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  Unstable supercritical discontinuous percolation transitions

Chen, W., Cheng, X., Zheng, Z., Chung, N. N., D'Souza, R. M., & Nagler, J. (2013). Unstable supercritical discontinuous percolation transitions. Physical Review E, 88: 042152. doi:10.1103/PhysRevE.88.042152.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0029-0F9F-1 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0029-0FA0-C
Genre: Journal Article

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 Creators:
Chen, Wei, Author
Cheng, Xueqi, Author
Zheng, Zhiming, Author
Chung, Ning Ning, Author
D'Souza, Raissa M., Author
Nagler, Jan1, Author              
Affiliations:
1Department of Nonlinear Dynamics, Max Planck Institute for Dynamics and Self-Organization, Max Planck Society, ou_2063286              

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 Abstract: The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a “Devil's staircase” of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdős-Rényipercolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdős-Rényiand other processes exhibiting global continuity, including models exhibiting an “explosive” transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions—a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.

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Language(s): eng - English
 Dates: 2013-10-31
 Publication Status: Published in print
 Pages: -
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 Table of Contents: -
 Rev. Method: Peer
 Identifiers: eDoc: 673710
DOI: 10.1103/PhysRevE.88.042152
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Title: Physical Review E
Source Genre: Journal
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Pages: - Volume / Issue: 88 Sequence Number: 042152 Start / End Page: - Identifier: -