We propose a multinomial logistic regression model for link prediction in a time series of directed binary networks. To account for the dynamic nature of the data,we employ a dynamic model for the model parameters that is strongly connected with the fused lasso penalty. In addition to promoting sparseness, this prior allows us to explore the presence of change points in the structure of the network. We introduce fast computational algorithms for estimation and prediction using both optimization and Bayesian approaches. The performance of the model is illustrated using simulated data and data from a financial trading network in the NYMEX natural gas futuresmarket. Supplementary material containing the trading network dataset and code to implement the algorithms is available online.