Abstract
We define the (random) \(k\)-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon [14] except now a node must be cut \(k\) times before it is destroyed. The first order terms of the expectation and variance of \(\mathcal{X}_n\), the \(k\)-cut number of a path of length \(n\), are proved. We also show that \(\mathcal{X}_n\), after rescaling, converges in distribution to a limit \(\mathcal{B}_{k}\), which has a complicated representation. The paper then briefly discusses the \(k\)-cut number of general graphs. We conclude by some analytic results which may be of interest.
This work is supported by the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the Ragnar Söderbergs foundation.
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Addario-Berry, L., Broutin, N., Holmgren, C.: Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24(6), 2297–2339 (2014)
Ahsanullah, M.: Record Values-Theory and Applications. University Press of America Inc., Lanham (2004)
Cai, X.S., Devroye, L., Holmgren, C., Skerman, F.: \(k\)-cut on paths and some trees. ArXiv e-prints, January 2019
Cai, X.S., Holmgren, C.: Cutting resilient networks - complete binary trees. arXiv e-prints, November 2018
Chandler, K.N.: The distribution and frequency of record values. J. R. Stat. Soc. Ser. B. 14, 220–228 (1952)
Dagon, D., Gu, G., Lee, C.P., Lee, W.: A taxonomy of botnet structures. In: Twenty-Third Annual Computer Security Applications Conference (ACSAC 2007), pp. 325–339 (2007)
Drmota, M., Iksanov, A., Moehle, M., Roesler, U.: A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct. Algorithms 34(3), 319–336 (2009)
Durrett, R.: Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, 4th edn. Cambridge University Press, Cambridge (2010)
Holmgren, C.: Random records and cuttings in binary search trees. Combin. Probab. Comput. 19(3), 391–424 (2010)
Holmgren, C.: A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. Appl. Probab. 43(1), 151–177 (2011)
Iksanov, A., Möhle, M.: A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Comm. Probab. 12, 28–35 (2007)
Janson, S.: Random records and cuttings in complete binary trees. In: Mathematics and Computer Science III, Trends Math, pp. 241–253. Birkhäuser, Basel (2004)
Janson, S.: Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29(2), 139–179 (2006)
Meir, A., Moon, J.W.: Cutting down random trees. J. Austral. Math. Soc. 11, 313–324 (1970)
Meir, A., Moon, J.: Cutting down recursive trees. Math. Biosci. 21(3), 173–181 (1974)
Rényi, A.: Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand No. 8, 7–13 (1962)
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Cai, X.S., Devroye, L., Holmgren, C., Skerman, F. (2019). \(k\)-cuts on a Path. In: Heggernes, P. (eds) Algorithms and Complexity. CIAC 2019. Lecture Notes in Computer Science(), vol 11485. Springer, Cham. https://doi.org/10.1007/978-3-030-17402-6_10
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