Mapping the Structure of Directed Networks: Beyond the "Bow-tie" Diagram

TL;DR

This paper uncovers a hierarchical multilayer structure in directed networks beyond the traditional 'bow-tie' diagram, revealing how network components reorganize under damage and proposing algorithms to analyze this organization.

Contribution

It introduces a new multilayer hierarchical framework for directed networks and efficient algorithms to map their complex structure beyond the classic 'bow-tie' model.

Findings

Networks exhibit multilayer tendril and tube structures.

Damage reduces giant component size and increases tendril layers.

Networks become more susceptible to damage as structure complexity increases.

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 \emph{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.