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We present a scalable approach for simplifying the stability analysis ofcluster synchronization patterns on directed networks. When a network hasdirectional couplings, decomposition of the coupling matrix into independentblocks (which in turn decouples the variational equation) is no longer adequateto reveal the full relations among perturbation modes. Instead, it is oftennecessary to introduce directional dependencies among the blocks and establishhierarchies among perturbation modes. For this purpose, we develop an algorithmthat finds the simultaneous block upper triangularization of sets of asymmetricmatrices, which generalizes the Jordan canonical decomposition from a singlematrix to an arbitrary number of matrices. The block upper triangularizationorders subspaces of the variational equation in a directional manner, allowingthe stability of perturbation modes to be analyzed in sequence. We show thatour algorithm gives the greatest possible simplification under mildassumptions, both in terms of the sizes of the blocks and in terms of thenumber of nonzero upper triangular entries linking the blocks.

Forget Partitions: Cluster Synchronization in Directed Networks Generate Hierarchies