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Strong-majority bootstrap percolation on regular graphs with low dissemination threshold

Author

Listed:
  • Mitsche, Dieter
  • Pérez-Giménez, Xavier
  • Prałat, Paweł

Abstract

Consider the following model of strong-majority bootstrap percolation on a graph. Let r≥1 be some integer, and p∈[0,1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v)+r)/2 of its neighbours are active. Given any arbitrarily small p>0 and any integer r, we construct a family of d=d(p,r)-regular graphs such that with high probability all vertices become active in the end. In particular, the case r=1 answers a question and disproves a conjecture of Rapaport et al. (2011).

Suggested Citation

  • Mitsche, Dieter & Pérez-Giménez, Xavier & Prałat, Paweł, 2017. "Strong-majority bootstrap percolation on regular graphs with low dissemination threshold," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 3110-3134.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:3110-3134
    DOI: 10.1016/j.spa.2017.02.001
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    References listed on IDEAS

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    1. Cerf, R. & Manzo, F., 2002. "The threshold regime of finite volume bootstrap percolation," Stochastic Processes and their Applications, Elsevier, vol. 101(1), pages 69-82, September.
    2. Candellero, Elisabetta & Fountoulakis, Nikolaos, 2016. "Bootstrap percolation and the geometry of complex networks," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 234-264.
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