Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-11T08:58:20.833Z Has data issue: false hasContentIssue false
  • English
  • Français

A Counter-Intuitive Correlation in a Random Tournament

Published online by Cambridge University Press:  13 May 2010

SVEN ERICK ALM  and
SVANTE LINUSSON
SVEN ERICK ALM
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06, Uppsala, Sweden (e-mail: sea@math.uu.se)
SVANTE LINUSSON
Affiliation:
Department of Mathematics, KTH – Royal Institute of Technology, SE-100 44, Stockholm, Sweden (e-mail: linusson@math.kth.se)
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a randomly oriented graph G = (V, E) and let a, s and b be three distinct vertices in V. We study the correlation between the events {a → s} and {s → b}. We show that, counter-intuitively, when G is the complete graph Kn, n ≥ 5, then the correlation is positive. (It is negative for n = 3 and zero for n = 4.) We briefly discuss and pose problems for the same question on other graphs.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 1 , January 2011 , pp. 1 - 9
Copyright
Copyright © Cambridge University Press 2010

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]Alm, S. E. and Linusson, S. A counter-intuitive correlation in a random tournament (early version of the present paper). arXiv:0906.0240Google Scholar
[2]Alm, S. E. and Linusson, S. (2009) Correlations for paths in random orientations of G(n, p). Preprint. arXiv:0906.0720Google Scholar
[3]van den Berg, J. and Kahn, J. (2001) A correlation inequality for connection events in percolation. Ann. Probab. 29 123–126.Google Scholar
[4]van den Berg, J., Häggström, O. and Kahn, J. (2006) Some conditional correlation inequalities for percolation and related processes. Random Struct. Alg. 29 417–435.CrossRefGoogle Scholar
[5]Grimmett, G. R. (1999) Percolation, Springer, Berlin.CrossRefGoogle Scholar
[6]Grimmett, G. R. (2001) Infinite paths in randomly oriented lattices. Random Struct. Alg. 18 257–266.CrossRefGoogle Scholar
[7]Häggström, O. (2003) Probability on Bunkbed Graphs. InProc. FPSAC'03: Formal Power Series and Algebraic Combinatorics (Linköping, Sweden). Available at http://www.fpsac.org/FPSAC03/ARTICLES/42.pdf.Google Scholar
[8]Linusson, S. (2009) A note on correlations in randomly oriented graphs. Preprint. arXiv:0905.2881.Google Scholar
[9]Linusson, S. (2010) On percolation and the bunkbed conjecture. Combin. Probab. Comput. XX.Google Scholar
[10]McDiarmid, C. (1981) General percolation and random graphs. Adv. Appl. Probab. 13 40–60.CrossRefGoogle Scholar