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Correlations for Paths in Random Orientations of G(n, p) and G(n, m)
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2011 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 39, no 4, 486-506 p.Article in journal (Refereed) Published
Abstract [en]

We study random graphs, both G(n, p) and G(n, m), with random orientations on the edges. For three fixed distinct vertices s, a, b we study the correlation, in the combined probability space, of the events {a -> s} and {s -> b}. For G(n, p), we prove that there is a p(c) = 1/2 such that for a fixed p < p(c) the correlation is negative for large enough n and for p > p(c) the correlation is positive for large enough n. We conjecture that for a fixed n >= 27 the correlation changes sign three times for three critical values of p. For G(n, m) it is similarly proved that, with p = m/((n)(2)), there is a critical p(c) that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any l directed edges in G(n, m), is thought to be of independent interest. We present exact recursions to compute P(a -> s) and P(a -> s, s -> b). We also briefly discuss the corresponding question in the quenched version of the problem.

Place, publisher, year, edition, pages
2011. Vol. 39, no 4, 486-506 p.
Keyword [en]
random directed graphs, correlation, directed paths, annealed, quenched
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-162440DOI: 10.1002/rsa.20358ISI: 000296716500002OAI: oai:DiVA.org:uu-162440DiVA: diva2:460923
Available from: 2011-12-01 Created: 2011-11-30 Last updated: 2017-12-08Bibliographically approved

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Alm, Sven ErickJanson, Svante

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