# Numerical assessment of the percolation threshold using complement networks

## TL;DR

This paper investigates the relationship between empirical and model-predicted percolation thresholds in networks, revealing a linear relation and proposing a correction method using complement graphs for more accurate estimates.

## Contribution

It introduces a universal linear relation between empirical and theoretical percolation thresholds and develops a correction method based on complement networks.

## Key findings

- Empirical and model percolation thresholds are linearly related across various networks.
- A correction method using complement graphs improves threshold estimates.
- The approach enhances understanding of finite size effects and loops in real networks.

## Abstract

Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold $p_c$ and its model-predicted value $\pi_c$. Here we show the existence of an empirical linear relation between $p_c$ and $\pi_c$ across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of $\pi_c$. We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, $p_c$, and that of its complement, $\bar{p}_c$.