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Automata and Complexity pp 179–201Cite as

Computing the Probability of Getting Infected: On the Counting Complexity of Bootstrap Percolation

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Part of the Emergence, Complexity and Computation book series (ECC,volume 42)

Abstract

Consider a network where each node has one over two possible states, namely healthy or infected. Given an initial configuration, the network evolves in discrete time-steps picking uniformly at random a single node and updating its state according to the following rule: if the node is infected, it remains infected. If the node is healthy it switches its state to the one of the strict majority of its neighbors. We address, from the point of view of the computational complexity, the problem of computing the probability that a given healthy node becomes infected in at most a given number of time-steps, given as input network and an initial configuration. We show that this problem is \(\#\)P-Complete in general, and solvable in polynomial time when the input graph is of degree at most 4.

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Acknowledgements

Eric Goles has been a mentor, colleague, and a dear friend for both of us. This chapter is not only dedicated to Eric’s 70th birthday, but also to the passion, love, and enthusiasm that he has taught us for this beautiful research topic. For both of us working with Eric has been a wonderful experience in many aspects. As he likes to say: we have a good time, and in the meantime, we do some mathematics. We also want to acknowledge the financial support given by: ANID via PAI + Convocatoria Nacional Subvención a la Incorporación en la Academia Año 2017 + PAI77170068 (P.M.), FONDECYT 11190482 (P.M.), FONDECYT 1200006 (P.M.), STIC- AmSud CoDANet project 88881.197456/2018-01 (P.M.), ANID via PFCHA/DOCTORADO NACIONAL/2018 – 21180910 + PIA AFB 170001 (M.R.W).

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Authors and Affiliations

  1. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile

    Pedro Montealegre

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

    Martín Ríos-Wilson

  3. Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

    Martín Ríos-Wilson

Authors
  1. Pedro Montealegre
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  2. Martín Ríos-Wilson
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Corresponding author

Correspondence to Pedro Montealegre .

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Editors and Affiliations

  1. Unconventional Computing Centre, University of the West of England, Bristol, UK

    Prof. Andrew Adamatzky

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Montealegre, P., Ríos-Wilson, M. (2022). Computing the Probability of Getting Infected: On the Counting Complexity of Bootstrap Percolation. In: Adamatzky, A. (eds) Automata and Complexity. Emergence, Complexity and Computation, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-92551-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-92551-2_12

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