  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.
