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The algebraic method in tree percolation

Fatemeh Mohammadi (UGent) , Eduardo Sáenz-de-Cabezón and Henry P. Wynn
(2016) SIAM JOURNAL ON DISCRETE MATHEMATICS. 30(2). p.1193-1212
Author
Organization
Abstract
We apply the methods of algebraic reliability to the study of percolation on trees. To a complete k-ary tree T-k,T-n of depth n we assign a monomial ideal I-k,I-n on Sigma(n)(i = 1) k(i) variables and k(n) minimal monomial generators. We give explicit recursive formulae for the Betti numbers of I-k,I-n and their Hilbert series, which allow us to study explicitly percolation on T-k,T-n. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.
Keywords
percolation, Betti numbers, monomial ideals, Hilbert series, MONOMIAL IDEALS, RELIABILITY, DIVISORS, GRAPHS

Citation

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MLA
Mohammadi, Fatemeh, et al. “The Algebraic Method in Tree Percolation.” SIAM JOURNAL ON DISCRETE MATHEMATICS, vol. 30, no. 2, 2016, pp. 1193–212, doi:10.1137/151003647.
APA
Mohammadi, F., Sáenz-de-Cabezón, E., & Wynn, H. P. (2016). The algebraic method in tree percolation. SIAM JOURNAL ON DISCRETE MATHEMATICS, 30(2), 1193–1212. https://doi.org/10.1137/151003647
Chicago author-date
Mohammadi, Fatemeh, Eduardo Sáenz-de-Cabezón, and Henry P. Wynn. 2016. “The Algebraic Method in Tree Percolation.” SIAM JOURNAL ON DISCRETE MATHEMATICS 30 (2): 1193–1212. https://doi.org/10.1137/151003647.
Chicago author-date (all authors)
Mohammadi, Fatemeh, Eduardo Sáenz-de-Cabezón, and Henry P. Wynn. 2016. “The Algebraic Method in Tree Percolation.” SIAM JOURNAL ON DISCRETE MATHEMATICS 30 (2): 1193–1212. doi:10.1137/151003647.
Vancouver
1.
Mohammadi F, Sáenz-de-Cabezón E, Wynn HP. The algebraic method in tree percolation. SIAM JOURNAL ON DISCRETE MATHEMATICS. 2016;30(2):1193–212.
IEEE
[1]
F. Mohammadi, E. Sáenz-de-Cabezón, and H. P. Wynn, “The algebraic method in tree percolation,” SIAM JOURNAL ON DISCRETE MATHEMATICS, vol. 30, no. 2, pp. 1193–1212, 2016.
@article{8667474,
  abstract     = {{We apply the methods of algebraic reliability to the study of percolation on trees. To a complete k-ary tree T-k,T-n of depth n we assign a monomial ideal I-k,I-n on Sigma(n)(i = 1) k(i) variables and k(n) minimal monomial generators. We give explicit recursive formulae for the Betti numbers of I-k,I-n and their Hilbert series, which allow us to study explicitly percolation on T-k,T-n. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.}},
  author       = {{Mohammadi, Fatemeh and Sáenz-de-Cabezón, Eduardo and Wynn, Henry P.}},
  issn         = {{0895-4801}},
  journal      = {{SIAM JOURNAL ON DISCRETE MATHEMATICS}},
  keywords     = {{percolation,Betti numbers,monomial ideals,Hilbert series,MONOMIAL IDEALS,RELIABILITY,DIVISORS,GRAPHS}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{1193--1212}},
  title        = {{The algebraic method in tree percolation}},
  url          = {{http://doi.org/10.1137/151003647}},
  volume       = {{30}},
  year         = {{2016}},
}
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