Information diffusion mechanisms based on social influence models are mainly studied using likelihood of adoption when active neighbors expose a user to a message. The problem arises primarily from the fact that for the most part, this explicit information of who-exposed-whom among a group of active neighbors in a social network, before a susceptible node is infected is not available. In this paper, we attempt to understand the diffusion process through information cascades by studying the temporal network structure of the cascades. In doing so, we accommodate the effect of exposures from active neighbors of a node through a network pruning technique that leverages network motifs to identify potential infectors responsible for exposures from among those active neighbors. We attempt to evaluate the effectiveness of the components used in modeling cascade dynamics and especially whether the additional effect of the exposure information is useful. Following this model, we develop an inference algorithm namely InferCut, that uses parameters learned from the model and the exposure information to predict the actual parent node of each potentially susceptible user in a given cascade. Empirical evaluation on a real world dataset from Weibo social network demonstrate the significance of incorporating exposure information in recovering the exact parents of the exposed users at the early stages of the diffusion process.