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Graph Hopfield Networks: Energy-Based Node Classification with Associative Memory

Abinav Rao, Alex Wa, Rishi Athavale

TL;DR

Graph Hopfield Networks is introduced, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification, and Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.

Abstract

We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield retrieval with Laplacian propagation. Memory retrieval provides regime-dependent benefits: up to 2.0~pp on sparse citation networks and up to 5 pp additional robustness under feature masking; the iterative energy-descent architecture itself is a strong inductive bias, with all variants (including the memory-disabled NoMem ablation) outperforming standard baselines on Amazon co-purchase graphs. Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.

Graph Hopfield Networks: Energy-Based Node Classification with Associative Memory

TL;DR

Graph Hopfield Networks is introduced, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification, and Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.

Abstract

We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield retrieval with Laplacian propagation. Memory retrieval provides regime-dependent benefits: up to 2.0~pp on sparse citation networks and up to 5 pp additional robustness under feature masking; the iterative energy-descent architecture itself is a strong inductive bias, with all variants (including the memory-disabled NoMem ablation) outperforming standard baselines on Amazon co-purchase graphs. Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.
Paper Structure (27 sections, 15 theorems, 57 equations, 4 figures, 11 tables)

This paper contains 27 sections, 15 theorems, 57 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Graph Hopfield Networks
  3. Energy Function and Update Rule
  4. Gated Memory Retrieval
  5. Hierarchical Memory
  6. Experiments
  7. Setup
  8. Node Classification
  9. Robustness to Graph Corruption
  10. Heterophilous Graphs
  11. Discussion and Future Directions
  12. Experimental Details
  13. Related Work
  14. Additional Results
  15. Edge Deletion on Planetoid
  16. ...and 12 more sections

Key Result

Proposition D.1

If $\mathbf{A}$ is symmetric and $\mathbf{D}_{vv}=d_v>0$, then Hence $\mathbf{L}\succeq 0$ and the Laplacian term is convex.

Figures (4)

  • Figure 1: One iteration of the GHN update: node features $\mathbf{x}_v$ are blended with (i) memory retrieval from pattern bank $\mathbf{M}$, and (ii) graph Laplacian smoothing over neighbors. Damping $\alpha$ controls the step size.
  • Figure 2: Accuracy under progressive edge deletion on Amazon and Planetoid datasets. GHN variants degrade more gradually on Amazon; Planetoid details in Appendix \ref{['app:edge_robustness']}.
  • Figure 3: Accuracy under feature masking across five datasets. Tuning selects $K{=}64$ on Planetoid and $K{=}256$ on Amazon. All models degrade at comparable rates on Planetoid; on Amazon, memory-enabled variants show substantially greater robustness (Table \ref{['tab:amazon_corruption']}).
  • Figure 4: Phase diagram of LSE vs. LSR energy on Amazon Photo. Each cell shows mean test accuracy over 10 seeds. High variance in some cells reflects bimodal behavior (seeds either converge to ${\sim}94\%$ or collapse). At $K{=}64$, LSR is stable for moderate $\beta$ while LSE is bimodal for all tested $\beta$.

Theorems & Definitions (32)

  • Proposition D.1: Normalized Laplacian quadratic form
  • proof
  • Proposition D.2: Gradient formula
  • proof
  • Lemma D.3: Retrieval Jacobian
  • proof
  • Lemma D.4: Softmax covariance bound
  • proof
  • Proposition D.5: Convexity and strong convexity regime
  • proof
  • ...and 22 more