Reply to comment on "Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks"

TL;DR

This paper is a brief reply to a comment on a previous study about the limitations of the simultaneous block diagonalization technique in analyzing synchronization in random networks, clarifying that 'failure' refers to lack of simplification, not incorrect decoupling.

Contribution

The authors clarify their previous findings, emphasizing that the 'failure' of the SBD technique pertains to its inability to simplify the stability analysis, not to its correctness in decoupling.

Findings

SBD does not always produce a simplified system for analysis.

The paper clarifies the interpretation of 'failure' in the context of SBD.

The original indices measure success or failure of the technique in decoupling.

Abstract

We respond briefly to a comment [1, arXiv:2110.15493] recently posted online on our paper [2, arXiv:2108.07893]. Complete and cluster synchronization of random networks is undoubtedly a topic of interest in the Physics, Engineering, and Nonlinear Dynamics literature. In [3] we study both complete and cluster synchronization of networks and introduce indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. Our usage of the word `failure' indicates that the technique does not produce a decomposition which results in a system which is easier to analyze, not that the technique fails in correctly decoupling these problems.