# Weighted percolation on directed networks

## TL;DR

This paper develops a theoretical criterion for network disintegration under node removal in directed networks, based on the largest eigenvalue of a weighted adjacency matrix, applicable to various removal strategies without needing a Markov model.

## Contribution

It introduces a novel eigenvalue-based percolation criterion for directed networks that does not require a Markov network model, broadening applicability.

## Key findings

- The percolation threshold is predicted by the largest eigenvalue condition.
- Numerical tests confirm the accuracy of the theoretical criterion.
- The method applies to various node removal strategies in directed networks.

## Abstract

We present an analysis of the percolation transition for general node removal strategies valid for locally tree-like directed networks. On the basis of heuristic arguments we predict that, if the probability of removing node $i$ is $p_i$, the network disintegrates if $p_i$ is such that the largest eigenvalue of the matrix with entries $A_{ij}(1-p_i)$ is less than 1, where $A$ is the adjacency matrix of the network. The knowledge or applicability of a Markov network model is not required by our theory, thus making it applicable to situations not covered by previous works. We test our predicted percolation criterion against numerical results for different networks and node removal strategies.