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Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

Part of: Graph theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Maria Deijfen  and
Ronald Meester
Maria Deijfen*
Affiliation:
Chalmers University of Technology
Ronald Meester*
Affiliation:
Vrije Universiteit Amsterdam
*
∗ Current address: DIAM, Mekelweg 4, 2628 CD Delft, The Netherlands. Email address: mia@math.su.se
∗∗ Postal address: Divisie Wiskunde en Informatica, Faculteit der Exacte Wetenshappen, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: rmeester@cs.vu.nl
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Abstract

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Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

Keywords

Random graphdegree distributionstationary algorithmrandom walkfinitary isomorphism

MSC classification

Primary: 05C80: Random graphs 60G50: Sums of independent random variables; random walks
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 287 - 298
Copyright
Copyright © Applied Probability Trust 2006 

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