Hierarchical Longitudinal Models of Relationships in Social Networks
A. James O'Malley, Harvard Medical School 
*Sudeshna Paul, Harvard Medical School 

Keywords: Dyad, Health traits,Transitivity, Bayesian, Longitudinal, Hierarchical, Social Network.

We develop a new longitudinal model for transitions in the relationship status of pairs of individuals (dyads) in a social network. We first specify a model for the relationship state of two individuals or “dyads” in the network, which in the case of binary-valued relationships follows a four-component multinomial distribution. To account for complexities due to the dependence of observations among dyads, which in general requires the specification of a joint model for the whole network, we assume dyads are conditionally independent given latent variables (random effects) of the actors in a dyad and lagged covariates judiciously chosen to account for important inter-dyad dependencies (e.g., transitivity – “a friend of a friend is a friend”). Model parameters are estimated using Bayesian analysis implemented via Markov chain Monte Carlo (MCMC). The model is applied to the friendship networks from excerpts of the Teenage Health and Lifestyle Study (a small network) and the Framingham Heart Study (a large network). Results of both analyses indicate a strong dependence across time, high reciprocation of ties between individuals, and extensive triadic clustering.Model fit revealed that our model successfully captured the most important features of the data.