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

TL;DR

This paper clarifies that the limitations observed in the SBD technique for network synchronization are due to inappropriate network models used in prior tests, reaffirming SBD's effectiveness in suitable scenarios.

Contribution

The authors demonstrate that the SBD technique's perceived failure is caused by poor benchmark choices, emphasizing its robustness and optimality in analyzing synchronization patterns.

Findings

SBD always finds the optimal reduction for any network and pattern

Poor benchmarks can misrepresent SBD's effectiveness

SBD is effective beyond symmetry-based analysis

Abstract

In their recent preprint [arXiv:2108.07893v1], S. Panahi, N. Amaya, I. Klickstein, G. Novello, and F. Sorrentino tested the simultaneous block diagonalization (SBD) technique on synchronization in random networks and found the dimensionality reduction to be limited. Based on this observation, they claimed the SBD technique to be a failure in generic situations. Here, we show that this is not a failure of the SBD technique. Rather, it is caused by inappropriate choices of network models. SBD provides a unified framework to analyze the stability of synchronization patterns that are not encumbered by symmetry considerations, and it always finds the optimal reduction for any given synchronization pattern and network structure [SIAM Rev. 62, 817-836 (2020)]. The networks considered by Panahi et al. are poor benchmarks for the performance of the SBD technique, as these systems are often intrinsically irreducible, regardless of the method used. Thus, although the results in Panahi et al. are technically valid, their interpretations are misleading and akin to claiming a community detection algorithm to be a failure because it does not find any meaningful communities in Erd\H{o}s-R\'enyi networks.