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

Abinav Rao; Alex Wa; Rishi Athavale

cs.LGMar 3, 2026

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

Authors: Abinav Rao, Alex Wa, Rishi Athavale

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.

Figures & tables

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.

GHN variants
	Cora	CiteSeer	PubMed	Photo	Computers
LSE	$82.0{\scriptstyle\pm 0.6}$	$70.9{\scriptstyle\pm 0.8}$	$77.9{\scriptstyle\pm 0.4}$	$94.5{\scriptstyle\pm 0.3}$	$91.3{\scriptstyle\pm 0.2}$
LSR	$82.0{\scriptstyle\pm 0.6}$	$70.9{\scriptstyle\pm 1.0}$	$78.1{\scriptstyle\pm 0.4}$	$94.4{\scriptstyle\pm 0.2}$	$91.3{\scriptstyle\pm 0.4}$
Hier(8)	$82.3{\scriptstyle\pm 0.4}$	$70.3{\scriptstyle\pm 1.1}$	$77.6{\scriptstyle\pm 0.8}$	$94.2{\scriptstyle\pm 0.2}$	$91.3{\scriptstyle\pm 0.2}$
NoMem	$80.3{\scriptstyle\pm 0.4}$	$69.8{\scriptstyle\pm 1.1}$	$77.2{\scriptstyle\pm 0.6}$	$\mathbf{94.7{\scriptstyle\pm 0.2}}$	$\mathbf{91.5{\scriptstyle\pm 0.2}}$
Baselines

Table 1: Node classification accuracy (%) with per-model HP tuning. Bold : best per column.

\dagger

: bimodal (

>

10% std).

GHN variants
	Clean	Edge drop	Feat. mask	Feat. noise
LSE	$\mathbf{94.4{\scriptstyle\pm 0.3}}$	$93.7{\scriptstyle\pm 0.4}$	$89.4{\scriptstyle\pm 1.8}$	$93.9{\scriptstyle\pm 0.3}$
LSR	$94.3{\scriptstyle\pm 0.2}$	$93.7{\scriptstyle\pm 0.5}$	$91.0{\scriptstyle\pm 2.4}$	$94.1{\scriptstyle\pm 0.2}$
Hier(8)	$94.2{\scriptstyle\pm 0.2}$	$93.7{\scriptstyle\pm 0.5}$	$\mathbf{91.9{\scriptstyle\pm 1.1}}$	$93.9{\scriptstyle\pm 0.1}$
NoMem	$\mathbf{94.4{\scriptstyle\pm 0.3}}$	$\mathbf{94.0{\scriptstyle\pm 0.5}}$	$86.9{\scriptstyle\pm 3.1}$	$\mathbf{94.2{\scriptstyle\pm 0.3}}$
Baselines

Table 2: Accuracy (%) under 50% corruption on Amazon Photo. Clean baselines may differ from Table 1 due to a separate HP sweep optimized for robustness. Bold : best per column.

Figure 2: Accuracy under progressive edge deletion on Amazon and Planetoid datasets. GHN variants degrade more gradually on Amazon; Planetoid details in Appendix C.1 .

GHN variants
	Texas	Wisconsin	Cornell	Actor
LSE	$\mathbf{81.4{\scriptstyle\pm 1.5}}$	$82.2{\scriptstyle\pm 0.6}$	$78.1{\scriptstyle\pm 3.2}$	$\mathbf{38.0{\scriptstyle\pm 0.5}}$
LSR	$80.8{\scriptstyle\pm 1.5}$	$82.5{\scriptstyle\pm 1.1}$	$78.9{\scriptstyle\pm 1.7}$	$37.9{\scriptstyle\pm 0.5}$
Hier(8)	$80.5{\scriptstyle\pm 3.3}$	$82.2{\scriptstyle\pm 1.7}$	$78.1{\scriptstyle\pm 2.0}$	$37.8{\scriptstyle\pm 0.6}$
NoMem	$80.5{\scriptstyle\pm 1.7}$	$82.4{\scriptstyle\pm 0.9}$	$78.6{\scriptstyle\pm 2.4}$	$\mathbf{38.0{\scriptstyle\pm 0.7}}$
Heterophily-aware baselines

Table 3: Node classification (%) on heterophilous benchmarks. Bold : best per column.

\dagger

Published results ( Chien et al., 2021 ) on same splits; Actor not reported.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

GHN (all variants)
Parameter	Value / Grid
Hidden dimension	$\{64,128\}$
Number of patterns $K$	$\{64,256\}$ (LSE/LSR)
Inverse temperature $\beta$ (init)	1.0 (learnable)
Laplacian weight $\lambda$	0.3
Damping $\alpha$	0.3

Appendix

Table 4: Hyperparameters for GHN and training protocol. Per-model HP tuning (Table 1 ) selects the best config per dataset from the grid ranges shown.

	50% edge drop			100% edge drop
	Cora	CiteSeer	PubMed	Cora	CiteSeer	PubMed
GHN -LSE	$-6.5$	$-10.2$	$\mathbf{-1.8}$	$-15.5$	$-19.5$	$\mathbf{-4.6}$
GHN -Hier(8)	$\mathbf{-5.6}$	$-8.1$	$-2.5$	$-13.5$	$-14.2$	$-6.1$
GCN	$-5.9$	$-5.5$	$-2.2$	$-13.6$	$-9.7$	$-4.8$
APPNP	$-6.1$	$\mathbf{-3.0}$	$-2.3$	$\mathbf{-12.5}$	$\mathbf{-6.3}$	$-5.5$
GIN	$-12.7$	$-16.5$	$-10.3$	$-44.1$	$-49.9$	$-30.5$

Appendix

Table 5: Relative accuracy drop (%) under edge deletion on Planetoid. Smaller magnitude = greater robustness. Bold : most robust per dataset.

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 2 ).

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 $\beta$ .

Dataset	$0$	$0.01$	$0.05$	$0.1$	$0.2$	$0.3$	$0.5$	$1.0$
Cora	57.7	62.7	71.7	76.1	80.2	81.8	81.6	80.6
CiteSeer	55.6	59.1	65.1	67.6	70.2	69.6	68.9	69.4
PubMed	73.0	73.2	74.5	75.2	78.2	77.7	76.9	76.7
Texas	74.1	71.6	70.0	64.9	66.8	67.0	66.8	64.9
Wisconsin	79.8	80.8	72.2	67.1	60.0	55.1	45.3	44.9
Cornell	73.5	72.4	63.8	46.2	41.1	44.3	42.2	41.4

Appendix

Table 6: GHN-LSE test accuracy (%) as a function of

\lambda

. Bold : best

\lambda

per dataset.

		$\lambda{=}{-}0.3$	$\lambda{=}{-}0.1$	$\lambda{=}{-}0.05$	$\lambda{=}{-}0.01$
Texas	LSE	75.9	77.3	78.6	74.6
Texas	LSR	75.9	77.3	79.5	78.6
Wisconsin	LSE	76.9	81.0	81.8	79.0
Wisconsin	LSR	78.8	80.8	80.4	79.0
Cornell	LSE	61.9	66.5	74.6	77.0
Cornell	LSR	63.2	69.5	74.3	77.8

Appendix

Table 7: GHN accuracy (%) with negative

\lambda

on heterophilous datasets. Best result per dataset-model in bold .

	$T{=}1$	$T{=}2$	$T{=}4$	$T{=}8$	$T{=}16$
Cora	$74.5{\scriptstyle\pm 1.1}$	$78.4{\scriptstyle\pm 0.5}$	$\mathbf{81.8{\scriptstyle\pm 1.4}}$	$56.1{\scriptstyle\pm 26.1}$	$30.9{\scriptstyle\pm 2.0}$
Texas	$\mathbf{73.5{\scriptstyle\pm 2.8}}$	$72.2{\scriptstyle\pm 2.9}$	$67.0{\scriptstyle\pm 1.7}$	$65.9{\scriptstyle\pm 1.4}$	$65.4{\scriptstyle\pm 1.1}$
Wisconsin	$\mathbf{63.7{\scriptstyle\pm 1.0}}$	$58.6{\scriptstyle\pm 2.3}$	$55.1{\scriptstyle\pm 5.9}$	$44.1{\scriptstyle\pm 1.9}$	$46.3{\scriptstyle\pm 1.4}$

Appendix

Table 8: GHN-LSE accuracy (%) vs. iterations

T

with

\lambda{=}0.3

. Bold : best

T

	$H{=}1$	$H{=}2$	$H{=}4$	$H{=}8$
Cora	$81.5{\scriptstyle\pm 0.9}$	$81.6{\scriptstyle\pm 1.0}$	$81.8{\scriptstyle\pm 1.4}$	$80.3{\scriptstyle\pm 0.4}$
CiteSeer	$69.6{\scriptstyle\pm 1.4}$	$70.0{\scriptstyle\pm 1.6}$	$69.6{\scriptstyle\pm 0.7}$	$69.5{\scriptstyle\pm 1.5}$
PubMed	$77.6{\scriptstyle\pm 0.5}$	$77.6{\scriptstyle\pm 0.6}$	$77.7{\scriptstyle\pm 0.9}$	$77.9{\scriptstyle\pm 1.0}$
Texas	$64.9{\scriptstyle\pm 1.8}$	$65.9{\scriptstyle\pm 2.9}$	$67.0{\scriptstyle\pm 1.7}$	$67.6{\scriptstyle\pm 0.0}$

Appendix

Table 9: GHN-LSE accuracy (%) vs. number of memory heads

H

Mask %	0%	10%	30%	50%	70%
$\bar{g}$	$0.727{\scriptstyle\pm 0.039}$	$0.727{\scriptstyle\pm 0.039}$	$0.726{\scriptstyle\pm 0.038}$	$0.725{\scriptstyle\pm 0.038}$	$0.723{\scriptstyle\pm 0.037}$
Accuracy (%)	$93.6{\scriptstyle\pm 0.5}$	$93.6{\scriptstyle\pm 0.5}$	$93.2{\scriptstyle\pm 0.5}$	$90.6{\scriptstyle\pm 1.3}$	$67.9{\scriptstyle\pm 12.8}$

Appendix

Table 10: Mean gate value

\bar{g}

of GHN-LSE on Amazon Photo under increasing feature masking. Higher

g

means more weight on memory retrieval.

Dataset	$\beta$	$\\|\mathbf{M}\\|_{\sigma}^{2}$	$\beta\\|\mathbf{M}\\|_{\sigma}^{2}$
Cora	1.11	1.94	2.15
CiteSeer	1.09	1.71	1.88
PubMed	1.36	5.24	6.54
Amazon Photo	0.99	36.37	35.96
Amazon Comp.	1.12	2.11	2.36
Texas	1.25	3.88	4.65

Appendix

Table 11: Learned

\beta\|\mathbf{M}\|_{\sigma}^{2}

at trained operating points (mean over 5 seeds). The convexity condition requires this product to be

{<}\,2

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Graph Memory: Spectral Associative Memory via Dirichlet Energy

Zhaoyang Shi

Center for Applied Mathematics, Fudan University Shanghai, China

Associative Memory Graph-Based Retrieval Spectral Methods

Open paper AskPDF arXiv

Dense associative memories have traditionally focused on storing and retrieving vector-valued patterns. Many modern machine learning problems, however, are naturally graph-structured, requiring memory mechanisms for relational patterns, graph diffusion geometries, community structures, and graph-based inductive biases. We propose a spectral dense associative memory for storage and retrieval of graph data, extending the classical vector-valued memories. Retrieval is performed through a log-sum-exp energy induced by Dirichlet energy with spectral norm distances, producing a softmax-weighted average of the stored Laplacians that remains a valid graph Laplacian. We prove exponential storage capacity and exponentially decaying retrieval error. Beyond graph retrieval, we establish theoretical guarantees for spectral quantities central to graph learning, including eigenvalues, eigenspaces, and diffusion operators. Experiments on synthetic graph data, real-world airline network, protein conformation data and wearable sensor data demonstrate robust graph retrieval while preserving the graph geometry of the data. Our framework provides a new associative memory paradigm for graph-structured data and bridges dense associative memory with modern graph learning and generative AI.

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