Robustness of the emergence of synchronized clusters in branching hierarchical systems under parametric noise

The dynamical robustness of networks in the presence of noise is of utmost fundamental and applied interest. In this work, we explore the effect of parametric noise on the emergence of synchronized clusters in diffusively coupled Chaté-Manneville maps on a branching hierarchical structure. We consider both quenched and dynamically varying parametric noise. We find that the transition to a synchronized fixed point on the maximal cluster is robust in the presence of both types of noise. We see that the small sub-maximal clusters of the system, which coexist with the maximal cluster, exhibit a power-law cluster size distribution. This power-law scaling of synchronized cluster sizes is robust against noise in a broad range of coupling strengths. However, interestingly, we find a window of coupling strength where the system displays markedly different sensitivities to noise for the maximal cluster and the small clusters, with the scaling exponent for the cluster distribution for small clusters exhibiting clear dependence on noise strength, while the cluster size of the maximal cluster of the system displays no significant change in the presence of noise. Our results have implications for the observability of synchronized cluster distributions in real-world hierarchical networks, such as neural networks, power grids, and communication networks, that necessarily have parametric fluctuations.