Advances in Applied Probability

Distance between two random k-out digraphs, with and without preferential attachment

Nicholas R. Peterson and Boris Pittel

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Abstract

A random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a 'preferential attachment' rule: the current vertex selects an image i with probability proportional to a given parameter α = α(n) plus the number of times i has already been selected. Intuitively, the larger α becomes, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α = Θ(n1/2) is the threshold for α growing 'fast enough' to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We also determine an exact limit for this distance for the α = β n1/2 case.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 3 (2015), 858-879.

Dates
First available in Project Euclid: 8 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1444308885

Digital Object Identifier
doi:10.1239/aap/1444308885

Mathematical Reviews number (MathSciNet)
MR3406611

Zentralblatt MATH identifier
1325.05068

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Keywords
Random graphs random digraphs preferential attachment uniform k-out digraphs total variation distance local limit theorem

Citation

Peterson, Nicholas R.; Pittel, Boris. Distance between two random k -out digraphs, with and without preferential attachment. Adv. in Appl. Probab. 47 (2015), no. 3, 858--879. doi:10.1239/aap/1444308885. https://projecteuclid.org/euclid.aap/1444308885


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