization in general directed networks. Our firstobservation is that, in directed networks, a cluster A of nodes might beone-way dependent on another cluster B: in this case, A may remain synchronizedprovided that B is stable, but the opposite does not hold. The maincontribution of this paper is a method to transform the cluster stabilityproblem in an irreducible form. In this way, we decompose the original probleminto subproblems of the lowest dimension, which allows us to immediately detectinter-dependencies among clusters. We apply our analysis to two examples ofinterest, a human network of violin players executing a musical piece for whichdirected interactions may be either activated or deactivated by the musicians,and a multilayer neural network with directed layer-to-layer connections.

One-way dependent clusters and stability of cluster synchronization in directed networks