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

Part of: Limit theorems Combinatorial probability Graph theory

Published online by Cambridge University Press:  21 March 2016

Nicholas R. Peterson  and
Boris Pittel
Nicholas R. Peterson*
Affiliation:
The Ohio State University
Boris Pittel*
Affiliation:
The Ohio State University
*
∗ Postal address: Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210, USA.
∗ Postal address: Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210, USA.
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Abstract

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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 = βn1/2 case.

Keywords

Random graphsrandom digraphspreferential attachmentuniformk-out digraphstotal variation distancelocal limit theorem

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 858 - 879
Copyright
Copyright © Applied Probability Trust 2015 

Footnotes

The authors gratefully acknowledge support from the NSF (grant no. DMS-1101237).

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