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Percolation in networks with voids and bottlenecks
Amir Haji-Akbari1, Robert M Ziff
1Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA. hajakbar@umich.edu
Summary
A new method predicts percolation thresholds in bottleneck networks. This approach was validated on checkerboard and triangular lattices, yielding accurate asymptotic threshold values.
Area of Science:
- Statistical Mechanics
- Network Theory
- Computational Physics
Background:
- Percolation theory studies the formation of connected components in random networks.
- Predicting the percolation threshold in complex networks with bottlenecks is challenging.
- Existing methods may not accurately capture asymptotic behavior as network mesh size approaches zero.
Purpose of the Study:
- To propose a general method for predicting the asymptotic percolation threshold of networks with bottlenecks.
- To validate the proposed method using bond percolation on specific lattice structures.
- To determine the critical corner-connection probability for these lattices.
Main Methods:
- Developed a general method for predicting asymptotic percolation thresholds in bottleneck networks.
- Applied bond percolation to filled checkerboard and "stack-of-triangle" lattices.
- Estimated thresholds using the gradient percolation method and exact triangle-triangle transformations.
- Calculated critical corner-connection probabilities.
Main Results:
- The proposed method accurately predicts asymptotic percolation thresholds.
- Thresholds for checkerboard lattices approached 0.64222 as mesh size decreased.
- Thresholds for triangular lattices approached 0.53993 with finer mesh.
- Results are consistent with direct determinations based on predicted critical probabilities.
Conclusions:
- The general method provides a reliable way to determine asymptotic percolation thresholds for networks with bottlenecks.
- The study confirms the validity of the method on standard lattice models.
- This work contributes to a deeper understanding of percolation phenomena in complex systems.
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