The full range of activity in a temporal network is captured in its edge activity data -- time series encoding the tie strengths or on-off dynamics of each edge in the network. However, in many practical applications, edge-level data are unavailable, and the network analyses must rely instead on node activity data which aggregates the edge-activity data and thus is less informative. This raises the question: Is it possible to use the static network to recover the richer edge activities from the node activities? Here we show that recovery is possible, often with a surprising degree of accuracy given how much information is lost, and that the recovered data are useful for subsequent network analysis tasks. Recovery is more difficult when network density increases, either topologically or dynamically, but exploiting dynamical and topological sparsity enables effective solutions to the recovery problem. We formally characterize the difficulty of the recovery problem both theoretically and empirically, proving the conditions under which recovery errors can be bounded and showing that, even when these conditions are not met, good quality solutions can still be derived. Effective recovery carries both promise and peril, as it enables deeper scientific study of complex systems but in the context of social systems also raises privacy concerns when social information can be aggregated across multiple data sources.
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