Abstract: Large-scale directed social network data often include degree heterogeneity, reciprocity, and transitivity properties. Thus, a sensible network-generating model should consider these features. To this end, we propose a popularity-scaled latent space model for large-scale directed network structure formulations. This model assumes each node occupies a position in a hypothetically assumed latent space. Then, the nodes close to (far away from) each other should have a higher (lower) probability of being connected. Thus, reciprocity and transitivity can be derived analytically. In addition, we assume a popularity parameter for each node. Nodes with larger (smaller) popularity are more (less) likely to be followed. By assuming different distributions for the popularity parameters, we model various types of degree heterogeneity. Based on the proposed model, we construct a comprehensive probabilistic index for link prediction. We demonstrate the performance of the proposed model using simulation studies and a Sina Weibo data set. The results show that the performance of the model is competitive.

Key words and phrases: Degree heterogeneity, large-scale social network, latent space model, link prediction, reciprocity, transitivity.