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Unstable supercritical discontinuous percolation transitions.
Wei Chen1, Xueqi Cheng, Zhiming Zheng
1Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, Peking University, Beijing, China and University of California, Davis, California 95616, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 16, 2013
Summary
Discontinuous percolation transitions in random networks can exhibit complex behavior. This study finds the largest jump doesn't always align with the percolation threshold, revealing novel phenomena in supercritical regimes.
Area of Science:
- Network Science
- Statistical Physics
- Graph Theory
Background:
- Percolation transitions in random networks are crucial for understanding network properties.
- Discontinuous percolation transitions, marked by macroscopic jumps, have been recently introduced.
- These transitions can display exotic behaviors like Devil's staircases in the supercritical regime.
Purpose of the Study:
- To investigate the relationship between the largest jump in component size and the percolation transition point.
- To analyze this relationship across various graph evolution processes, including Erdős-Rényi percolation and models with edge competition or growth by overtaking.
- To explore the rich behaviors of discontinuous percolation transitions in supercritical regimes.
Main Methods:
- Analysis of graph evolution processes leading to discontinuous percolation.
- Comparison of the location of the largest component size jump with the percolation threshold.
- Investigation of Erdős-Rényi percolation, percolation via edge competition, and growth by overtaking models.
- Examination of models with Devil's staircase behavior and the generalized Bohman-Frieze-Wormald model.
Main Results:
- For globally continuous processes (Erdős-Rényi, explosive transitions), the largest jump asymptotically coincides with the percolation transition.
- For genuinely discontinuous processes with a staircase order parameter, the largest discontinuity typically does not align with the percolation transition.
- The generalized Bohman-Frieze-Wormald model shows parameter-dependent behavior, with some regimes exhibiting unstable discontinuous transitions in the supercritical phase.
Conclusions:
- The location of the largest jump relative to the percolation threshold varies significantly depending on the nature of the discontinuity.
- Discontinuous percolation transitions exhibit richer behavior in the supercritical regime than previously appreciated.
- Novel phenomena, such as unstable discontinuous transitions, are identified in supercritical percolation models.
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