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Predicting Long-term Dynamics of Complex Networks via Identifying Skeleton in Hyperbolic Space

Ruikun Li, Huandong Wang, Jinghua Piao, Qingmin Liao, Yong Li

TL;DR

DiskNet introduces a skeleton-based framework to predict long-term dynamics on complex networks by identifying a low-dimensional skeleton in hyperbolic space via a renormalization-group-inspired embedding, modeling skeleton dynamics with a neural ODE, and lifting predictions back to the original network using degree-based super-resolution. The method integrates physics-informed initialization with adaptive hyperbolic embeddings to capture both topological and dynamical similarities, enabling accurate long-horizon forecasting across multiple dynamics and networks. Empirical results show DiskNet outperforms state-of-the-art baselines on 120-step predictions, with notable improvements in stability and efficiency; ablations confirm the value of the initialization and skeleton-learning components. The work highlights the utility of dynamics-aware skeletons and hyperbolic geometry for scalable, accurate modeling of complex network dynamics, and points to future avenues for automatic reduction-ratio selection and broader applications.

Abstract

Learning complex network dynamics is fundamental for understanding, modeling, and controlling real-world complex systems. Though great efforts have been made to predict the future states of nodes on networks, the capability of capturing long-term dynamics remains largely limited. This is because they overlook the fact that long-term dynamics in complex network are predominantly governed by their inherent low-dimensional manifolds, i.e., skeletons. Therefore, we propose the Dynamics-Invariant Skeleton Neural Net}work (DiskNet), which identifies skeletons of complex networks based on the renormalization group structure in hyperbolic space to preserve both topological and dynamics properties. Specifically, we first condense complex networks with various dynamics into simple skeletons through physics-informed hyperbolic embeddings. Further, we design graph neural ordinary differential equations to capture the condensed dynamics on the skeletons. Finally, we recover the skeleton networks and dynamics to the original ones using a degree-based super-resolution module. Extensive experiments across three representative dynamics as well as five real-world and two synthetic networks demonstrate the superior performances of the proposed DiskNet, which outperforms the state-of-the-art baselines by an average of 10.18\% in terms of long-term prediction accuracy. Code for reproduction is available at: https://github.com/tsinghua-fib-lab/DiskNet.

Predicting Long-term Dynamics of Complex Networks via Identifying Skeleton in Hyperbolic Space

TL;DR

DiskNet introduces a skeleton-based framework to predict long-term dynamics on complex networks by identifying a low-dimensional skeleton in hyperbolic space via a renormalization-group-inspired embedding, modeling skeleton dynamics with a neural ODE, and lifting predictions back to the original network using degree-based super-resolution. The method integrates physics-informed initialization with adaptive hyperbolic embeddings to capture both topological and dynamical similarities, enabling accurate long-horizon forecasting across multiple dynamics and networks. Empirical results show DiskNet outperforms state-of-the-art baselines on 120-step predictions, with notable improvements in stability and efficiency; ablations confirm the value of the initialization and skeleton-learning components. The work highlights the utility of dynamics-aware skeletons and hyperbolic geometry for scalable, accurate modeling of complex network dynamics, and points to future avenues for automatic reduction-ratio selection and broader applications.

Abstract

Learning complex network dynamics is fundamental for understanding, modeling, and controlling real-world complex systems. Though great efforts have been made to predict the future states of nodes on networks, the capability of capturing long-term dynamics remains largely limited. This is because they overlook the fact that long-term dynamics in complex network are predominantly governed by their inherent low-dimensional manifolds, i.e., skeletons. Therefore, we propose the Dynamics-Invariant Skeleton Neural Net}work (DiskNet), which identifies skeletons of complex networks based on the renormalization group structure in hyperbolic space to preserve both topological and dynamics properties. Specifically, we first condense complex networks with various dynamics into simple skeletons through physics-informed hyperbolic embeddings. Further, we design graph neural ordinary differential equations to capture the condensed dynamics on the skeletons. Finally, we recover the skeleton networks and dynamics to the original ones using a degree-based super-resolution module. Extensive experiments across three representative dynamics as well as five real-world and two synthetic networks demonstrate the superior performances of the proposed DiskNet, which outperforms the state-of-the-art baselines by an average of 10.18\% in terms of long-term prediction accuracy. Code for reproduction is available at: https://github.com/tsinghua-fib-lab/DiskNet.
Paper Structure (38 sections, 14 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 38 sections, 14 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Method
  4. Hyperbolic Renormalization Group
  5. Learnable hyperbolic embeddings.
  6. Adaptive assignment matrix.
  7. Node dynamics aggregation.
  8. Physics-informed initialization.
  9. Neural Dynamics on Skeleton
  10. ODE function
  11. Solve
  12. Lift to Original Nodes
  13. Degree-based clustering.
  14. Super-resolution.
  15. Training
  16. ...and 23 more sections

Figures (7)

  • Figure 1: Long-term prediction via the skeleton of complex network dynamics.
  • Figure 2: Overall framework of DiskNet: (1) Hyperbolic Renormalization Group, which identifies the representation and skeleton of network dynamics; (2) Neural Dynamics on Skeleton, which models the dynamics of super-nodes on the skeleton; and (3) Degree-based Super-Resolution, which lifts the predicted values of super-nodes to the original nodes.
  • Figure 3: Percentage improvement in MAE as a function of the horizon for different dynamics.
  • Figure 4: Identified skeleton for different dynamics on the BA-500 networks of DiskNet and static RG model. The bottom half of each figure depicts the average evolution trajectory of all nodes.
  • Figure 5: Time cost per iteration on different network topologies for Hindmarsh-Rose dynamics.
  • ...and 2 more figures