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Convergence Of Directed Random Graphs To The Poisson-Weighted Infinite TreePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, 463-474 p.Article in journal (Refereed) PublishedText
##### Abstract [en]

##### Place, publisher, year, edition, pages

2016. Vol. 53, no 2, 463-474 p.
##### Keyword [en]

Directed random graph, Poisson-weighted infinite tree, rooted geometric graph
##### National Category

Probability Theory and Statistics
##### Identifiers

URN: urn:nbn:se:uu:diva-299908DOI: 10.1017/jpr.2016.13ISI: 000378598700012OAI: oai:DiVA.org:uu-299908DiVA: diva2:950354
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Available from: 2016-07-29 Created: 2016-07-29 Last updated: 2016-07-29Bibliographically approved

We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n(-1), whenever i < j, independently of all other edges. Moreover, to each edge (i, j) we assign weight n(-1) (j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n -> infinity. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most rho from the root.

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