title: Mapping the Structure of Directed Networks: Beyond the Bow-Tie Diagram.
abstract: 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.
authors.affiliation: Departamento de Física, Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
authors.forename: G
authors.initials: G
authors.lastname: Timár
authors.affiliation: Departamento de Física, Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.|A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
authors.forename: A V
authors.initials: AV
authors.lastname: Goltsev
authors.affiliation: Departamento de Física, Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.|A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
authors.forename: S N
authors.initials: SN
authors.lastname: Dorogovtsev
authors.affiliation: Departamento de Física, Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
authors.forename: J F F
authors.initials: JF
authors.lastname: Mendes