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Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold

Aldric Labarthe, Roland Bouffanais, Julien Randon-Furling

TL;DR

This work tackles the mismatch between graph diffusion geometry and node-attribute manifolds by reframing misalignment as a diagnostic signal rather than a nuisance. It introduces a two-phase VAE that first learns the attribute manifold and then deforms it to align with the graph’s Heat Kernel diffusion geometry, with a differentiable geodesic computation to backpropagate through geometry. The Kernel Alignment Objective quantifies the required geometric deformation, turning misalignment into a measurable signal that improves anomaly detection and reveals structural patterns beyond conventional methods. Across synthetic and real datasets, the approach demonstrates that geometric tension encodes meaningful information about homophily, connectivity shortcuts, and region-level structure, offering a principled diagnostic tool for geometry-aware graph analysis.

Abstract

The standard approach to representation learning on attributed graphs -- i.e., simultaneously reconstructing node attributes and graph structure -- is geometrically flawed, as it merges two potentially incompatible metric spaces. This forces a destructive alignment that erodes information about the graph's underlying generative process. To recover this lost signal, we introduce a custom variational autoencoder that separates manifold learning from structural alignment. By quantifying the metric distortion needed to map the attribute manifold onto the graph's Heat Kernel, we transform geometric conflict into an interpretable structural descriptor. Experiments show our method uncovers connectivity patterns and anomalies undetectable by conventional approaches, proving both their theoretical inadequacy and practical limitations.

Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold

TL;DR

This work tackles the mismatch between graph diffusion geometry and node-attribute manifolds by reframing misalignment as a diagnostic signal rather than a nuisance. It introduces a two-phase VAE that first learns the attribute manifold and then deforms it to align with the graph’s Heat Kernel diffusion geometry, with a differentiable geodesic computation to backpropagate through geometry. The Kernel Alignment Objective quantifies the required geometric deformation, turning misalignment into a measurable signal that improves anomaly detection and reveals structural patterns beyond conventional methods. Across synthetic and real datasets, the approach demonstrates that geometric tension encodes meaningful information about homophily, connectivity shortcuts, and region-level structure, offering a principled diagnostic tool for geometry-aware graph analysis.

Abstract

The standard approach to representation learning on attributed graphs -- i.e., simultaneously reconstructing node attributes and graph structure -- is geometrically flawed, as it merges two potentially incompatible metric spaces. This forces a destructive alignment that erodes information about the graph's underlying generative process. To recover this lost signal, we introduce a custom variational autoencoder that separates manifold learning from structural alignment. By quantifying the metric distortion needed to map the attribute manifold onto the graph's Heat Kernel, we transform geometric conflict into an interpretable structural descriptor. Experiments show our method uncovers connectivity patterns and anomalies undetectable by conventional approaches, proving both their theoretical inadequacy and practical limitations.
Paper Structure (39 sections, 19 equations, 8 figures, 4 tables)

This paper contains 39 sections, 19 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. The geometric misalignment problem
  4. Kernel influence
  5. Methods
  6. VAE for manifold learning
  7. Training Algorithm
  8. Phase 1: Manifold Learning.
  9. Phase 2: Geometric Alignment.
  10. Differentiable Geodesic Computation.
  11. Results
  12. Synthetic Dataset
  13. Experimental setup
  14. Manifold Recovery (Phase 1)
  15. Detecting Misalignment (Phase 2)
  16. ...and 24 more sections

Figures (8)

  • Figure 1: Geometric Alignment VAE. (Black) Phase 1 learns the intrinsic attribute manifold $\mathcal{M}$ via standard reconstruction. (Blue) Phase 2 freezes the encoder and deforms the manifold geometry to align the geometry of $\hat{\mathcal{M}}$ with the kernel geometry.
  • Figure 2: Synthetic dataset latent space. Panel A: theoretical latent space $\mathcal{Z}$; Panel B: estimated latent space after Phase 1; Panel C: estimated latent space after Phase 2; Panel D: estimated curvature changes between Phases 1 and 2. Nodes are plotted in dark purple for normal nodes, and in yellow for perturbed ones.
  • Figure 3: Node Geometric Distortion Scores$S_i$ (from eq. \ref{['eq:modified_z_scores']}). Non-perturbed nodes are plotted in blue, perturbed nodes in red.
  • Figure 4: Phase 1 Latent manifold clustering Points are colored according to the log. robust $Z$-score of the variation between Phase 1 and Phase 2 of the total pairwise distances with respect to other nodes.
  • Figure 5: Cluster Interaction Network. Nodes represent the five socioeconomic functional regions identified in Phase 1 (0: Rural, 1: Residential, 2: Dense Urban, 3: Metro Center, 4: Productive). Edges are colored by the Average Manifold-Graph Distortion (modified $Z$-Score), measuring the tension between the attribute manifold and the transport topology. Red edges (positive $Z$) identify structural "shortcuts" where transit reduces the distances between points placed according to socio-demographic proximity.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Definition 1: Estimated Attribute Geometry
  • Definition 2: Kernel Alignment Objective
  • Definition 3: Linear interpolation distance approximation