Riemannian Liquid Spatio-Temporal Graph Network

TL;DR

RLSTG addresses geometric distortion in Euclidean continuous-time graph models by evolving node states on a product Riemannian manifold with a liquid-time-constant ODE. It extends LTCs to the curved setting via a Geodesic Decay solver, proving stability and universal approximation in the Riemannian domain and quantifying expressivity through trajectory length on manifolds. The framework demonstrates superior performance on irregularly-sampled, non-Euclidean graphs across node regression and link prediction benchmarks, validating the practical value of geometry-aware continuous-time graph learning. By bridging continuous-time dynamics with geometric deep learning, RLSTG offers a principled approach for faithful representation of complex graph structures and dynamics, with potential for automated manifold selection and efficiency improvements in future work.

Abstract

Liquid Time-Constant networks (LTCs), a type of continuous-time graph neural network, excel at modeling irregularly-sampled dynamics but are fundamentally confined to Euclidean space. This limitation introduces significant geometric distortion when representing real-world graphs with inherent non-Euclidean structures (e.g., hierarchies and cycles), degrading representation quality. To overcome this limitation, we introduce the Riemannian Liquid Spatio-Temporal Graph Network (RLSTG), a framework that unifies continuous-time liquid dynamics with the geometric inductive biases of Riemannian manifolds. RLSTG models graph evolution through an Ordinary Differential Equation (ODE) formulated directly on a curved manifold, enabling it to faithfully capture the intrinsic geometry of both structurally static and dynamic spatio-temporal graphs. Moreover, we provide rigorous theoretical guarantees for RLSTG, extending stability theorems of LTCs to the Riemannian domain and quantifying its expressive power via state trajectory analysis. Extensive experiments on real-world benchmarks demonstrate that, by combining advanced temporal dynamics with a Riemannian spatial representation, RLSTG achieves superior performance on graphs with complex structures. Project Page: https://rlstg.github.io

Figures (5)

• Figure 1: Euclidean continuous-time models, including Liquid networks, distort non-Euclidean graph structures. RLSTG reduces distortion by embedding graphs onto suitable Riemannian manifolds. The system's state evolves on this curved space via a Liquid ODE formulated in the manifold's tangent spaces, enabling a more faithful representation.
• Figure 2: GD ODE solver employs an operator splitting technique to decouple the stiff dynamics. (1) Driving step. An explicit Euler update advances the state from $\mathbf{x}(t)$ to an intermediate point $\mathbf{x}^*$ based on the non-stiff driving input. (2) GD: The intermediate state $\mathbf{x}^*$ is analytically pulled towards the origin $\mathbf{o}$ to yield the final state $\mathbf{x}(t+\Delta t)$.
• Figure 3: Expressive power analysis. (a) Mean trajectory length versus tree depth (D) in Euclidean and Hyperbolic spaces. (b) Mean trajectory length versus cycle size (n) in Euclidean and Spherical spaces. (c, d) 2D Principal Component Analysis (PCA, described in hasani2021liquid) projections of the system trajectory for a tree graph (D=10) in Euclidean and Hyperbolic space. (e, f) 2D PCA projections for a cycle graph (n=10) in Euclidean and Spherical space. (g-j) Variance explained by the first four principal components (bars) and their cumulative contribution (black line). Each plot corresponds to an experimental setting: (g) Tree in Euclidean, (h) Tree in Hyperbolic, (i) Cycle in Euclidean, and (j) Cycle in Spherical space. In (c-f), the color gradient on the trajectory line indicates the temporal progression from the start to the end state.
• Figure 4: Geometric analysis of benchmark datasets. Tree-likeness is measured by $\frac{1}{\delta}$. Loops is measured by $\ln(1+\beta_1)$. $\delta$ denotes Gromov hyperbolicity and $\beta_{1}$ the first Betti number.
• Figure 5: Trajectories on $\mathbb{S}^2$ for stiff system with decay phase (toward the black point) followed by driven phase (away from it). The trajectory generated by our solver (red) is visually almost indistinguishable from the ground truth. It faithfully captures the intricate path along the sphere, demonstrating its ability to handle both the stiff decay forces and the driving input within the curved geometry with high precision and stability.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma E.1
• Lemma E.2
• Lemma E.3
• Lemma E.4: Universal approximation, vector form