Latent-Hysteresis Graph ODEs: Modeling Coupled Topology-Feature Evolution via Continuous Phase Transitions
For graph learning practitioners, HGODE addresses the fundamental collapse problem in continuous-time graph neural networks, enabling long-range propagation without information loss.
Graph ODEs with strictly positive irreducible mixing operators suffer from a monostability trap leading to information leakage and global consensus. The proposed Hysteresis Graph ODE (HGODE) couples feature evolution with a latent topological potential using a double-well edge potential and bipolarized gate, achieving state-of-the-art results on graph benchmarks.
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.