Skip to main content Accessibility help
×
Home

Disparity of clustering coefficients in the Holme‒Kim network model

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

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.

Copyright

Corresponding author

* 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.
*** Email address: rodrigo-matematica@ufmg.br
**** Email address: rsanchis@mat.ufmg.br

References

Hide All
[1]Barabási, A.-L. and Álbert, R. (1999). Emergence of scaling in random networks. Science 286, 509512.
[2]Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.
[3]Bollobás, B. and Riordan, O. M. (2003). Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks, Wiley-VCH, Weinheim, pp. 134.
[4]Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290.
[5]Buckley, P. G. and Osthus, D. (2004). Popularity based random graph models leading to a scale-free degree sequence. Discrete Math. 282, 5368.
[6]Chung, F. and Lu, L. (2006). Complex Graphs and Networks. American Mathematical Society, Providence, RI.
[7]Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.
[8]Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6, 290297.
[9]Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Prob. 3, 100118.
[10]Holme, P. and Kim, B. J. (2002). Growing scale-free networks with tunable clustering. Phys. Rev. E 65, 026107.
[11]Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. John Wiley, New York.
[12]Kumar, R., et al. (2000). Stochastic models for the web graph. In Proceedings of the 41st Annual Symposium on the Foundations of Computer Science, IEEE, pp. 5765.
[13]Newman, M. E. J. (2010). Networks: An Introduction. Oxford University Press.
[14]Ostroumova, L., Ryabchenko, A. and Samosvat, E. (2013). Generalized preferential attachment: Tunable power-law degree distribution and clustering coefficient. In Algorithms and Models for the Web Graph, Springer, Cham, pp. 185202.
[15]Van der Hofstad, R. (2009). Random graphs and complex networks. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.html.
[16]Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature 393, 440442.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed