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Published on: October 4, 2018
Finite-size effects for percolation on Apollonian networks
Daniel M Auto1, André A Moreira, Hans J Herrmann
1Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970 Fortaleza, Ceará, Brazil.
Summary
We studied percolation on Apollonian networks, finding ultraslow convergence to the thermodynamic limit. This highlights the importance of finite-size effects in hierarchical real-world systems.
Area of Science:
- Network Science
- Statistical Physics
- Complex Systems
Background:
- Apollonian networks exhibit small-world, scale-free, and hierarchical properties common in real-world systems.
- Percolation theory analyzes the connectivity and behavior of networks under random processes.
Purpose of the Study:
- To investigate the percolation problem specifically on the Apollonian network model.
- To analyze the convergence properties of percolation on hierarchical networks.
Main Methods:
- Utilized the real-space renormalization-group technique.
- Derived exact iterative equations relating percolation properties across different network scales.
Main Results:
- Percolation probability and average cluster mass exhibit logarithmic convergence towards the thermodynamic limit.
- Demonstrated ultraslow convergence, potentially characteristic of hierarchical networks.
Conclusions:
- Finite-size effects are crucial for understanding percolation in real, finite, and hierarchical network systems.
- The ultraslow convergence observed suggests a fundamental property of hierarchical network structures.
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