A latent variable approach for modeling relational data with multiple receivers

Directional relational event data, such as email data, often contain unicast messages (i.e., messages of one sender toward one receiver) and multicast messages (i.e., messages of one sender toward multiple receivers). The Enron email data that is the focus in this paper consists of 31% multicast messages. Multicast messages contain important information about the roles of actors in the network, which is needed for better understanding social interaction dynamics. In this paper a multiplicative latent factor model is proposed to analyze such relational data. For a given message, all potential receiver actors are placed on a suitability scale, and the actors are included in the receiver set whose suitability score exceeds a threshold value. Unobserved heterogeneity in the social interaction behavior is captured using a multiplicative latent factor structure with latent variables for actors (which differ for actors as senders and receivers) and latent variables for individual messages. The model is referred to as the multicast additive and multiplicative effects network (mc-amen) model. A Bayesian computational algorithm, which relies on Gibbs sampling, is proposed for model fitting. Model assessment is done using posterior predictive checks. Numerical simulations show that the model is widely applicable for various scenarios involving multicast messages. Furthermore, a mc-amen model with a two-dimensional latent variable can accurately capture the empirical distribution of the cardinality of the receiver set and the composition of the receiver sets for commonly observed messages in the Enron email data. In the Enron network, actors have a comparable (but not identical) role as a sender and as a receiver in the network.