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Graph Additive Models disentangling feature, topology, and interactions

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

arXiv:2610.20662v1stat.MLcs.LG

TL;DR

GNANs combine feature shape functions with distance reweighting, but their relevance scores can blur node contributions, their topology encoding can miss structural distinctions, and their multiplicative terms entangle effects. AMIGA uses mined motifs and graph-balanced centered feature, structural, and interaction channels. Across ten benchmarks, it outperforms GNAN on eight datasets and G-NAMRFF on nine, while black-box GNNs remain stronger on most real-world tasks.

  • Problem

    GNAN node relevance can mix contributions across nodes, shortest-path distances can miss structural distinctions, and multiplicative feature–distance terms entangle effects.

    Source: §1 · p. 2, §3.2 · p. 3

  • Method

    AMIGA mines motifs from training graphs and decomposes node contributions into graph-balanced centered feature, structural, and feature–structure interaction channels.

    Source: §1 · p. 2, §4 · p. 4, §4.1 · p. 4

  • Results

    Across ten benchmarks, AMIGA outperforms GNAN on eight datasets and G-NAMRFF on nine, while black-box GNNs remain stronger on most real-world tasks.

    Source: §1 · p. 2

  • Takeaways & Limitations

    AMIGA provides an exact decomposition of each prediction that can be inspected across feature, structural, and interaction contributions.

    Source: §7 · p. 9

    Factorized Motif Explanations describe model computation and should not be interpreted as causal effects of deleting a motif.

    Source: §4.4 · p. 6

Abstract

from arXiv · show

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.

1 INTRODUCTION

GNANs offer interpretable graph learning, but their node relevance, distance-only topology, and multiplicative feature–distance terms limit localization and disentanglement. AMIGA replaces distance summaries with mined motifs and uses graph-balanced centering to produce unique additive channels; across ten benchmarks, it outperforms GNAN on eight datasets and G-NAMRFF on nine.

1 INTRODUCTION

  • GNAN node-wise relevance blurs each node’s own contribution, shortest-path distances can miss informative topology, and multiplicative feature–distance terms entangle effects.These limitations can hinder downstream performance and interpretability.Source: §1 · p. 2
  • Graph-balanced centering makes the feature, structural, and interaction channels a unique canonical decomposition.The reference distribution assigns equal mass to each training graph.Source: §1 · p. 2
  • AMIGA replaces distance-based aggregation with automatically mined motifs, allowing nodes to contribute according to their features and motif participation.Its three additive channels are feature, structural, and feature–structure interaction.Source: §1 · p. 2
  • Across ten benchmarks, AMIGA outperforms GNAN on eight datasets and G-NAMRFF on nine, while black-box GNNs remain stronger on most real-world tasks.AMIGA narrows the performance gap in eight of ten datasets while retaining an exact decomposition of every prediction.Source: §1 · p. 2
  • On two synthetic datasets with known ground-truth explanations, AMIGA recovers task composition from label supervision and achieves explanation accuracy of 0.99 and AUCROC of 1.00.These controlled results accompany the benchmark comparisons.Source: §1 · p. 2
  • Black-box GNNs remain stronger on most real-world benchmarks, although AMIGA narrows the gap while preserving exact prediction decomposition.The decomposition separates feature, structural, and interaction contributions.Source: §1 · p. 2

2 RELATED WORKS

Related work covers interpretable-by-design graph models and motif-aware representations. AMIGA builds on graph additive models while keeping feature and interaction effects separate.

Interpretable Graph Models

  • Prototype-based and self-explainable GNNs make explanations part of model computation, but both families can be unfaithful.Logical-rule GNN layers are limited to classification and may produce long, deeply nested formulas on complex tasks.Source: §2 · p. 2

Motif- and Subgraph-Based Graph Representations

  • Motif-aware graph methods encode higher-order structure through local counts, subgraph processing, or motif vocabularies.Such methods address cases where message-passing GNNs bounded by the 1-WL test can miss simple substructures such as triangles.Source: §2 · p. 2
  • AMIGA uses a motif vocabulary while keeping feature and interaction effects separate for model transparency by design.This resembles prior motif-vocabulary approaches while maintaining the separation of effects.Source: §2 · p. 2

3 PRELIMINARIES

GNANs combine per-feature shape functions with shortest-path-based reweighting, then sum node representations for graph prediction. The section identifies blurred node relevance, limited topology encoding, and entangled feature and distance effects as limitations shared in part with G-NAMRFF.

3.1 GRAPH NEURAL ADDITIVE MODELS

  • GNANs learn one univariate shape function per feature and use a shared univariate distance function to weight information across the graph.Graph-level prediction sums node-wise representations before a task-specific activation.Source: §3.1 · p. 3
  • G-NAMRFF replaces GNAN’s all-pairs distance weighting with a polynomial graph filter, then applies univariate responses to filtered feature coordinates.Both models aggregate node-wise representations for graph-level prediction.Source: §3.1 · p. 3

3.2 LIMITATIONS OF CURRENT GRAPH NEURAL ADDITIVE MODELS

  • GNAN relevance scores aggregate feature contributions from every node, so a node with no own contribution can receive nonzero relevance from a nearby contributing node.G-NAMRFF has a similar issue because it mixes neighboring features within its receptive field before assigning relevance.Source: §3.2 · p. 3
  • Shortest-path profiles can assign identical scores to K3,3 and the triangular prism even though only the prism contains a triangle.With constant node features, the two 3-regular graphs are also mapped identically by G-NAMRFF’s Laplacian filtering.Source: §3.2 · p. 3
  • GNAN’s multiplicative feature and distance functions combine constant, feature-only, distance-only, and joint terms without separating them.Because the factors have no fixed scale, their explanations can change while the prediction remains the same.Source: §3.2 · p. 4

4 AMIGA: ATTRIBUTE-MOTIF INTERACTION GRAPH ADDITIVE MODEL

AMIGA replaces distance-based structural summaries with mined motifs and decomposes node contributions into feature, structural, and interaction channels. Graph-balanced centering makes these channels unique under the chosen reference, and Factorized Motif Explanations reconstruct the score exactly.

4.1 MOTIF DISCOVERY AND NODE PARTICIPATION

  • Each node’s normalized motif-membership counts complement its attributes, while log compression and standardization control count scale.Raw counts and occurrences are retained to localize explanations.Source: §4 · p. 4, §4.1 · pp. 4–5

4.2 FEATURE, STRUCTURAL, AND INTERACTION CHANNELS

  • AMIGA decomposes node contributions into feature, structural, and feature–motif interaction channels built from centered univariate shape functions.The factorized interaction uses d + M learned shapes while exposing dM feature–motif pairs for inspection.Source: §4 · p. 4, §4.2 · p. 5

4.1 MOTIF DISCOVERY AND NODE PARTICIPATION

  • AMIGA mines motifs from training graphs and freezes the resulting vocabulary for validation and test graphs.It retains motifs enriched against degree-preserving rewired replicas after Benjamini–Hochberg correction.Source: §4.1 · p. 4

4.3 GRAPH-BALANCED CANONICALIZATION

  • Graph-balanced centering averages within each training graph, then across graphs, so larger graphs do not receive greater reference weight.The product of graph-balanced feature and motif marginals is a centering reference; it does not assume observed features and motifs are independent.Source: §4.3 · p. 5
  • Centering yields unique feature, structural, and interaction channel functions on the empirical product support, but not unique coordinate-level shape functions.The fitted feature–motif terms describe that parameterization, while equivalent parameterizations may allocate interaction differently across pairs.Source: §4.3 · pp. 5–6

4.4 FACTORIZED MOTIF EXPLANATIONS

  • Factorized Motif Explanations reconstruct graph scores and channels exactly, then distribute each node’s motif-participation change equally across its motif occurrences.The allocations remain valid for overlapping occurrences and are equivariant to vertex relabeling; they are not causal effects of motif deletion.Source: §4.4 · p. 6

5 EXPERIMENTAL EVALUATION

AMIGA is evaluated for effect recovery, explanation plausibility, and predictive performance across graph classification and regression tasks. It recovers known effects and performs strongly against interpretable baselines, while black-box GNNs remain stronger on most real-world tasks.

5 EXPERIMENTAL EVALUATION

  • The evaluation spans four graph-classification and six graph-regression datasets, using AUROC for classification and MAE for regression across five seeds.The comparison includes standard black-box GNNs and interpretable GNAN and G-NAMRFF baselines.Source: §5 · pp. 6–7

5.1 RECOVERING KNOWN EFFECTS

  • AMIGA recovers the sinusoidal feature effect, saturating path effect, and sign-changing triangle interaction, with inactive effects near zero.Its test MAE is 1.02 ± 0.42; recovered feature, structural, and interaction contributions have test-node RMSEs of 0.088, 0.037, and 0.065, respectively.Source: §5.1 · p. 7, Figure 3 · p. 7

5.2 EXPLANATION PLAUSIBILITY

  • AMIGA achieves explanation AUROC of 0.99 ± 0.02 on BA-Reg 1 and 1.00 ± 0.00 on BA-Reg 2, outperforming both interpretable baselines and two post-hoc methods.This plausibility evaluation is restricted to the two datasets with annotated ground-truth explanations.Source: §5.2 · pp. 7–8, Table 1 · p. 7

5.3 PREDICTIVE PERFORMANCE

  • AMIGA improves on GNAN on eight of ten benchmarks and on G-NAMRFF on nine, while black-box GNNs remain stronger on most real-world tasks.It has the lowest overall MAE on BA-Reg 1 and BA-Reg 2 and narrows the performance gap in eight of ten datasets while exactly decomposing every prediction.Source: §5.3 · p. 8, Table 2 · p. 8

6 FROM GLOBAL RULES TO A LOCAL EXPLANATION

AMIGA’s global rules separate feature, structural, and interaction effects relative to a graph-balanced reference, then apply those rules to individual atoms. For the example molecule, the interaction channel dominates the exact graph-score decomposition.

Local Explanations

  • Zero is the common reference for centered channels, with positive and negative contributions raising and lowering the model score relative to it.The graph-balanced training marginals give every training graph equal weight, regardless of graph size.Source: §6 · p. 8

Global Decision Rules

  • For motif m3, the structural effect rises with participation, but its interaction becomes positive for fluorine and negative for iodine, bromine, and sulfur.Thus the motif’s effect on the model score depends on the participating atom.Source: §6 · p. 8

Local Explanations

  • Figure 4 plots centered atom-type effects, structural effects by motif participation, and the m3 interaction by atom identity and participation.Figure 5 organizes the molecular decomposition by atom rows and feature, motif, interaction, and node-total columns; its bottom row aggregates components across nodes.Source: Figure 4 · p. 9, Figure 5 · p. 9
  • The interaction channel contributes ΦI(G) = −2.25, and adding β0 = 0.67 to the node totals reconstructs the exact graph score s(G) = −2.16.Feature and structural channels contribute ΦF(G) = −0.70 and ΦS(G) = 0.12, respectively.Source: §6 · p. 9, Figure 5 · p. 9

7 CONCLUSION

AMIGA replaces distance-only structural summaries with mined motifs and gives predictions canonical feature, structural, and interaction decompositions. Across ten benchmarks it outperforms GNAN and G-NAMRFF on eight and nine datasets, respectively; its current motif vocabulary and learned allocations leave scope for extensions and causal testing.

7 CONCLUSION

  • Across ten benchmarks, AMIGA outperformed GNAN and G-NAMRFF on eight and nine datasets, respectively, and recovered known generative effects from graph-level supervision.The paper concludes that richer structural modeling can coexist with exact model transparency.Source: §7 · p. 9
  • The current exact enumeration procedure supports a small vocabulary of induced graphlets; scalable counting or sampling-based estimates could extend it.The paper also identifies a remaining accuracy gap with black-box GNNs on several real-world benchmarks.Source: §7 · pp. 9–10
  • Factorized Motif Explanations faithfully decompose the learned score and provide a starting point for motif-effect hypotheses, but do not estimate causal effects.Testing those hypotheses through controlled graph interventions with explicit causal assumptions is described as a promising direction.Source: §7 · p. 10
  • Codex assisted with interpreting model outputs, drafting and editing manuscript text, and drafting portions of proofs; authors developed the research and verified the AI-assisted work.The authors report that generative AI did not implement methods, run experiments, or produce reported experimental results.Source: AI USE STATEMENT · p. 11
  • The reproducibility statement directs readers to Appendix A for proofs, Appendix B for experimental protocols, and Appendix C for channel ablations.Sections 4.1–4.4 define the model, canonicalization procedure, and explanation identities.Source: REPRODUCIBILITY STATEMENT · p. 11

A.1 PROOF OF THEOREM 1

The proof establishes uniqueness of the centered decomposition, validates occurrence allocations under overlap and vertex permutations, and explains how sparsity regularization balances equivalent interaction factorizations.

A.1 PROOF OF THEOREM 1

  • The four decomposition components are pairwise orthogonal under the graph-balanced measure, with zero-mean identities separating each nonconstant component from the constant.The proof applies the inner product defined by expectation under the graph-balanced distribution.Source: §A.1 · p. 14
  • Any two decompositions satisfying the constraints coincide on empirical support, establishing uniqueness.The proof subtracts the decompositions, takes the graph-balanced expectation of the squared difference, and uses nonnegativity of every term.Source: §A.1 · p. 14

A.2 OCCURRENCE ALLOCATION AND PERMUTATION INVARIANCE

  • Occurrence allocations remain valid for overlapping motifs, while vertex permutations preserve graph-level contributions and transport node and occurrence explanations equivariantly.The allocation argument applies separately to each motif type.Source: §A.2 · p. 15

A.3 SPARSITY PENALTY AND RECIPROCAL-SCALE BALANCING

  • The sparsity penalty promotes sparse feature and structural main effects and selects a magnitude within the reciprocal-rescaling class of factorized interactions.Rescaling one interaction factor by γ and the other by 1/γ leaves the interaction unchanged.Source: §A.3 · p. 15
  • Among equivalent factorizations, the penalty equalizes the two penalized factors; the global sign remains unidentified but cancels in their product.If either factor's penalty is zero, the interaction vanishes almost everywhere under the reference distribution, so its scale is immaterial.Source: §A.3 · p. 15

B.1 DATASETS AND PREPROCESSING

The experiments use classification and regression benchmarks with training-split-specific preprocessing, then select enriched graphlets as a frozen motif vocabulary. The appendix also reports the enumeration runtime.

B.1 DATASETS AND PREPROCESSING

  • The benchmarks include four binary-classification datasets and six regression targets, including planted-motif targets, Wildman–Crippen log P, and three QM9 properties.BA-Reg 1 targets the sum of node features inside planted house motifs; BA-Reg 2 targets the number of planted houses.Source: §B.1 · p. 16
  • Each seed uses a 70/15/15 train/validation/test split, with feature statistics, motif discovery, and normalization fitted on training data and frozen for evaluation.Numerical features are standardized using training-node statistics, while regression targets are standardized on training graphs and returned to original units for MAE.Source: §B.1 · p. 16

B.2 MOTIF DISCOVERY PROTOCOL

  • Motif discovery enumerates 29 connected unlabeled graphlets of sizes three to five and retains candidates enriched over degree-preserving rewired training graphs after Benjamini–Hochberg correction.The selected vocabulary and its normalization statistics are frozen for validation and test graphs; synthetic recovery instead uses a fixed path-and-triangle vocabulary.Source: §B.2 · pp. 16–17
  • Enumerating all 29 candidate graphlets takes less than 0.6 seconds on every dataset except QM9, which processes 91,581 training graphs in 2.52 seconds.The reported times cover exact enumeration over each complete training split and are averaged across seed-specific splits.Source: §B.2 · p. 17, Table 5 · p. 17

B.3 TRAINING AND MODEL SELECTION

The appendix reports training efficiency and details the training, explanation-evaluation, and synthetic-recovery protocols. AMIGA has the lowest mean training runtime on most datasets, while the synthetic setup evaluates channel recovery and graph-level accuracy separately.

B.3 TRAINING AND MODEL SELECTION

  • Models use AdamW for at most 500 epochs with early stopping, and the selected configuration is chosen from 30 sampled configurations per model family and dataset.Configurations are ranked by mean best-validation score, then retrained from scratch for each of five evaluation seeds.Source: §B.3 · pp. 17–18
  • AMIGA has the lowest mean runtime on nine of ten datasets; Graph Transformer is faster only on BA-Reg 2.On QM9, AMIGA trains roughly one order of magnitude faster than the baselines; measurements exclude hyperparameter search and motif discovery.Source: §B.3 · p. 18, Table 6 · p. 18

B.4 EXPLANATION EVALUATION PROTOCOL

  • Node-level plausibility is evaluated against planted-motif nodes on BA-Reg 1 and BA-Reg 2, with AUROC averaged equally across test graphs and then across five seeds.AMIGA nodes are ranked by native contribution magnitude without allocating the graph intercept; the compared baselines use their respective node-level relevance scores.Source: §B.4 · p. 18

B.5 SYNTHETIC RECOVERY DETAILS

  • Synthetic recovery uses five independent 1,400/300/300 splits, a fixed 1–32–1 AMIGA configuration, and independently generated target noise.The noise standard deviation is 10% of the clean training-target population standard deviation.Source: §B.5 · p. 19
  • Synthetic evaluation re-anchors components at raw feature and motif count zero, preserves predictions exactly, and measures channel recovery by RMSE and noisy-target graph accuracy by MAE.Noise is applied only to the graph target; each graph has 12 nodes for the stated intercept transformation.Source: §B.5 · p. 19

C ADDITIONAL EXPERIMENTS

The ablation compares feature, structural, and feature–motif interaction channels. Its results show strong but task-dependent channel effects, with the full model best on most datasets.

C ADDITIONAL EXPERIMENTS

  • The full model has the best mean performance on nine of ten datasets; on real-world benchmarks, combining channels generally performs best, with NCI1 as the exception.On NCI1, F + I marginally outperforms the full model.Source: C ADDITIONAL EXPERIMENTS · p. 19, Table 7 · p. 20
  • On BA-Reg 1, adding interaction reduces MAE from 449.50 to 10.44, and combining all channels further reduces it to 4.13.BA-Reg 1 depends on node features inside planted motifs.Source: C ADDITIONAL EXPERIMENTS · p. 19
  • On BA-Reg 2, adding structure reduces MAE from 5.1492 to 0.0097, while F + I does not improve over the feature-only model.BA-Reg 2 depends directly on the number of planted motifs.Source: C ADDITIONAL EXPERIMENTS · p. 19

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