Restoring Network Evolution from Static Structure

TL;DR

A transferable machine learning framework that infers network evolutionary trajectories solely from present topology is presented, demonstrating its robustness and transferability and demonstrating that a substantial fraction of evolutionary information is encoded within static network architecture.

Abstract

The dynamical evolution of complex networks underpins the structure-function relationships in natural and artificial systems. Yet, restoring a network's formation from a single static snapshot remains challenging. Here, we present a transferable machine learning framework that infers network evolutionary trajectories solely from present topology. By integrating graph neural networks with transformers, our approach unlocks a latent temporal dimension directly from the static topology. Evaluated across diverse domains, the framework achieves high transfer accuracy of up to 95.3%, demonstrating its robustness and transferability. Applied to the Drosophila brain connectome, it restores the formation times of over 2.6 million neural connections, revealing that early-forming links support essential behaviors such as mating and foraging, whereas later-forming connections underpin complex sensory and social functions. These results demonstrate that a substantial fraction of evolutionary information is encoded within static network architecture, offering a powerful, general tool for elucidating the hidden temporal dynamics of complex systems.

Figures (7)

• Figure 1: Network evolution and machine learning framework for restoring structure evolution. (a) Diagram of network formation, where new edges are added at each time step, with the goal of restoring the edge formation sequence from the final network structure alone. (b) Overview of the machine learning framework for inferring network evolution. Node embeddings are initialized using topological features like degree, clustering coefficient, and betweenness (see Supplementary Sec. 1.1), then processed by GNN and transformer modules to capture local and global structural patterns. Edge embeddings, formed by combining endpoint node embeddings, are fed into a RankNet model to predict relative edge formation times ($R_i$), enabling restoration of the network’s evolutionary trajectory.
• Figure 2: Comparison of different models in terms of accuracy, computational cost, and model size. (a–f) Pairwise edge prediction accuracy on the test set under different training ratios of edge pairs. Error bars represent standard deviations over 50 runs. Green: CPNN; Blue: R; Orange: GR; Pink: GTR. Results are shown for three representative network categories: protein–protein interaction network, scientific collaboration network, and animal interaction network. See Supplementary Fig. S3 for results on additional network categories. (g) Few-shot learning performance of the GTR model. Accuracy across the number of training edges on four representative networks. See Supplementary Fig. S4 for results on additional networks and models. (h) Comparison of accuracy and training time between GTR and CPNN on four representative networks, as shown in (g). Experiments were conducted on a workstation equipped with an Intel i7-13700KF CPU, 80GB RAM, and an NVIDIA RTX 4070Ti GPU (see Supplementary Fig. S5 for results on additional networks and models). (i) Model parameter size comparison of the four models.
• Figure 3: Comparison between true and predicted edge formation times. Results of collaboration and PPI networks are shown in the left and right columns, respectively. Panels (a,b) correspond to the supervised setting, where models are trained and tested on the same network: (a) Collaboration (Fluctuations$\rightarrow$Fluctuations); (b) PPI (Fruit fly$\rightarrow$Fruit fly). Panels (c,d) correspond to the cross-network transfer setting, where models are trained and tested on different networks: (c) Collaboration (Chaos$\rightarrow$Fluctuations); (d) PPI (Human$\rightarrow$Fruit fly). Rectangles highlight edges that are formed within the same snapshot in the true formation times. Here, $\alpha$ and $\hat{\alpha}$ denote the ground-truth and predicted formation time, respectively, both normalized by the total number of edges $E$. The prediction error is quantified by $\varepsilon$ (see definition in Materials and Methods). Results for additional networks are reported in Table \ref{['tltab_error']}. The GTR model is used for all results.
• Figure 4: Results of cross-network transferability. (a) Accuracy matrix of cross-network transferability for the GTR model, using isomorphism-invariant features as initial node embeddings. The color intensity of each matrix cell reflects the transfer accuracy between source and target networks. The Y-axis denotes the source network (where the model is trained), and the X-axis denotes the target network (where the model is tested). Networks are clustered by category, and blocks from the same category are marked with black squares and corresponding labels. Network labels: A: Thermodynamics, B: Complex networks, C: Phase transitions, D: Fluctuations, E: Chaos, F: Worm, G: Fruit fly, H: Human, I: Fungi, J: Ferry, K: Weaver, L: World Trade Web. Results of the synthetic networks, please refer to Supplementary Sec. 7. (b) Comparison of transfer accuracy across three models with node embeddings initialized by isomorphism-invariant features. Each $X\text{-}Y$ pair indicates a model trained on network $X$ and transferred to network $Y$. (c) Scatter plot of transfer accuracy versus cross-network distance (see Supplementary Sec. 3 for definition) using the GTR model. The pink line shows a linear fit with a slope of $-1.86\pm0.24$, and the light pink shaded area indicates the 95% confidence interval. Both accuracy and distance values are Min-Max normalized: $x_{norm} = (x - x_{min}) / (x_{max} - x_{min})$. (d–e) Two-dimensional projection of node embeddings via t-SNE hge02: (d) isomorphism-invariant embedding; (e) random walk embedding. Each dot represents a node, colored by its network type. Labels indicate the corresponding network names. (f) Comparison of average transfer accuracy across seven initialization methods of node embeddings using the GTR model. Average accuracy refers to the mean pairwise edge prediction accuracy of a model trained on one source network and transferred to other target networks within the same category. Details of seven embedding methods please refer to Materials and Methods and Supplementary Sec. 1.1.
• Figure 5: Predicted connection formation time and functional organization of the Drosophila olfactory circuit. Violin plots of predicted formation times of neural connections, grouped by functional connection type in the Drosophila olfactory circuit. Each color denotes a distinct category of connections (text labels), white dots mark the mean formation time, and vertical bars indicate the standard error. Inset: spatial map of glomeruli where neural connections form, with highlighted regions corresponding to the same connection categories and gray areas indicating other glomeruli. Details please refer to Supplementary Sec. 2.6. The Drosophila illustration is from the DataBase Center for Life Science.