Mapping the Structure of Directed Networks: Beyond the Bow-Tie Diagram

We reveal a hierarchical, multilayer organization of finite components—i.e., tendrils and tubes—around the giant connected components in directed networks and propose efficient algorithms allowing one to uncover the entire organization of key real-world directed networks, such as the World Wide Web, the neural network of Caenorhabditis elegans, and others. With increasing damage, the giant components decrease in size while the number and size of tendril layers increase, enhancing the susceptibility of the networks to damage.