The death of a relationship

Users in online social networks ostensibly have relationships with a large number of other users. This has prompted many to comment that the nature of friendship in the online world is different from the offline world. However, even though a user may connect with many others, a majority of such connections become inactive after a period of time, and therefore, many relationships cease to exist. A challenge in modeling such connections is that death is unobserved, as the connection remains, but is not active. As a consequence, models that ignore the death of relationships overestimate the density of a network and can bias measures of user influence. We model interactivity among users in an online social network and explicitly account for unobserved relationship death via extensions of the well known Pareto-NBD and BG-NBD models to social network settings. We estimate our model on a network of musicians using hierarchical Bayesian methods and find that accounting for the death of relationships is beneficial for predicting future interactivity.