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Disparity of clustering coefficients in the Holme‒Kim network model

  • R. I. Oliveira (a1), R. Ribeiro (a2) and R. Sanchis (a2)


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.


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* Postal address: IMPA, Estrada Da. Castorina, 110 CEP 22460-320 Rio de Janeiro, RJ, Brazil. Email address:
** 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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