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In this paper, we study the problem of using representation learning to assist information diffusion prediction on graphs. In particular, we aim at estimating the probability of an inactive node to be activated next in a cascade. Despite the success of recent deep learning methods for diffusion, we find that they often underexplore the cascade structure. We consider a cascade as not merely a sequence of nodes ordered by their activation time stamps; instead, it has a richer structure indicating the diffusion process over the data graph. As a result, we introduce a new data model, namely diffusion topologies, to fully describe the cascade structure. We find it challenging to model diffusion topologies, which are dynamic directed acyclic graphs (DAGs), with the existing neural networks. Therefore, we propose a novel topological recurrent neural network, namely Topo-LSTM, for modeling dynamic DAGs. We customize Topo …