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

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.