Forget Partitions: Cluster Synchronization in Directed Networks Generate Hierarchies

TL;DR

This paper introduces a scalable method for analyzing the stability of cluster synchronization in directed networks by hierarchically decomposing the coupling matrix, enabling sequential stability analysis of perturbation modes.

Contribution

It develops an algorithm for simultaneous block upper triangularization of multiple asymmetric matrices, generalizing Jordan decomposition to reveal hierarchical relations among perturbation modes.

Findings

Algorithm achieves maximal simplification of stability analysis.

Hierarchical decomposition reveals directional dependencies among modes.

Method applicable to large directed networks with complex coupling structures.

Abstract

We present a scalable approach for simplifying the stability analysis of cluster synchronization patterns on directed networks. When a network has directional couplings, decomposition of the coupling matrix into independent blocks (which in turn decouples the variational equation) is no longer adequate to reveal the full relations among perturbation modes. Instead, it is often necessary to introduce directional dependencies among the blocks and establish hierarchies among perturbation modes. For this purpose, we develop an algorithm that finds the simultaneous block upper triangularization of sets of asymmetric matrices, which generalizes the Jordan canonical decomposition from a single matrix to an arbitrary number of matrices. The block upper triangularization orders subspaces of the variational equation in a directional manner, allowing the stability of perturbation modes to be analyzed in sequence. We show that our algorithm gives the greatest possible simplification under mild assumptions, both in terms of the sizes of the blocks and in terms of the number of nonzero upper triangular entries linking the blocks.