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Disparity of clustering coefficients in the Holme‒Kim network model

Part of: Special processes Discrete mathematics in relation to computer science Graph theory

Published online by Cambridge University Press:  16 November 2018

R. I. Oliveira ,
R. Ribeiro  and
R. Sanchis
R. I. Oliveira*
Affiliation:
IMPA
R. Ribeiro*
Affiliation:
Universidade Federal de Minas Gerais
R. Sanchis*
Affiliation:
Universidade Federal de Minas Gerais
*
* Postal address: IMPA, Estrada Da. Castorina, 110 CEP 22460-320 Rio de Janeiro, RJ, Brazil. Email address: rimfo@impa.br
** Postal address: Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627 C.P. 702 CEP 30123-970 Belo Horizonte-MG, Brazil.
** Postal address: Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627 C.P. 702 CEP 30123-970 Belo Horizonte-MG, Brazil.
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Abstract

The Holme‒Kim random graph process is a variant of the Barabási‒Álbert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that the local clustering coefficient typically remains positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

Keywords

Random graphcomplex networkconcentration boundclustering coefficientpreferential attachment

MSC classification

Primary: 05C82: Small world graphs, complex networks
Secondary: 60K40: Other physical applications of random processes 68R10: Graph theory (including graph drawing)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 918 - 943
Copyright
Copyright © Applied Probability Trust 2018 

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