Abstract:Hierarchical networks are frequently encountered in animal groups, gene networks, artificial engineering systems like multiple robots, unmanned vehicle systems, smartgrids, wind farm networks, etc. The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower to higher level nodes, like lagging birds' howl in a flock, or opinions of low-level individuals feeding back to the higher-level ones in a social group. This study reveals that for most of the large-scale real hierarchical networks, the majority of the reverse edges do not affect the synchronization process of the entire network; they are influenced only by a small part of these along specific paths. More surprisingly, a single effective reverse edge can slow down the synchronization of a huge hierarchical network by over 60%. The effect of such edges depends not on the network size, but merely on the average in-degree of the involved subnetwork. The overwhelming majority of active reverse edges turn out to have some kind of "bunching" effect on the information flows of hierarchical networks, which slows down the synchronization processes. This finding refines the understanding of the role of reverse edges on many natural, social and engineering hierarchical networks, which might be beneficial to precisely tune the synchronization rhythms of these networks. Our study also proposes an effective way to attack a hierarchical network by adding a malicious reverse edge to it and provides some guidance for protecting a network by screening out the specific small proportion of vulnerable nodes.