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Typical Distances in Ultrasmall Random Networks

Part of: Combinatorial probability Graph theory Operations research and management science

Published online by Cambridge University Press:  04 January 2016

Steffen Dereich ,
Christian Mönch  and
Peter Mörters
Steffen Dereich*
Affiliation:
Philipps-Universität Marburg
Christian Mönch*
Affiliation:
University of Bath
Peter Mörters*
Affiliation:
University of Bath
*
∗ Postal address: Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straβe, 35032 Marburg, Germany.
∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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Abstract

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We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ − 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

Keywords

Scale-free networkpreferential attachmentconfiguration modelpower-law graphconditionally Poissonian graphgraph distancediameter

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 583 - 601
Copyright
© Applied Probability Trust 

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