Latent-Hysteresis Graph ODEs: Modeling Coupled Topology-Feature Evolution via Continuous Phase Transitions

2026-04-27

AI Summary

This work addresses the tendency of conventional Graph Ordinary Differential Equations (GODEs) to fall into monostable traps during long-term evolution, which leads to a loss of diversity in node representations. To overcome this limitation, the authors propose Hysteresis Graph ODE (HGODE), which introduces, for the first time, a hysteresis-based double-well potential mechanism into the graph ODE framework. HGODE couples node feature dynamics with continuous phase transitions of edge states through learnable pairwise-force-driven latent topological potentials, enabling dual-phase polarization between connected and insulating states. This approach transcends monostability by supporting differentiable dynamic topological phase transitions, thereby effectively mitigating representation collapse. Theoretical analysis and experiments demonstrate that HGODE significantly enhances long-range dependency modeling and representation stability on both synthetic and real-world graph benchmarks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Graph neural ordinary differential equations (Graph ODEs) extend graph learning from discrete message-passing layers to continuous-time representation flows. While it supports adaptive long-range propagation, we show that Graph ODEs with strictly positive irreducible mixing operators face an inherent \emph{monostability trap}: in the long-time regime, information leakage is unavoidable and the dynamics converge to a single global consensus attractor. We propose the \textbf{Hysteresis Graph ODE (HGODE)}, which couples feature evolution with a latent topological potential driven by a learned pairwise force. A double-well edge potential and bipolarized gate allow edge states to polarize into connected or insulated phases while preserving differentiability. We provide asymptotic analysis of the collapse mechanism and the proposed hysteretic topology dynamics, and validate HGODE on theory-driven synthetic diagnostics and real-world graph benchmarks.

Problem

• Graph ODEs
• monostability trap
• topology-feature evolution
• information leakage
• global consensus

Innovation

• Hysteresis Graph ODE
• latent topological potential
• double-well edge potential
• bipolarized gate
• continuous phase transitions