The diffusion of information, rumors, and diseases are assumed to be probabilistic processes over some network structure. An event starts at one node of the network, and then spreads to the edges of the network. In most cases, the underlying network structure that generates the diffusion process is unobserved, and we only observe the times at which each node is altered/influenced by the process. This paper proposes a probabilistic model for inferring the diffusion network, which we call Probabilistic Latent Network Visualization (PLNV); it is based on cascade data, a record of observed times of node influence. An important characteristic of our approach is to infer the network by embedding it into a low-dimensional visualization space. We assume that each node in the network has latent coordinates in the visualization space, and diffusion is more likely to occur between nodes that are placed close together. Our model uses maximum a posteriori estimation to learn the latent coordinates of nodes that best explain the observed cascade data. The latent coordinates of nodes in the visualization space can 1) enable the system to suggest network layouts most suitable for browsing, and 2) lead to high accuracy in inferring the underlying network when analyzing the diffusion process of new or rare information, rumors, and disease.