Clustering Coefficients of Random Intersection Graphs

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. E. Godehardt ( ) Clinic of Cardiovascular Surgery, Heinrich Heine University, 40225 Düsseldorf, Germany e-mail: [email protected] J. Jaworski K. Rybarczyk Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland e-mail: [email protected]; [email protected] W. Gaul et al. (eds.), Challenges at the Interface of Data Analysis, Computer Science, and Optimization, Studies in Classification, Data Analysis, and Knowledge Organization, DOI 10.1007/978-3-642-24466-7 25, © Springer-Verlag Berlin Heidelberg 2012 243 244 E. Godehardt et al.