# Sensitivity of directed networks to the addition and pruning of edges and vertices

## TL;DR

This paper introduces the concept of susceptibility to measure how directed networks respond to edge and vertex modifications, revealing critical points and structural peculiarities in various real and synthetic networks.

## Contribution

It defines susceptibility for directed networks, analyzes its divergence at percolation transitions, and proposes it as a new tool for studying network structure and robustness.

## Key findings

- Susceptibility diverges at the directed percolation critical point.
- Different network regions exhibit varying sensitivities to modifications.
- Structural peculiarities cause non-monotonous sensitivity behaviors.

## Abstract

We study the sensitivity of directed complex networks to the addition and pruning of edges and vertices and introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the \emph{Caenorhabditis elegans} neural network, directed Erd\H{o}s-R\'enyi graphs, and others. We reveal a non-monotonous dependence of the sensitivity to random pruning of edges or vertices in the case of \emph{Caenorhabditis elegans} and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.