Fig. 4. Underlying growth mechanism in the restored network evolution processes.

Cumulative PA function κ(k) for a PPI network (Fungi), b World Trade Web, and c Collaboration network (Interfaces). In each figure, the yellow circles and blue triangles are the results of the ground-truth and restored evolution processes, respectively. If the growth of a network follows the PA rule, the rate at which a node with degree k acquires new edges should be positively correlated with k and the cumulative PA function κ(k) is expected to grow superlinearly (see Supplementary Section 9 for details). So, the solid gray line with slope = 1 represents the case in which PA is absent. d Adjacency matrices of the evolution process for the protein function network generated by the PPI network (Fungi). Proteins with the same function in the network are treated as a single node to form a simplified protein function network where the edges represent the interactions between proteins with weights being the number of protein interactions. The upper row shows the results based on our restored temporal edge sequence while the lower row shows those based on a simulation study assuming the pure PA rule. The simulation is performed by adding edges according to the PA rule and keeping the average node degree consistent with the real network (details are provided in Supplementary Section 9). e, f Visualizations of the protein function network in d when the number of edges are E = 2000 and E = 5425. Letters marking the nodes denote the protein functions (with specific meanings listed in Supplementary Table S8), and the self-connected and non-self-connected edges are respectively displayed in blue and red. g The modularity⁵⁴ of the PPI network (Fungi). The yellow triangles represent results computed at the real snapshots of the networks. The blue solid lines and pink dashed lines are results based on edge generation order by our reconstruction method and the pure PA rule, respectively. h, i Adjacency matrix and protein function network of the PPI network (Fungi) obtained at the first real snapshot (i.e., E = 5425).