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Strict Majority Bootstrap Percolation on Augmented Tori and Random Regular Graphs: Experimental Results

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Cellular Automata and Discrete Complex Systems (AUTOMATA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8996))

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Abstract

We study the strict majority bootstrap percolation process on graphs. Vertices may be active or passive. Initially, active vertices are chosen independently with probability \(p\). Each passive vertex \(v\) becomes active if at least \(\lceil \frac{deg(v)+1}{2} \rceil \) of its neighbors are active (and thereafter never changes its state). If at the end of the process all vertices become active then we say that the initial set of active vertices percolates on the graph. We address the problem of finding graphs for which percolation is likely to occur for small values of \(p\). For that purpose we study percolation on two topologies. The first is an \(n\times n\) toroidal grid augmented with a universal vertex. Also, each vertex \(v\) in the torus is connected to all nodes whose distance to \(v\) is less than or equal to a parameter \(r\). The second family contains all random regular graphs of even degree, also augmented with a universal node. We compare our computational results to those obtained in previous publications for \(r\)-rings and random regular graphs.

This work has been partially supported by CONICYT via Basal in Applied Mathematics (I.R.), Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003 (I.R).

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Author information

Authors and Affiliations

  1. Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile

    P. Moisset de Espanés & I. Rapaport

  2. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    I. Rapaport

  3. Centro de Biotecnología y Bioingeniería, Universidad de Chile, Santiago, Chile

    P. Moisset de Espanés

Authors
  1. P. Moisset de Espanés
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  2. I. Rapaport
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Correspondence to P. Moisset de Espanés .

Editor information

Editors and Affiliations

  1. University of Hyogo, Himeji, Japan

    Teijiro Isokawa

  2. Hiroshima University, Hiroshima, Japan

    Katsunobu Imai

  3. University of Hyogo, Himeji, Japan

    Nobuyuki Matsui

  4. National Institute of Information and Communications Technology, Osaka, Japan

    Ferdinand Peper

  5. Osaka Electro-Communication University, Osaka, Japan

    Hiroshi Umeo

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de Espanés, P.M., Rapaport, I. (2015). Strict Majority Bootstrap Percolation on Augmented Tori and Random Regular Graphs: Experimental Results. In: Isokawa, T., Imai, K., Matsui, N., Peper, F., Umeo, H. (eds) Cellular Automata and Discrete Complex Systems. AUTOMATA 2014. Lecture Notes in Computer Science(), vol 8996. Springer, Cham. https://doi.org/10.1007/978-3-319-18812-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-18812-6_8

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-18812-6

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