Percolation in Networks with Voids and Bottlenecks

TL;DR

This paper introduces a method to predict the percolation threshold in complex networks with voids and bottlenecks, validated on specific lattice models, showing convergence to known asymptotic values.

Contribution

A novel general approach for estimating asymptotic percolation thresholds in networks with bottlenecks and voids, validated on lattice models with different mesh sizes.

Findings

Thresholds approach known asymptotic values as mesh size decreases.

Method accurately predicts critical corner-connection probabilities.

Validation on checkerboard and triangular lattices confirms effectiveness.

Abstract

A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the sub-net mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and "stack-of-triangle" lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993 respectively as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.