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Correlations for Paths in Random Orientations of G(n, p) and G(n, m)
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-6339-2230
2011 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 39, no 4, p. 486-506Article 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, p. 486-506
Keywords [en]
random directed graphs, correlation, directed paths, annealed, quenched
Mathematics
Identifiers
ISI: 000296716500002Scopus ID: 2-s2.0-80054838779OAI: oai:DiVA.org:kth-51413DiVA, id: diva2:464535
Funder
Knut and Alice Wallenberg Foundation
Note
QC 20111213Available from: 2011-12-13 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved

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