  The focus of this work is on developing probabilistic models for user
activity in social networks by incorporating the social network influence as
perceived by the user. For this, we propose a coupled Hidden Markov Model,
where each user's activity evolves according to a Markov chain with a hidden
state that is influenced by the collective activity of the friends of the user.
We develop generalized Baum-Welch and Viterbi algorithms for model parameter
learning and state estimation for the proposed framework. We then validate the
proposed model using a significant corpus of user activity on Twitter. Our
numerical studies show that with sufficient observations to ensure accurate
model learning, the proposed framework explains the observed data better than
either a renewal process-based model or a conventional uncoupled Hidden Markov
Model. We also demonstrate the utility of the proposed approach in predicting
the time to the next tweet. Finally, clustering in the model parameter space is
shown to result in distinct natural clusters of users characterized by the
interaction dynamic between a user and his network.
