Generalized Random Dot Product graph

The Random Dot Product model for social network was introduced in Nickel (2007) and extended by Young and Scheinerman (2007), where each asymptotic results such as degree distribution, clustering and diameter on both dense and sparse cases were derived. Young and Scheinerman (2007) explored two generalizations of the model in the dense case and obtained similar asymptotic results. In this paper, we consider a generalization of the Random Dot Product model and derive its theoretical properties under the dense, sparse and intermediate cases. In particular, properties such as the size of the largest component and connectivity can be derived by applying recent results on inhomogeneous random graphs (Bollobás et al., 2007; Devroye and Fraiman, 2014).