A Threshold For Clusters in Real-World Random Networks

Recent empirical work [Leskovec2009] has suggested the existence of a size threshold for the existence of clusters within many real-world networks. We give the first proof that this clustering size threshold exists within a real-world random network model, and determine the asymptotic value at which it occurs. More precisely, we choose the Community Guided Attachment (CGA) random network model of Leskovek, Kleinberg, and Faloutsos [Leskovec2005]. The model is non-uniform and contains self-similar communities, and has been shown to have many properties of real-world networks. To capture the notion of clustering, we follow Mishra et. al. [Mishra2007], who defined a type of clustering for real-world networks: an (\alpha,\beta)-cluster is a set that is both internally dense (to the extent given by the parameter \beta), and externally sparse (to the extent given by the parameter \alpha) . With this definition of clustering, we show the existence of a size threshold of (\ln n)^{1/2} for the existence of clusters in the CGA model. For all \epsilon>0, a.a.s. clusters larger than (\ln n)^{1/2-\epsilon} exist, whereas a.a.s. clusters larger than (\ln n)^{1/2+\epsilon} do not exist. Moreover, we show a size bound on the existence of small, constant-size clusters.