Graph Additive Models disentangling feature, topology, and interactions

Clara Alvarez-Rodriguez; Steve Azzolin; Vincenzo Marco De Luca; Filippo Maria Bianchi; Andrea Passerini; Antonio Longa

stat.MLOct 14, 2026

Graph Additive Models disentangling feature, topology, and interactions

Authors: Clara Alvarez-Rodriguez, Steve Azzolin, Vincenzo Marco De Luca, Filippo Maria Bianchi, Andrea Passerini, Antonio Longa

Organizations: The Arctic University of Norway

Abstract

Graph Neural Additive Networks (GNANs) have emerged as effective, interpretable-by-design architectures for graph learning. They achieve transparency by combining per-feature shape functions with simple distance-based reweighting. However, GNANs have three limitations. First, node-wise relevance scores aggregate information from all graph nodes, allowing irrelevant nodes to inherit spurious relevance from relevant ones. Second, topology is represented only through shortest-path distances, so structurally distinct graphs may share identical distance profiles. Third, feature and distance information are combined multiplicatively, entangling feature, distance, and interaction effects in a single rescalable term. We introduce AMIGA, which represents topology through a vocabulary of mined motifs and decomposes each node contribution into three additive channels: feature, structure, and feature-structure interaction. The channels are centered around a graph-balanced training baseline, making the decomposition well-defined and unique under the chosen reference distribution. Across ten graph classification and regression benchmarks, AMIGA outperforms GNAN on most datasets and narrows the gap with standard black-box GNNs. Because each channel is centered and assigned to a single node, AMIGA produces node-local explanations that reconstruct the model output exactly and support interpretation at multiple levels of granularity.

Figures & tables

Panel of Figure 1 — Figure 1: Two limitations of GNAN : (a) node relevance mixes neighboring feature contributions; (b) shortest-path profiles cannot distinguish $K_{3,3}$ from the triangular prism.

Figure 2: Overview of

\mathsf{AMIGA}

. Left: motifs are selected when they occur significantly more often than in randomly rewired training graphs with unchanged node degrees. Each node is then described by its participation in the selected motifs. Center: each node contribution is decomposed into feature, structural, and feature–motif interaction channels, and then pooled into the graph score. Right: this decomposition represents the model-level explanation of

\mathsf{AMIGA}

, and enables full model transparency by independently inspecting feature, structural, and interaction contributions, allowing the model’s computations to be exactly reconstructed.

Figure 3: $\mathsf{AMIGA}$ can recover known effects from only label supervision. (a) True and learned node-level channel contributions. (b) Graph-level channel totals for one test graph. (c) Node-level total contributions for the same graph. (d–g) Active and inactive main effects. (h–i) True and learned $I_{22}^{(0)}$ , shown with common axes and color scale.

Model	BA-Reg 1	BA-Reg 2
GIN + GNNExpl.	$0.94\pm 0.11$	$0.72\pm 0.19$
GIN + IG	$0.88\pm 0.11$	$0.22\pm 0.11$
GNAN	$0.71\pm 0.12$	$0.45\pm 0.16$
G-NAMRFF	$0.50\pm 0.11$	$0.54\pm 0.08$
$\mathsf{AMIGA}$ (ours)	$\mathbf{0.99\pm 0.02}$	$\mathbf{1.00\pm 0.00}$

Table 1: Explanation AUROC on BA-Reg 1 and BA-Reg 2, averaged over 5 seeds.

Graph classification: AUROC $\uparrow$
	Black-box models			Interpretable models
Dataset	GCN	GIN	GTrans	GNAN	G-NAMRFF	$\mathsf{AMIGA}$ (ours)
Mutagenicity	$0.851\pm 0.005$	$\mathbf{0.868\pm 0.018}$	$0.849\pm 0.010$	$0.751\pm 0.002$	$0.698\pm 0.068$	$\mathbf{0.828\pm 0.015}$
NCI1	$0.840\pm 0.017$	$\mathbf{0.868\pm 0.013}$	$0.807\pm 0.007$	$0.651\pm 0.045$	$0.734\pm 0.042$	$\mathbf{0.768\pm 0.020}$
PROTEINS	$0.791\pm 0.010$	$\mathbf{0.811\pm 0.012}$	$0.697\pm 0.020$	$\mathbf{0.821\pm 0.008}$	$0.786\pm 0.036$	$0.733\pm 0.046$
PTC-MR	$0.524\pm 0.037$	$0.518\pm 0.044$	$\mathbf{0.593\pm 0.056}$	$0.501\pm 0.029$	$0.555\pm 0.056$	$\mathbf{0.574\pm 0.026}$

Table 2: Predictive performance averaged over five seeds. Classification rows report test AUROC (

\uparrow

); regression rows report test MAE (

\downarrow

). Bold marks the best result within each model group.

Figure 4: Global response functions learned by $\mathsf{AMIGA}$ on Mutagenicity. Left: centered atom-type effects $f(\mathrm{atom})$ . Center: structural effects $s_{m}(k)$ against the number $k$ of motif occurrences containing a node. Right: interaction $I_{3}(\mathrm{atom},k)$ between atom identity and motif $m_{3}$ . Positive and negative values raise and lower the positive-class logit relative to the graph-balanced reference.

Figure 5: Example of the exact $\mathsf{AMIGA}$ decomposition for one molecular graph . Rows correspond to atoms; columns report the feature contribution, motif contributions, atom–motif interaction contributions, and node total $z_{v}$ . The bottom row aggregates each component across nodes. Adding the bias $\beta_{0}$ reconstructs $s(G)$ exactly. Outlined cells mark the contributions discussed in the text.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Task	Train/Val/Test	Avg. nodes	Avg. edges	Features
Mutagenicity	B	3035/651/651	30.32	30.77	$14\rightarrow 1$
NCI1	B	2877/616/617	29.87	32.30	$37\rightarrow 1$
PROTEINS	B	779/167/167	39.06	72.82	$4\rightarrow 2$
PTC-MR	B	240/52/52	14.29	14.69	$18\rightarrow 1$
BA-Reg 1	R	700/150/150	25.00	25.00	$10\rightarrow 10$
BA-Reg 2	R	700/150/150	120.00	128.90	$10\rightarrow 10$

Appendix

Table 3: Dataset statistics. B and R denote binary classification and regression. Edges count undirected pairs; feature dimensions are shown before and after

\mathsf{AMIGA}

encoding.

Dataset	Selected	Dataset	Selected
Mutagenicity	4	BA-Reg 1	7
NCI1	7	BA-Reg 2	5
PROTEINS	21	Crippen	4
PTC-MR	6	QM9 (all targets)	7

Appendix

Table 4: Selected motif-vocabulary sizes across the five seed-specific training splits. The three QM9 targets share the same seed-specific vocabulary because they use identical graph indices within each seed.

Dataset	Selected motifs	Enumeration time (s)
Mutagenicity	4	0.148
NCI1	7	0.128
PROTEINS	21	0.164
PTC-MR	6	0.005
BA-Reg 1	7	0.044
BA-Reg 2	5	0.580

Appendix

Table 5: Wall-clock time required to enumerate all 29 candidate graphlets, averaged over the five seed-specific training splits. The QM9 time is averaged over the three equivalent measurements.

Dataset	GCN	GIN	GTrans	GNAN	G-NAMRFF	$\mathsf{AMIGA}$
Mutagenicity	$108\pm 25$	$60\pm 22$	$51\pm 8$	$193\pm 61$	$838\pm 443$	$\mathbf{19\pm 9}$
NCI1	$99\pm 35$	$93\pm 23$	$81\pm 19$	$132\pm 46$	$958\pm 440$	$\mathbf{14\pm 3}$
PROTEINS	$21\pm 9$	$16\pm 7$	$19\pm 7$	$60\pm 9$	$125\pm 41$	$\mathbf{7\pm 1}$
PTC-MR	$2\pm 0$	$3\pm 1$	$4\pm 1$	$3\pm 1$	$8\pm 3$	$\mathbf{2\pm 1}$
BA-Reg 1	$43\pm 2$	$38\pm 8$	$28\pm 9$	$100\pm 29$	$119\pm 30$	$\mathbf{25\pm 8}$
BA-Reg 2	$97\pm 0$	$75\pm 23$	$\mathbf{37\pm 15}$	$187\pm 136$	$115\pm 41$	$56\pm 8$

Appendix

Table 6: Wall-clock training time in seconds, averaged over the five final runs and rounded to the nearest second. Hyperparameter search and motif discovery are excluded. Lower is better; the best result is bold.

Classification: test AUROC (%), $\uparrow$
Dataset	$F$	$F+S$	$F+I$	$F+S+I$
Mutagenicity	$75.48\pm 0.06$	$80.00\pm 0.37$	$80.09\pm 0.26$	$\mathbf{82.81\pm 1.50}$
NCI1	$64.24\pm 0.46$	$75.76\pm 0.29$	$\mathbf{78.17\pm 0.68}$	$76.81\pm 2.01$
PROTEINS	$64.90\pm 0.95$	$70.36\pm 0.84$	$65.44\pm 1.08$	$\mathbf{73.32\pm 4.63}$
PTC-MR	$49.69\pm 3.99$	$54.09\pm 3.74$	$49.69\pm 3.99$	$\mathbf{57.43\pm 2.62}$
Regression: test MAE, $\downarrow$

Appendix

Table 7: Channel ablation across five independently generated test splits. Entries report the mean

\pm

standard deviation over five seeds, each defining both the data split and model training. The

F+S+I

column reuses the full-model runs reported in the main results. Bold indicates the best mean in each row.

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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.

Abstract

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