Abstract
Two general random intersection graph models (active and passive) were introduced by Godehardt and Jaworski (Exploratory Data Analysis in Empirical Research, Springer, Berlin, Heidelberg, New York, pp.68–81, 2002). Recently the models have been shown to have wide real life applications. The two most important ones are: non-metric data analysis and real life network analysis. Within both contexts, the clustering coefficient of the theoretical graph models is studied. Intuitively, the clustering coefficient measures how much the neighborhood of the vertex differs from a clique. The experimental results show that in large complex networks (real life networks such as social networks, internet networks or biological networks) there exists a tendency to connect elements, which have a common neighbor. Therefore it is assumed that in a good theoretical network model the clustering coefficient should be asymptotically constant. In the context of random intersection graphs, the clustering coefficient was first studied by Deijfen and Kets (Eng Inform Sci, 23:661–674, 2009). Here we study a wider class of random intersection graphs than the one considered by them and give the asymptotic value of their clustering coefficient. In particular, we will show how to set parameters – the sizes of the vertex set, of the feature set and of the vertices’ feature sets – in such a way that the clustering coefficient is asymptotically constant in the active (respectively, passive) random intersection graph.
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References
Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Modern Phys 74:47–97
Barrat A, Weigt M (2000) On the properties of small-world networks. Eur Phys J B 13:547–560
Bloznelis M (2008) Degree distribution of a typical vertex in a general random intersection graph. Lithuanian Math J 48:38–45
Bloznelis M, Jaworski J, Rybarczyk K (2009) Component evolution in a secure wireless sensor network. Networks 53:19–26
Bock HH (1996) Probabilistic models in cluster analysis. Comput Stat Data Anal 23:5–28
Deijfen M, Kets W (2009) Random intersection graphs with tunable degree distribution and clustering probability. Eng Inform Sci 23:661–674
Godehardt E (1990) Graphs as structural models. Vieweg, Braunschweig
Godehardt E, Jaworski J (1996) On the connectivity of a random interval graph. Random Struct Algorithm 9:137–161
Godehardt E, Jaworski J (2002) Two models of random intersection graphs for classification. In: Schwaiger M, Opitz O (eds) Exploratory data analysis in empirical research. Springer, Berlin, Heidelberg, New York, pp 68–81
Godehardt E, Jaworski J, Rybarczyk K (2007) Random intersection graphs and classification. In: Decker R, Lenz HJ (eds) Advances in data analysis. Springer, Berlin, Heidelberg, New York, pp 67–74
Godehardt E, Jaworski J, Rybarczyk K (2010) Isolated vertices in random intersection graphs. In: Fink A, Lausen B, Seidel W, Ultsch A (eds) Advances in data analysis, data handling and business intelligence. Springer, Berlin, Heidelberg, New York, pp 135–145
Guillaume JL, Latapy M (2004) Bipartite structure of all complex networks. Inform Process Lett 90:215–221
Newman MEJ (2003) Properties of highly clustered networks. Phys Rev 68(026121)
Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64(026118)
Rybarczyk K (2010) Random intersection graphs. analysis and modeling of networks structure. PhD thesis, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, URL http://hdl.handle.net/10593/386
Rybarczyk K (2011) The degree distribution in random intersection graphs. In: Gaul W, Geyer-Schulz A, Schmidt-Thieme L, Kunze J (eds) Challenges at the interface of data analysis, computer science, and optimization, studies in classification, data analysis, and knowledge organization. Springer, Heidelberg, Berlin
Strogatz SH, Watts DJ (1998) Collective dynamics of small-world networks. Nature 393:440–442
Acknowledgements
J. Jaworski acknowledges the support by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme. This work had been also supported by Ministry of Science and Higher Education, grant N N206 2701 33, 2007–2010.
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Godehardt, E., Jaworski, J., Rybarczyk, K. (2012). Clustering Coefficients of Random Intersection Graphs. In: Gaul, W., Geyer-Schulz, A., Schmidt-Thieme, L., Kunze, J. (eds) Challenges at the Interface of Data Analysis, Computer Science, and Optimization. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24466-7_25
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