Recovering the past states of growing trees

In principle one can reconstruct the past states of a growing network from only its current state. In practice, however, the extent to which this can be done is severely limited since existing methods are either inexact, inefficient, or both. Here we present methods for temporal reconstruction that are both exact and efficient on trees. We derive analytic expressions for the number of possible histories in which each node arrived at each time, and we present a Monte Carlo method to sample full histories in O(n log log n) operations for networks with n nodes. We demonstrate the use of these methods with a series of applications: seed finding, network interpolation, full history reconstruction, and model fitting. With these new tools one can directly fit growth models such as preferential attachment to static network data---testing models directly at the level of mechanism with only a single network snapshot.