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

TL;DR

This paper introduces Hysteresis Graph ODEs (HGODE), a novel continuous-time graph model that overcomes the monostability trap of traditional Graph ODEs by coupling feature and topology evolution through hysteretic dynamics.

Contribution

The paper proposes HGODE, integrating latent topological potential with feature evolution, enabling phase polarization and long-term diversity in graph dynamics.

Findings

HGODE can polarize edge states into connected or insulated phases.

The model overcomes the monostability trap of traditional Graph ODEs.

Validated on synthetic and real-world graph benchmarks.

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.