The Immense Impact of Reverse Edges on Large Hierarchical Networks

Hierarchical networks are frequently encountered in animal groups, gene networks, and artificial engineering systems such as multiple robots, unmanned vehicle systems, smart grids, wind farm networks, and so forth. The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower- to higher- level nodes, such as lagging birds' howl in a flock or the opinions of lower-level individuals feeding back to higher-level ones in a social group. This study reveals that, for most large-scale real hierarchical networks, the majority of the reverse edges do not affect the synchronization process of the entire network; the synchronization process is influenced only by a small part of these reverse edges 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 only 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 synchronization processes. This finding refines the current understanding of the role of reverse edges in many natural, social, and engineering hierarchical networks, which might be beneficial for precisely tuning 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.