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          Institute: MPI für Dynamik und Selbstorganisation     Collection: Nichtlineare Dynamik     Display Documents

ID: 673710.0, MPI für Dynamik und Selbstorganisation / Nichtlineare Dynamik
Unstable supercritical discontinuous percolation transitions
Authors:Chen, Wei; Cheng, Xueqi; Zheng, Zhiming; Chung, Ning Ning; D'Souza, Raissa M.; Nagler, Jan
Date of Publication (YYYY-MM-DD):2013-10-31
Title of Journal:Physical Review E
Sequence Number of Article:042152
Review Status:Peer-review
Audience:Experts Only
Abstract / Description: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.
External Publication Status:published
Document Type:Article
Communicated by:Folkert Müller-Hoissen
Affiliations:MPI für Dynamik und Selbstorganisation/Nichtlineare Dynamik
External Affiliations:Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
School of Mathematical Sciences, Peking University, Beijing, China
University of California, Davis, California 95616, USA
Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing University of Aeronautics and Astronautics, 100191 Beijing, China
Department of Physics, National University of Singapore, 117542, Singapore
Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Göttingen, Germany
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