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Convergence of directed random graphs to the Poisson-weighted infinite tree

Part of: Combinatorial probability Graph theory Limit theorems

Published online by Cambridge University Press:  21 June 2016

Katja Gabrysch
Katja Gabrysch*
Affiliation:
Uppsala University
*
* Postal address: Department of Mathematics, Uppsala University, PO Box 480, 751 06 Uppsala, Sweden. Email address: katja.gabrysch@math.uu.se
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Abstract

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 → ∞. 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 ρ from the root.

Keywords

Directed random graphPoisson-weighted infinite treerooted geometric graph

MSC classification

Primary: 05C80: Random graphs
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 463 - 474
Copyright
Copyright © Applied Probability Trust 2016 

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