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Ordinary percolation with discontinuous transitions

Author

Listed:
  • Stefan Boettcher

    (Emory University)

  • Vijay Singh

    (Emory University)

  • Robert M. Ziff

    (University of Michigan)

Abstract

Percolation on a one-dimensional lattice and fractals, such as the Sierpinski gasket, is typically considered to be trivial, because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a non-trivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here we provide a simple example in the form of a small-world network consisting of a one-dimensional lattice which, when combined with a hierarchy of long-range bonds, reveals many features of this transition in a mathematically rigorous manner.

Suggested Citation

  • Stefan Boettcher & Vijay Singh & Robert M. Ziff, 2012. "Ordinary percolation with discontinuous transitions," Nature Communications, Nature, vol. 3(1), pages 1-5, January.
    • Handle: RePEc:nat:natcom:v:3:y:2012:i:1:d:10.1038_ncomms1774
    DOI: 10.1038/ncomms1774
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    Cited by:

    1. Li, Hui & Li, Tianwei, 2018. "Recursive sequences in the Ford sphere packing," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 94-106.
    2. Bastas, N. & Giazitzidis, P. & Maragakis, M. & Kosmidis, K., 2014. "Explosive percolation: Unusual transitions of a simple model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 54-65.
    3. Hanlin Sun & Filippo Radicchi & Jürgen Kurths & Ginestra Bianconi, 2023. "The dynamic nature of percolation on networks with triadic interactions," Nature Communications, Nature, vol. 14(1), pages 1-9, December.
    4. Giacomo Rapisardi & Ivan Kryven & Alex Arenas, 2022. "Percolation in networks with local homeostatic plasticity," Nature Communications, Nature, vol. 13(1), pages 1-9, December.
    5. Han, Jihui & Zhang, Ge & Dong, Gaogao & Zhao, Longfeng & Shi, Yuefeng & Zou, Yijiang, 2024. "Exact analysis of generalized degree-based percolation without memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 642(C).
    6. Oh, S.M. & Son, S.-W. & Kahng, B., 2021. "Percolation transitions in growing networks under achlioptas processes: Analytic solutions," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).

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