Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study

A graph layout is normally a table of $N$ free coordinates. We optimise a function with a fixed number of parameters instead. This gives a drawing a sample complexity and an extensible domain. Force-directed algorithms remain the standard tools for graph drawing. The most accurate among them minimise stress in the Kamada-Kawai formulation by directly optimising the node coordinates, at a full objective cost of $O(N^2)$ in time and space. Here, we propose Fling (Field Layout via Implicit Neural Geometry), a small neural network mapping the distances of each node to a set of landmarks, positioning it in the plane by training on the layout energy. The full spring system then becomes tractable without its distance matrix, as rest lengths follow from a landmark bound in constant time per pair while a second network learns the majorisation sums from exact anchor rows, at $O(|\mathcal{A}|N)$ per step for $|\mathcal{A}|\ll N$ anchors. Unlike neural drawers that read the graph by message passing, we represent the drawing as a function of node features. An unseen node costs one forward pass, where sparse and low-rank majorisation remain transductive. As the unknowns are weights rather than coordinates, the energy only requires a small fraction of the nodes, and a field fitted that way outperforms PivotMDS, landmark MDS, and a kernel ridge trained on the same energy and features, when the task is fitting the energy of a graph from a sample of its nodes. In addition, the same parameterisation enables a stochastic pivot stress variant, an aesthetics-optimised variant carrying a neighbour-embedding energy with node-edge clearance and crossing terms on the same field, and conditioning on the weight between two energies gives a whole layout family from one run.

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.

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.