Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T22:09:31.850Z Has data issue: false hasContentIssue false
  • English
  • Français

Breakage and restoration in recursive trees

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Guillermo Tomás Tetzlaff
Guillermo Tomás Tetzlaff*
Affiliation:
Universidad de Buenos Aires and Universidad Nacional de La Plata
*
∗ Postal address: Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Pabelló I, Ciudad Universitaria, (1428) Buenos Aires, Argentina. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Uniform sequential tree-building aggregation of n particles is analyzed together with the effect of the avalanche that takes place when a subtree rooted at a uniformly chosen vertex is removed. For large n, the expected subtree size is found to be ≃ logn both for the tree of size n and the tree that remains after an avalanche. Repeated breakage-restoration cycles are seen to give independent avalanches which attain size k(1 ≤ k ≤ n-1) with probability (k(k+1))-1 and restored trees that are recursive.

Keywords

Aggregationrecursive treeavalancheself-organized criticality

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 2 , June 2002 , pp. 383 - 390
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bak, P., and Chen, K. (1991). Self-organized criticality. Sci. Amer. 264, 26–33.CrossRefGoogle Scholar
[2] Bak, P., and Sneppen, K. (1993). Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71, 4083–4086.Google Scholar
[3] Bergeron, F., Flajolet, P., and Salvy, B. (1992). Varieties of increasing trees. In Proc. 17th Coll. Trees in Algebra and Programming (Lecture Notes Comput. Sci. 581), ed. Raoult, J.-C., Springer, Berlin, pp. 24–48.Google Scholar
[4] Botet, R., and Jullien, R. (1985). Diffusion-limited aggregation with disaggregation. Phys. Rev. Lett. 55, 1943–1946.CrossRefGoogle ScholarPubMed
[5] Eden, M. (1961). A two-dimensional growth process. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. 4, ed. Neyman, F., University of California Press, Berkeley, pp. 223–239.Google Scholar
[6] Family, F., Meakin, P., and Deutch, J. M. (1986). Kinetics of coagulation with fragmentation: scaling behavior and fluctuations. Phys. Rev. Lett. 55, 1943–1946.Google Scholar
[7] Gastwirth, J. L., and Bhattacharya, P. K. (1984). Two probability models of pyramid or chain letter schemes demonstrating that their promotional claims are unreliable. Operat. Res. 32, 527–536.CrossRefGoogle Scholar
[8] Held, G. A., Solina, H., Keane, D. T., Haag, W. J., Horn, P. M., and Grinstein, G. (1990). Experimental study of critical-mass fluctuations in an evolving sandpile. Phys. Rev. Lett. 65, 1120–1123.Google Scholar
[9] Mahmoud, H. M. (1992). Distances in random plane-oriented recursive trees. J. Comput. Appl. Math. 41, 237–245.Google Scholar
[10] Meir, A., and Moon, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30, 997–1015 Google Scholar
[11] Najock, D., and Heyde, C. C. (1982). On the number of terminal vertices in certain random trees with an application to stemma construction in philology. J. Appl. Prob. 19, 675–680.Google Scholar
[12] Vicsek, T. (1989). Fractal Growth Phenomena. World Scientific, Teaneck, NJ.Google Scholar
[13] Witten, T. A., and Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403.Google Scholar