The distance backbone of directed networks

TL;DR

This paper extends a previously developed algebraic method to identify the distance backbone, or core structure, of weighted directed networks, revealing significant redundancy and robustness across various real-world systems.

Contribution

The paper generalizes an existing undirected graph methodology to directed graphs, enabling analysis of redundancy and robustness in directed complex networks.

Findings

Directed networks contain substantial redundancy similar to undirected ones.

The methodology effectively uncovers the distance backbone in diverse real-world networks.

Redundancy correlates with network robustness and structural importance.

Abstract

In weighted graphs the shortest path between two nodes is often reached through an indirect path, out of all possible connections, leading to structural redundancies which play key roles in the dynamics and evolution of complex networks. We have previously developed a parameter-free, algebraically-principled methodology to uncover such redundancy and reveal the distance backbone of weighted graphs, which has been shown to be important in transmission dynamics, inference of important paths, and quantifying the robustness of networks. However, the method was developed for undirected graphs. Here we expand this methodology to weighted directed graphs and study the redundancy and robustness found in nine networks ranging from social, biomedical, and technical systems. We found that similarly to undirected graphs, directed graphs in general also contain a large amount of redundancy, as measured by the size of their (directed) distance backbone. Our methodology adds an additional tool to the principled sparsification of complex networks and the measure of their robustness.