One can observe a dynamic process like an epidemic or a rumor spreading by tracking timestamps of when people become infected or start spreading a rumor. The edges of the underlying network in which this dynamic process unfolds are often hidden.
The goal is to recover the edges from such timestamp-information of several dynamic processes. 

We study the computational complexity of the corresponding combinatorial problem. Herein, given such observations from multiple dynamic processes, the task is to find a graph that is consistent with all these observations. If there is such a graph, then we want to find one minimizing the number of edges. If not, then we want to find the graph that is consistent with most observations.

We differentiate two propagation models: unanimity thresholds, where a person gets infected only after all its neighbors are infected, and constant thresholds, where a person gets infected when a constant number of its neighbors are infected. While the first model turns out to be mostly polynomial-time solvable, the second case is NP-complete but efficiently solvable in various restricted cases.