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Disparity of clustering coefficients in the Holme‒Kim network model

Published online by Cambridge University Press:  16 November 2018

R. I. Oliveira
Affiliation:
IMPA
R. Ribeiro
Affiliation:
Universidade Federal de Minas Gerais
R. Sanchis
Affiliation:
Universidade Federal de Minas Gerais

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.

Type
Original Article
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Barabási, A.-L. and Álbert, R. (1999). Emergence of scaling in random networks. Science 286, 509512.Google ScholarPubMed
[2]Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[5]Buckley, P. G. and Osthus, D. (2004). Popularity based random graph models leading to a scale-free degree sequence. Discrete Math. 282, 5368.CrossRefGoogle Scholar
[6]Chung, F. and Lu, L. (2006). Complex Graphs and Networks. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
[7]Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
[8]Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6, 290297.Google Scholar
[9]Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Prob. 3, 100118.CrossRefGoogle Scholar
[10]Holme, P. and Kim, B. J. (2002). Growing scale-free networks with tunable clustering. Phys. Rev. E 65, 026107.CrossRefGoogle ScholarPubMed
[11]Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. John Wiley, New York.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[13]Newman, M. E. J. (2010). Networks: An Introduction. Oxford University Press.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[15]Van der Hofstad, R. (2009). Random graphs and complex networks. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.html.Google Scholar
[16]Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature 393, 440442.CrossRefGoogle ScholarPubMed

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