Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks

TL;DR

This paper examines the limitations of the simultaneous block diagonalization technique in analyzing synchronization stability in random networks, showing it often reduces to trivial transformations without effective decoupling.

Contribution

It highlights the failure of SBD in simplifying stability analysis for random networks, providing indices to measure its success or failure.

Findings

SBD often reduces to trivial transformations in random networks

The technique does not effectively decouple stability problems

Indices can measure the success or failure of SBD application

Abstract

We discuss here the application of the simultaneous block diagonalization (SBD) of matrices to the study of the stability of both complete and cluster synchronization in random (generic) networks. For both problems, we define indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. We then see that in the case of random networks the extent of the dimensionality reduction achievable is the same as that produced by application of a trivial transformation.