HARMONIA: Interpretable Graph Learning through Mixtures of Neural Bases

Quan D. Bui; Nguyen Do; An Nguyen Dang; Huyen Nguyen; Nhu Duc Minh Nguyen; My T. Thai

stat.MLSep 27, 2026

HARMONIA: Interpretable Graph Learning through Mixtures of Neural Bases

Authors: Quan D. Bui, Nguyen Do, An Nguyen Dang, Huyen Nguyen, Nhu Duc Minh Nguyen, My T. Thai

Organizations: Center for AI Research, VinUniversity, Hanoi, Vietnam · University of Florida, Gainesville, USA · Posts and Telecommunications Institute of Technology, Hanoi, Vietnam · University of Illinois Chicago, Chicago, USA

Abstract

Existing interpretable graph additive models still face limitations in either computational scalability or modeling flexibility. In terms of structural modeling, previous approaches either face quadratic scaling costs or sacrifice explicit source-to-target contribution decomposition. In terms of feature components, they rely either on per-feature neural networks or on single shared bases with limited feature specialization. We address both problems by introducing HARMONIA: Interpretable Graph Learning through Mixtures of Neural Bases, an interpretable-by-design framework. For feature modeling, HARMONIA introduces a Mixture of Neural Bases (MoNB), which routes features to specialized basis experts, enabling parameter sharing without sacrificing feature-specific specialization. For structural modeling, HARMONIA uses Relative Random Walk Probabilities (RRWP) to capture multi-hop and multi-path relationships, and proposes Sparse RRWP Aggregation (SRA) to compute these interactions through sparse graph propagation without quadratic pairwise complexity. HARMONIA retains a simple additive form in which predictions decompose into feature responses modulated by structural influence. Empirically, HARMONIA achieves stronger explanation recovery than existing interpretable graph baselines while maintaining competitive predictive performance and scaling to graphs with millions of nodes. These results show that interpretable graph learning can remain both faithful and scalable without sacrificing predictive effectiveness.

Figures & tables

Figure 1: HARMONIA combines feature-dependent mixtures of neural bases with normalized random-walk aggregation. MoNB shares specialized functional spaces across features; SRA evaluates their structural contributions through sparse propagation. The resulting logits retain an exact feature–node–walk decomposition.

Type	Method	Cora	CiteSeer	Tolokers	ogbn-arxiv	IGB-Small	ogbn-products
Black-box	GCN	$\mathbf{81.23\pm 1.1}$	$71.20\pm 1.7$	$83.47\pm 0.7$	$71.74\pm 0.3$	$70.46\pm 1.17$	$75.64\pm 0.3$
	GAT	$80.32\pm 2.3$	$70.26\pm 2.3$	$83.76\pm 1.0$	$71.95\pm 0.6$	$70.93\pm 2.15$	$\mathbf{79.45\pm 0.5}$
	GraphSAGE	$79.94\pm 3.4$	$75.90\pm 5.0$	$82.58\pm 0.5$	$71.49\pm 0.2$	$\mathbf{75.49\pm 1.08}$	$75.63\pm 0.3$
	Graph Transformer	$80.70\pm 0.5$	$\mathbf{76.00\pm 0.9}$	$83.27\pm 0.9$	$70.13\pm 0.5$	$70.73\pm 1.26$	$74.74\pm 0.5$
Interpretable	NAM	$51.35\pm 2.3$	$55.40\pm 1.9$	$62.48\pm 2.1$	$56.12\pm 3.4$	OOM	OOM
	GP-NAM	$59.96\pm 3.2$	$60.30\pm 3.9$	$83.12\pm 0.4$	$62.35\pm 4.2$	$55.92\pm 3.32$	$60.13\pm 3.9$

Table 1: Node-classification accuracy (%). Methods are categorized as black-box or interpretable-by-design. Bold denotes the highest mean among all compared methods, underlining denotes the highest mean among interpretable methods. OOM(Out Of Memory)

Figure 2: Hop distinguishability decays with increasing walk length.

Panel of Figure 3 — Figure 3: HARMONIA univariate response recovery for 4 ground-truth functions.

	Feature and response recovery			Edge recovery		Node recovery
Method	Precision @4 $\uparrow$	NDCG @4 $\uparrow$	Effective NRMSE $\downarrow$	NDCG @20% $\uparrow$	Precision @20% $\uparrow$	Precision @4 $\uparrow$	NDCG @4 $\uparrow$	Signed NRMSE $\downarrow$
HARMONIA	$\mathbf{1.000}$	$\mathbf{1.000}$	$\mathbf{0.193}$	$\mathbf{0.9425}$	$\mathbf{0.8162}$	$\mathbf{0.8487}$	$\mathbf{0.9658}$	$\mathbf{0.2992}$
GNAN	$\mathbf{1.000}$	$0.996$	$0.435$	$0.6305$	$0.5005$	$0.7057$	$0.8861$	$0.5853$
G-NAMRFF	$0.783$	$0.897$	$0.735$	$0.7589$	$0.6430$	$0.6413$	$0.7975$	$0.8433$
GCN + GNNExplainer	$0.850$	$0.934$	N/A	$0.6259$	$0.5708$	$0.5312$	$0.6914$	N/A
GCN + GOAt	$0.950$	$0.967$	N/A	$0.7790$	$0.6597$	$0.6732$	$0.8294$	N/A

Table 2: Synthetic recovery means. Bold marks the best mean, including ties. Full results and available standard deviations are in Tables 10 , 11 , and 13 .

Panel of Figure 4 — Figure 4: UMAP visualizations of Cora features with different numbers of MoNB experts. Each point represents a feature, and its color indicates the expert assignment.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Meaning
Graph and prediction
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathbf{X}),\ n,\ d$	Input graph, number of nodes, and number of features.
$v_{i},\ x_{i,k}$	Node $i$ and its $k$ -th feature.
$f_{k},\ \mathbf{h}_{i}$	Feature- $k$ response function and propagated representation of node $i$ .
$\ell_{i,c},\ \beta_{c},\ w_{k,c}$	Class- $c$ logit, bias, and output coefficient for feature $k$ .
Mixture of Neural Bases (MoNB)

Appendix

Table 3: Summary of the main notation used in HARMONIA .

Dataset	# Nodes	# Edges	# Features	# Classes
Cora	2,708	10,556	1,433	7
CiteSeer	3,327	9,104	3,703	6
Tolokers	11,758	519,000	10	2
ogbn-arxiv	169,343	1,166,243	128	40
IGB-Small	1,000,000	12,070,502	1,024	19
ogbn-products	2,449,029	61,859,140	100	47

Appendix

Table 4: Statistics of node-classification datasets.

Dataset	# Graphs	Avg. Nodes	Avg. Edges	# Features	# Classes
MUTAG	188	17.93	19.79	7	2
PROTEINS	1,113	39.06	72.82	3	2
DD	1,178	284.32	715.66	89	2
AIDS	2,000	15.69	16.20	23	2
Mutagenicity	4,337	30.32	30.77	14	2
hERG-Karim	13,445	30.42	32.56	65	2

Appendix

Table 5: Statistics of graph-classification datasets.

Dataset	$C$	$B$	$q$	$m$	$T$	Expert depth / widths	Activation	Dropout / norm.
Cora	5	8	32	2	8	2 / [16, 8]	ReLU hidden; linear output	0.4 / none
CiteSeer	5	8	32	2	8	2 / [16, 8]	ReLU hidden; linear output	0.5 / none
Tolokers	3	8	8	1	4	2 / [16, 8]	ReLU hidden; linear output	0.2 / none
ogbn-arxiv	4	8	16	2	8	2 / [16, 8]	ReLU hidden; linear output	0.1 / none
IGB-Small	6	8	32	2	4	2 / [16, 8]	ReLU hidden; linear output	0.1 / none
ogbn-products	4	8	16	2	4	2 / [16, 8]	ReLU hidden; linear output	0.1 / none

Appendix

Table 6: Selected HARMONIA architecture by dataset. Here

C

is the number of experts,

B

the bases per expert,

q

the feature-embedding dimension,

m

the number of active experts, and

T

the number of RRWP channels including

t=0

Dataset	Optimizer	Learning rate	Weight decay	Batch size	Max. epochs	Patience	Selection metric
Cora	AdamW	$5\times 10^{-4}$	$5\times 10^{-4}$	full graph	800	150	validation accuracy ( val_acc )
CiteSeer	AdamW	$1\times 10^{-3}$	$1\times 10^{-4}$	full graph	800	150	validation accuracy ( val_acc )
Tolokers	AdamW	$1\times 10^{-3}$	$5\times 10^{-5}$	full graph	300	50	validation accuracy ( val_auc )
ogbn-arxiv	Adam	$2\times 10^{-2}$	$5\times 10^{-5}$	full graph	600	100	validation accuracy ( val_acc )
IGB-Small	AdamW	$1\times 10^{-3}$	$5\times 10^{-5}$	16384	500	50	validation accuracy ( val_acc )
ogbn-products	AdamW	$1\times 10^{-3}$	$5\times 10^{-5}$	3584	500	50	validation accuracy ( val_acc )

Appendix

Table 7: Selected HARMONIA optimization settings by dataset. For full-batch training, report the batch-size entry as “full graph.”

Method	MUTAG	PROTEINS	DD	AIDS	Mutagenicity	hERG
GCN	$68.07\pm 6.34$	$\mathbf{70.97\pm 4.67}$	$75.66\pm 2.36$	$\mathbf{99.25\pm 0.63}$	$\mathbf{75.69\pm 0.94}$	$73.06\pm 2.08$
GAT	$67.20\pm 3.45$	$69.92\pm 4.03$	$\mathbf{77.30\pm 3.68}$	$99.00\pm 0.75$	$69.40\pm 1.26$	$65.04\pm 1.07$
GraphSAGE	$64.12\pm 2.46$	$67.35\pm 2.38$	$75.78\pm 3.98$	$98.20\pm 1.05$	$69.25\pm 3.94$	$71.08\pm 2.05$
Graph Transformer	$\mathbf{73.30\pm 5.36}$	$69.76\pm 3.27$	$76.24\pm 3.26$	$99.10\pm 0.60$	$73.10\pm 0.93$	$\mathbf{74.58\pm 1.84}$
NAM	$63.12\pm 9.15$	$62.45\pm 4.28$	$58.07\pm 3.56$	$72.51\pm 2.53$	$67.35\pm 2.57$	$64.57\pm 2.54$
GP-NAM	$64.30\pm 8.46$	$65.68\pm 4.13$	$62.07\pm 3.24$	$79.35\pm 2.28$	$65.46\pm 2.25$	$68.06\pm 2.37$

Appendix

Table 8: Graph-classification accuracy (%). Bold denotes the highest mean among all compared methods; underlining denotes the highest mean among interpretable methods, including HARMONIA . The markings indicate mean rankings, not statistical significance.

Profile	$\theta_{0,k}^{\star}$	$\theta_{1,k}^{\star}$	$\theta_{2,k}^{\star}$	$\theta_{3,k}^{\star}$	Dominant hop
A	$0.70$	$0.20$	$0.08$	$0.02$	$0$
B	$0.05$	$0.15$	$0.65$	$0.15$	$2$
C	$0.05$	$0.10$	$0.20$	$0.65$	$3$
D	$0.10$	$0.55$	$0.25$	$0.10$	$1$

Appendix

Table 9: Ground-truth feature-specific structural profiles. Every profile is nonnegative and sums to one. The four profiles occur in every data seed.

Method	Precision @4 $\uparrow$	NDCG @4 $\uparrow$	Feature Spearman $\uparrow$	Effective NRMSE $\downarrow$	Hop L1 $\downarrow$	Top-hop Acc. $\uparrow$	Probe Coverage $\uparrow$
HARMONIA	$\mathbf{1.000}$	$\mathbf{1.000}$	$\mathbf{0.699}$	$\mathbf{0.193}$	$\mathbf{0.468}$	$\mathbf{0.950}$	$\mathbf{1.000}$
GNAN	$\mathbf{1.000}$	$0.996$	$0.698$	$0.435$	$0.822$	$0.250$	$0.983$
G-NAMRFF	$0.783$	$0.897$	$0.568$	$0.735$	$1.129$	$0.289$	$0.640$
GCN + GNNExplainer	$0.850$	$0.934$	$0.619$	N/A	$0.766$	$0.493$	$0.547$
GCN + GOAt	$0.950$	$0.967$	$0.669$	N/A	$0.766$	$0.493$	$0.547$
GCN + UCExplainer	$0.767$	$0.912$	$0.586$	N/A	$0.766$	$0.493$	$0.547$

Appendix

Table 10: Complete feature–hop recovery results, averaged over five data seeds and three model seeds per data seed. Feature explainers use the same 128 test targets per run. N/A indicates that the explainer does not define an explicit additive response curve. The GCN rows share structural-probe results by construction. Bold denotes the best reported mean, including ties; no dispersion estimates are available for this table.

Method	Edge Spearman $\uparrow$	Edge NDCG@20% $\uparrow$	Precision@20% $\uparrow$
HARMONIA	$\mathbf{0.7822\pm 0.0489}$	$\mathbf{0.9425\pm 0.0102}$	$\mathbf{0.8162\pm 0.0402}$
GNAN	$0.4546\pm 0.0581$	$0.6305\pm 0.0805$	$0.5005\pm 0.0766$
G-NAMRFF	$0.6006\pm 0.0282$	$0.7589\pm 0.0262$	$0.6430\pm 0.0478$
GCN + GNNExplainer	$0.5959\pm 0.0187$	$0.6259\pm 0.0527$	$0.5708\pm 0.0530$
GCN + GOAt	$0.5871\pm 0.0371$	$0.7790\pm 0.0424$	$0.6597\pm 0.0712$
GCN + UCExplainer	$0.0194\pm 0.0559$	$0.2970\pm 0.0517$	$0.2594\pm 0.0294$

Appendix

Table 11: Complete instance-level structural recovery results. Entries are mean

\pm

sample standard deviation across five data-seed means; each mean averages three model seeds, each evaluated on the same 128 targets. All methods share candidate edges, exact edge-deletion ground truth, and the rounded-up top-20% budget. Bold denotes the best reported mean.

Data index	Feature seed	Graph seed	Signal columns	Profile assignment
$0$	$20260916$	$123$	$19,17,5,2$	A, B, C, D
$1$	$20260917$	$223$	$18,3,7,1$	B, C, D, A
$2$	$20260918$	$323$	$1,14,18,0$	C, D, A, B

Appendix

Table 12: Source-node evaluation cohort. Signal-column indices are zero-based. Profile assignments follow the response order

(2x,\text{quadratic},\sin,\tanh)

. Each data seed is paired with model seeds

0,1,2

Method	Spearman $\uparrow$	Precision@4 $\uparrow$	NDCG@4 $\uparrow$	Signed NRMSE $\downarrow$	SignAgree@4 $\uparrow$
HARMONIA	$\mathbf{0.9601\pm 0.0464}$	$\mathbf{0.8487\pm 0.0254}$	$\mathbf{0.9658\pm 0.0122}$	$\mathbf{0.2992\pm 0.0626}$	$\mathbf{0.9792\pm 0.0066}$
GNAN	$0.5319\pm 0.0730$	$0.7057\pm 0.0344$	$0.8861\pm 0.0409$	$0.5853\pm 0.1196$	$0.8487\pm 0.0478$
G-NAMRFF	$0.9309\pm 0.0761$	$0.6413\pm 0.0412$	$0.7975\pm 0.0334$	$0.8433\pm 0.0888$	$0.7157\pm 0.0436$
GCN + GNNExplainer	$0.8893\pm 0.1106$	$0.5312\pm 0.0584$	$0.6914\pm 0.0812$	N/A	N/A
GCN + GOAt	$0.9363\pm 0.0669$	$0.6732\pm 0.0384$	$0.8294\pm 0.0507$	N/A	N/A
GCN + UCExplainer	$-0.0264\pm 0.0285$	$0.1120\pm 0.0098$	$0.2076\pm 0.0100$	N/A	N/A

Appendix

Table 13: Source-node contribution recovery. Entries are mean

\pm

sample standard deviation across three data-seed means, each averaging three model seeds and 128 targets per run. All methods rank the same 64 candidates against the same oracle intervention effects. N/A denotes unavailable signed logit-scale scores. Bold marks the best mean in each recovery column.

Explainer	Nodes with nonzero score/change
GCN + GNNExplainer	$25.83\pm 8.89$
GCN + UCExplainer	$64.00\pm 0.00$

Appendix

Table 14: Auxiliary source-node explanation diagnostics on the same cohort. UCExplainer changes at least one feature at every node under the

10^{-6}

threshold. These counts describe the full explanation before selecting its top-four nodes.

Panel of Figure 5 — Figure 5: HARMONIA univariate response recovery for twelve ground-truth functions. Panel annotations show the reported curve NRMSE summaries. This additional function-recovery study is separate from the feature–hop benchmark in Table 10 ; its curve errors are defined in Eq. equation 130 .

Panel of Figure 6 — Figure 6: Functional specialization under increasing heterogeneity. (a) Centered NRMSE of Global NBM, MoNB, and independent per-feature networks. The annotated values $\Delta=E_{\mathrm{Global}}-E_{\mathrm{MoNB}}$ measure the approximation gain of MoNB over global basis sharing. (b) Mean feature-to-expert gating weights in the high-heterogeneity regime.

Panel of Figure 7 — Figure 7: Effect of expert count and basis size under low, medium, and high functional heterogeneity. Each cell reports centered NRMSE with $m=\min(2,C)$ , where lower is better. Additional experts provide larger gains in more heterogeneous regimes, while increasing $C$ beyond $5$ provides no further improvement in this sweep.

Panel of Figure 8 — Figure 8: Routing quality and specialization gain. (a) Numerical validation of $E_{\mathrm{route}}-E_{\mathrm{global}}=R_{\mathrm{route}}-\Gamma_{B,C}$ . (b) Approximation error as a function of routing accuracy under nearest-wrong, random-wrong, and farthest-wrong corruption policies. Similar routing accuracies can produce substantially different errors. (c) Approximation damage as a function of the cost-weighted routing penalty $R_{\mathrm{route}}$ . Different corruption policies follow nearly the same relationship when routing errors are measured by their functional cost.

Figure 9: Hop recovery under varying structural conditioning at $\epsilon=0.01$ . Coefficient error $e_{\theta}$ measures recovery of the hop allocation, while operator error $e_{\mathrm{op}}$ measures recovery of the induced structural transformation. Only numerically non-singular settings with $\gamma_{T}>10^{-12}$ are shown. As $\gamma_{T}$ decreases, operator recovery remains accurate while coefficient recovery becomes increasingly unstable.

Panel of Figure 10 — Figure 10: Empirical validation of Theorem 2 . (a) Consecutive-hop difference $D_{t}$ across graph families with different mixing regimes. (b) Empirical horizon $H_{\epsilon}$ versus the spectral upper bound $\widehat{H}_{\epsilon}$ ; all settings remain below the identity line.

Model Variant	Graph Classification (Acc. %)			Node Classification (Acc. %)
Model Variant	Mutag	Proteins	AIDS	Cora	Citeseer	ogbn-arxiv
HARMONIA (DNN RRWP)	71.25 $\pm$ 1.8	72.10 $\pm$ 1.4	98.15 $\pm$ 0.9	80.52 $\pm$ 0.9	73.64 $\pm$ 0.8	OOM
HARMONIA (Linear RRWP)	69.50 $\pm$ 1.5	70.76 $\pm$ 1.2	97.75 $\pm$ 1.1	79.60 $\pm$ 1.3	72.80 $\pm$ 1.1	78.14 $\pm$ 2.3

Appendix

Table 15: Comparison between HARMONIA (Linear RRWP) and HARMONIA (DNN RRWP) across graph and node classification benchmarks. OOM indicates Out-Of-Memory (

>24

GB VRAM).

Figure 11: Change in test accuracy after removal and retraining. The plot uses accuracy fractions; $-0.104$ corresponds to a decrease of $10.4$ percentage points. Printed summaries are reproduced from the experiment.

Figure 12: Five-seed visualization of feature importance. The left panel displays validation-margin importance for non-bias features

Figure 13: Parameter complexity in the synthetic multi-class setting with $K=47$ output classes as the input feature dimension $D$ increases.

Figure 14: Selected feature-response functions learned by HARMONIA on the Mutagenicity dataset. Each curve shows the contribution associated with one atom-related feature as its normalized feature value varies.

Panel of Figure 15 — Figure 15: Effect of the RRWP simplex constraint on synthetic explanation faithfulness. (a) Ground-truth and recovered hop profiles for the four informative features. Recovered profiles use the same orientation and $\ell_{1}$ normalization as the intervention-based hop evaluation. The simplex-normalized model reduces mean hop-profile $\ell_{1}$ error from $0.5491$ to $0.4409$ , corresponding to a $19.7\%$ relative reduction. (b) Agreement between predicted and oracle edge importance. The normalized model increases edge-ranking Spearman correlation from $0.7556$ to $0.7823$ and Precision@20% from $0.8053$ to $0.8329$ . Points denote individual data-seed means, lines connect matched seeds, and error bars show sample standard deviation across the three data seeds.

Level	Metric	Normalized $\theta$	Unnormalized $\theta=\alpha$	Change
Feature / Hop	Effective NRMSE $\downarrow$	0.2049 $\pm$ 0.0139	0.2279 $\pm$ 0.0072	$-10.1\%$
Feature / Hop	Hop L1 $\downarrow$	0.4409 $\pm$ 0.0221	0.5491 $\pm$ 0.0188	$-19.7\%$
Node	Signed NRMSE $\downarrow$	0.3125 $\pm$ 0.0302	0.3252 $\pm$ 0.0281	$-3.9\%$
Node	Precision@4 $\uparrow$	0.8509 $\pm$ 0.0195	0.8409 $\pm$ 0.0179	$+1.0$ pp
Edge	Spearman $\uparrow$	0.7823 $\pm$ 0.0112	0.7556 $\pm$ 0.0170	$+0.0267$
Edge	Precision@20% $\uparrow$	0.8329 $\pm$ 0.0253	0.8053 $\pm$ 0.0140	$+2.76$ pp

Appendix

Table 16: Representative explanation-faithfulness metrics for simplex-normalized and unconstrained RRWP coefficients. For error metrics, Change reports the relative reduction from the unconstrained variant; for precision metrics it reports the percentage point improvement; and for Spearman correlation it reports the absolute improvement. Results are mean

\pm

sample standard deviation across three fixed synthetic data seeds.

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HiSE: A Lightweight Hierarchical Semantic Explainer for Heterogeneous Graph Neural Networks

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Graph Neural Networks Explainability Semantic Representations

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Heterogeneous graph neural networks (HGNNs) have demonstrated remarkable performance in modeling complex relational data, however their interpretability in high-stakes applications remains a critical challenge. Existing explanation methods suffer from two major limitations: on the one hand, the generated explanations fail to reflect the inherent semantic hierarchy of HGNNs, resulting in a lack of fidelity to the model's internal decision-making mechanism; on the other hand, feature explanations often rely on complex search or perturbation mechanisms, leading to excessive computational complexity and poor efficiency. To address these issues, we propose HiSE, a lightweight feature-oriented interpretable model for HGNNs. HiSE achieves semantically aware feature explanations through hierarchical semantic modeling: at the semantic level, local surrogate models based on the Least Absolute Shrinkage and Selection Operator (LASSO) are employed to learn sparse feature representations under each semantic view; at the cross-semantic level, the contributions of different semantic views are adaptively characterized via KL divergence to produce a unified explanation. Extensive experiments demonstrate that HiSE outperforms existing methods in terms of fidelity, robustness, and cross-semantic explanation capability, while its lightweight framework incurs low computational overhead, enabling efficient application to large-scale, complex real-world heterogeneous graphs.

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