Recent research has explored the increasingly important role of social media by examining the dynamics of individual and group behavior, characterizing patterns of information diffusion, and identifying influential individuals. In this paper we suggest a measure of causal relationships between nodes based on the information-theoretic notion of transfer entropy, or information transfer. This theoretically grounded measure is based on dynamic information, captures fine-grain notions of influence, and admits a natural, predictive interpretation. Causal networks inferred by transfer entropy can differ significantly from static friendship networks because most friendship links are not useful for predicting future dynamics. We demonstrate through analysis of synthetic and real-world data that transfer entropy reveals meaningful hidden network structures. In addition to altering our notion of who is influential, transfer entropy allows us to differentiate between weak influence over large groups and strong influence over small groups.