Accessing the network through which a propagation dynamics diffuses is essential for understanding and controlling it. In a few cases, such information is available through direct experiments or thanks to the very nature of propagation data. In a majority of cases however, available information about the network is indirect and comes from partial observations of the dynamics, rendering the network reconstruction a fundamental inverse problem. Here we show that it is possible to reconstruct the whole structure of an interaction network and to simultaneously infer the complete time course of activation spreading, relying just on single epoch (i.e. snapshot) or time-scattered observations of a small number of activity cascades. The method that we present is built on a belief propagation approximation, that has shown impressive accuracy in a wide variety of relevant cases, and is able to infer interactions in the presence of incomplete time-series data by providing a detailed modelling of the posterior distribution of trajectories conditioned to the observations. Furthermore, we show by experiments that the information content of full cascades is relatively smaller than that of sparse observations or single snapshots.
