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Burning the Plane

Densities of the Infinite Cartesian Grid

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Abstract

Graph burning is a discrete-time process on graphs, where vertices are sequentially burned, and burned vertices cause their neighbours to burn over time. We consider extremal properties of this process in the new setting where the underlying graph is also changing at each time-step. The main focus is on the possible densities of burning vertices when the sequence of underlying graphs are growing grids in the Cartesian plane, centred at the origin. If the grids are of height and width \(2cn+1\) at time n, then all values in \(\left[ \frac{1}{2c^2} , 1 \right] \) are possible densities for the burned set. For faster growing grids, we show that there is a threshold behaviour: if the size of the grids at time n is \(\omega (n^{3/2})\), then the density of burned vertices is always 0, while if the grid sizes are \(\varTheta (n^{3/2})\), then positive densities are possible. Some extensions to lattices of arbitrary but fixed dimension are also considered.

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References

  1. Benevides, F.S., Przykucki, M.: On slowly percolating sets of minimal size in bootstrap percolation. Electron. J. Comb. 20, 1–20 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Benevides, F.S., Przykucki, M.: Maximum percolation time in two-dimensional bootstrap percolation. SIAM J. Discret. Math. 29, 224–251 (2015)

    Article  MathSciNet  Google Scholar 

  3. Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E.: Burning a graph is hard. Discret. Appl. Math. 232, 73–87 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E.: Bounds on the burning number. Discret. Appl. Math. 235, 16–22 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bonato, A., Janssen, J., Roshanbin, E.: Burning a graph as a model of social contagion. Lect. Notes Comput. Sci. 8882, 13–22 (2014)

    Article  MathSciNet  Google Scholar 

  6. Bonato, A., Janssen, J., Roshanbin, E.: How to burn a graph. Internet Math. 12, 85–100 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bonato, A., Kamali, S.: Approximation and algorithms for burning graphs. Lect. Notes Comput. Sci. 11436, 74–92 (2019)

    Article  Google Scholar 

  8. Bonato, A., Lidbetter, T.: Bounds on the burning numbers of spiders and path-forests. Theoret. Comput. Sci. 794, 12–19 (2019)

    Article  MathSciNet  Google Scholar 

  9. Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE Trans. Inf. Theory 52, 2508–2530 (2006)

    Article  MathSciNet  Google Scholar 

  10. Cai, L., Wang, W.: The surviving rate of a graph for the firefighter problem. SIAM J. Discret. Math. 23, 1814–1826 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Australas. J. Comb. 43, 57–77 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Fitzpatrick, S.L., Wilm, L.: Burning circulant graphs. Preprint (2018)

  13. Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 1420–1443 (1978)

  14. Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Neyman, J., LeCam, L.M. (eds.) Bernoulli 1713 Bayes 1763 Laplace 1813, pp. 61–110. Springer, Berlin (1965)

    Google Scholar 

  15. Harris, T.E.: Contact interactions on a lattice. Ann. Probab. 2, 969–988 (1974)

    Article  MathSciNet  Google Scholar 

  16. Hartnell, B.L.: Firefighter! an application of domination. In: Presentation at 25th Manitoba Conference on Combinatorial Mathematics and Computing, University of Manitoba in Winnipeg, MB (1995)

  17. Kramer, A.D.I., Guillory, J.E., Hancock, J.T.: Experimental evidence of massive-scale emotional contagion through social networks. Proc. Natl. Acad. Sci. USA 111, 8788–8790 (2014)

    Article  Google Scholar 

  18. Land, M., Lu, L.: An upper bound on the burning number of graphs. In: Algorithms and models for the web graph, pp. 1–8. Springer, Cham (2016)

  19. Mitsche, D., Prałat, P., Roshanbin, E.: Burning graphs—a probabilistic perspective. Graphs Comb. 33, 449–471 (2017)

    Article  MathSciNet  Google Scholar 

  20. Przykucki, M.: Maximal percolation times in hypercubes under 2-bootstrap percolation. Electron. J. Comb. 19, 1–13 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Richardson, D.: Random growth in a tessellation. Math. Proc. Camb. Philos. Soc. 74, 515–528 (1973)

    Article  MathSciNet  Google Scholar 

  22. Roshanbin, E.: Burning a graph as a model of social contagion. PhD thesis, Dalhousie University, Halifax, NS (2016)

  23. Schelling, T.: Micromotives and Macrobehavior. Norton, New York (1978)

    Google Scholar 

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Acknowledgements

The authors would like to thank the referee for their very careful reading of the paper and the many helpful suggestions which greatly improved a number of the proofs.

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Authors and Affiliations

  1. Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada

    Anthony Bonato

  2. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Karen Gunderson & Amy Shaw

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  1. Anthony Bonato
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  2. Karen Gunderson
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  3. Amy Shaw
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Correspondence to Karen Gunderson.

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AB and KG supported by NSERC Discovery grants.

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Bonato, A., Gunderson, K. & Shaw, A. Burning the Plane. Graphs and Combinatorics 36, 1311–1335 (2020). https://doi.org/10.1007/s00373-020-02182-9

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