Fig. 2. Performance of the ensemble model and the restored edge sequence.

a Test accuracy of the ensemble model as a function of the percentage of edge pairs used for training. Each data point with error bars marks the corresponding simulation results (average ± standard deviation of 100 simulations), the same for b. b Overall error $\mathcal{E}$ as a function of the accuracy x of the ensemble model for different numbers of edges E. The solid curves represent the theoretical results from Eq. (2) and the colored crosses stand for the simulation results using the E and x of five real-world networks. c Simulated distributions of D_i/E using the E and x of five real-world networks. Specifically, assuming the ground-truth sequence α = (1, 2, …, E), 100(1 − x)% of all edge pairs are randomly selected and artificially assigned the wrong generation order while the remaining edge pairs are assigned the correct one. Then, the restored edge sequence $\widehat{\mathit{\alpha }}$ is obtained by applying the ranking algorithm on the artificially predicted order of all edge pairs and D_i’s are calculated accordingly. d, e Comparisons between the real and simulated distributions of D_i/E based on the collaboration network (CN) and the PPI network (Fungi). f Diagram illustrating how the distributions in c–e are obtained. The left and right panels show the calculation of D_i under a real case when we only know the coarse-grained ground-truth sequence and a simulation when we know the fine-grained ground-truth sequence, respectively. For the real case, D_i cannot be calculated directly as ${\alpha }_i-{\widehat{\alpha }}_i$ so the idea is to consider an intermediate sequence α^* by randomly assigning fine-grained order to edges added within the same snapshot and D_i is calculated as ${\alpha }_i^{*}-{\widehat{\alpha }}_i$ instead. Then the distribution of D_i/E is obtained by averaging over 5000 α^*’s to take the randomness into account. For the simulation, the calculation of D_i follows a similar procedure to match with the real case. The results under the real case and simulation are labeled as “Real Data” and “Simulation” in d and e. See Algorithms 2-3 in the Methods section for more details.