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REVIEW 4 major objections 5 minor 59 references

Explaining a GNN faithfully requires capturing how edges work together, and SeeExplainer does this by first grouping edges into granular balls.

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2026-08-01 07:36 UTC pith:XBFTUSFQ

load-bearing objection New granular-ball grouping mechanism for GNN explanations, but the evaluation's stability result is a definitional artifact and fidelity comparisons don't control for explanation size, so the headline claims don't hold. the 4 major comments →

arxiv 2607.21381 v1 pith:XBFTUSFQ submitted 2026-07-23 cs.AI

Towards Faithful Graph Explanations with Synergistic Edge Effects via Granular Balls

Jiancu Chen , Shuyin Xia , Guan Wang , Degang Chen , Fan Chen This is my paper
classification cs.AI
keywords graph neural networksinstance-level explainabilitysynergistic edge effectsgranular-ball computingstructural graphfidelitystabilityparameter-free explainer
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that existing instance-level explainers for graph neural networks are limited because they score each edge in isolation, missing the synergistic effects that give functional groups and motifs their meaning. To capture these effects, the authors propose SeeExplainer, a parameter-free method that first decomposes the input graph into disjoint, adaptively sized granular balls and treats those balls as nodes of a coarsened structural graph. It then scores each ball and each inter-ball edge by the drop in the model's prediction when that component is removed, and keeps the components whose contribution exceeds the graph's average contribution. The paper reports that on ten graph classification datasets with GIN and GCN backbones, SeeExplainer attains higher fidelity and far lower stability variance than six baselines, including Eig-Search, PGExplainer, and GNNExplainer. A sympathetic reader would care because, if the results hold, explanations can be both simpler (parameter-free) and more aligned with how humans read meaningful substructures.

Core claim

On its own terms, the central discovery is that a graph can be re-organized into a structural graph whose nodes are granular balls — disjoint subgraphs of variable size produced by a density-based splitting rule — and that scoring these balls as units, rather than scoring original edges one by one, yields explanatory subgraphs that better reproduce the GNN's predictions. The splitting rule starts with the whole graph, splits into √n balls around highest-degree centers, then recursively splits any ball whose two children have greater combined edge density than the parent. The resulting balls preserve the original edges inside them, so each ball acts as a composite feature. Contributions are m

What carries the argument

The granular-ball structural graph — a coarsened representation in which disjoint, adaptively sized subgraphs (granular balls) become nodes and edges between them become structural edges — is the central object. It carries the argument because it moves the unit of attribution from a single edge to a group of edges that share vertices, so that a functional group's contribution is evaluated as a whole rather than as a sum of parts. The density-based splitting condition (split when the two children's edge densities sum to more than the parent's density) determines the balls; the average-contribution threshold I(G) selects the final explanatory subgraph parameter-free.

Load-bearing premise

The fidelity and stability evaluations assume the method's output can be compared at multiple sparsity levels even though SeeExplainer returns one fixed subgraph with no sparsity parameter; the paper never specifies how sparsity enters the fidelity equations for its method.

What would settle it

Run SeeExplainer and a strong edge-only baseline (e.g., Eig-Search) while forcing both to output exactly the same number of kept edges (or the same induced-subgraph size) on the same GNN and datasets. If SeeExplainer's fidelity advantage vanishes when explanation sizes are equal, the claimed synergy benefit is not what drives the reported gains.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • If SeeExplainer's claims hold, explanation methods no longer need a tunable explanation-size parameter; a threshold derived from the graph itself decides what to include.
  • Chemical and biological substructures that a GNN actually relies on could be extracted as granular balls, giving domain experts directly inspectable motifs rather than scattered edges.
  • Since the structural graph preserves original edges inside balls, the same decomposition can be reused across multiple explanation queries or for counterfactual analysis.
  • The reported stability improvement suggests group-level attribution is less sensitive to arbitrary sparsity cutoffs than edge-level attribution.
  • The method's complexity O((m+n)log n) makes synergy-aware explanation feasible for large graphs such as DD.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the reported fidelity gains may be confounded by explanation size — granular-ball subgraphs tend to be larger than edge-only masks, and larger subgraphs mechanically raise fidelity under the Fidelity+ metric; a size-controlled comparison is needed to isolate the synergy effect.
  • Editorial inference: the structural graph is built without regard to node features, yet GNNs with GIN/GCN read features; testing feature-aware splitting (e.g., by feature similarity) would reveal whether unsupervised geometry alone, or feature alignment, drives the improvement.
  • Editorial inference: because SeeExplainer returns a single fixed subgraph, the near-zero stability values in Table IV likely reflect constancy across sparsity levels rather than robustness of ranking; a meaningful stability test would re-run the explanation under input or model perturbations.
  • Editorial inference: the √n initial centers are an empirical constant; varying it between, say, 2 and n/2 and tracking fidelity would make the method's 'parameter-free' claim more credible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes SeeExplainer, a GNN instance-level explainer. It partitions an input graph into disjoint granular balls via degree-based BFS splitting and a quality criterion, abstracts these balls as nodes of a structural graph, computes node and edge contributions by measuring prediction differences under perturbation, and selects an explanatory subgraph by thresholding these contributions against the graph-level average. The central claims are that the granular-ball structural graph captures synergistic effects among edges and that SeeExplainer outperforms PGExplainer, DeepLIFT, GNNExplainer, GraphLime, GradCAM, and Eig-Search in fidelity and stability on ten graph-classification datasets.

Significance. The motivating observation—that edge importance can be non-additive when edges form functional substructures—is legitimate and relevant for GNN explainability. If the proposed decomposition were shown to yield faithful and compact explanations, the paper would make a useful contribution. The authors also provide a large experimental comparison, ablations, case studies, and a complexity analysis. However, the evaluation protocol is not valid for SeeExplainer as presented: the method has no sparsity parameter, so the stability metric is definitionally near zero, and the fidelity comparisons do not control for the size of the returned explanations. These issues affect every reported empirical claim, so the current evidence does not support the paper's main conclusions.

major comments (4)
  1. [§V-C, Eqs. (12)–(15), Algorithm 2] Algorithm 2 returns a single subgraph: a node or edge is included if its contribution exceeds I(G). There is no sparsity level s_k in the algorithm. Yet Eqs. (12)–(14) define fidelity at a sparsity level and Eq. (15) defines stability as the variation of fidelity across u sparsity levels. The paper never states how SeeExplainer is evaluated at s_k=0.5,…,0.9. If the same fixed explanation is used at every s_k, fidelity is constant and S(s_k) is zero up to floating-point round-off. This exactly matches the ~1e-10 values in Table IV. Thus the RQ2 stability result is a definitional artifact, not an empirical property.
  2. [Table I, §V-B] The fidelity comparison is not size-controlled. Baselines are evaluated with sparsity constraints that fix the number of retained edges, while SeeExplainer's threshold can return an arbitrary—and potentially much larger—subgraph. Because Fidelity+ in Eq. (12) measures the prediction change when a subgraph is removed, a larger retained subgraph tends to inflate Fidelity+, and the combined fidelity in Eq. (14) can be inflated as well. Without reporting the size distribution of SeeExplainer's explanations or matching explanation sizes across methods, the large gains in Table I cannot be attributed to better explanations.
  3. [§IV-B1, Eqs. (10)–(11)] The operations G\v'_i and G\e'_j are undefined. Here v'_i is a granular ball—a subset of original nodes—and e'_j is an inter-ball edge in the structural graph, not necessarily an edge in G. The paper states that the original structure is preserved, but it never defines how removing a structural node or edge acts on the original graph. This makes the contribution scores in Eqs. (10)–(11) and the resulting explanatory subgraph ill-defined.
  4. [§IV-B and §V-B] The selection rule and the evaluation metric share the same prediction-difference objective. Contributions I(v'_i) and I(e'_j) are measured by ϕ(G)−ϕ(G\·), and Fidelity+ measures ϕ(G)−ϕ(G\delete). SeeExplainer therefore selects components that score well on the very criterion used to evaluate it. The reported high fidelity is partly self-fulfilling. An independent metric—or at least a clear separation between selection and evaluation—is needed to support the claim of superior faithfulness.
minor comments (5)
  1. [Tables I, IV] Typos: 'GNNExpaliner' appears in Tables I and IV; 'F idelity' is broken in Eq. (12) and the Metrics paragraph; Table IV (GCN/BZR/DeepLIFT) reports '1.1E-0' with a missing digit.
  2. [Algorithm 1, Eq. (6)] A granular ball with one node cannot be split into two centers as required by Eq. (6); a termination condition for singleton balls is missing.
  3. [Eq. (4), §IV-A] Center selection should be stated as 'top-s nodes with largest degree'; the argmax notation is ambiguous. Also, s=√n is an empirical setting, so the paper should clarify whether 'parameter-free' means 'no learned parameters' or 'no user-specified hyperparameters.'
  4. [§IV-A] The term 'non-isomorphic factorization' is used but never defined; the BFS splitting procedure does not obviously implement it. This should be clarified or removed.
  5. [Tables I, IV, V, VI] Tables I, IV, V, and VI report only point estimates. Since models are trained 20 times, standard deviations or statistical significance tests should be reported for the fidelity and stability comparisons.

Circularity Check

1 steps flagged

SeeExplainer's near-zero stability is a definitional artifact: the method returns one fixed explanation with no sparsity parameter, so Eq. (15) is ~0 by construction.

specific steps
  1. self definitional [Section V-A (Metrics, Eqs. 12–15), Section V-C (Stability), Algorithm 2, Table IV]
    "Stability is defined as the difference between the average fidelity across all sparsity levels and the fidelity at a specific sparsity level. A smaller difference indicates stronger stability. Formally, S(s_k) = |1/u sum_{k=1}^u Fidelity(G,s_k) - Fidelity(G,s_k)|. ... The reason is that SeeExplainer generates explanatory subgraphs directly from the structural graph and is therefore unaffected by parameter settings."

    Algorithm 2 takes no sparsity input: it thresholds I(G), I(v'_i), and I(e'_j) and returns a single fixed explanatory subgraph. Yet Eqs. (12)–(15) define fidelity and stability as functions of a sparsity level s_k, and Table I reports averages over s_k = 0.5,...,0.9. If the same fixed explanation is used at every sparsity level, Fidelity(G,s_k) is identical for all k, so Eq. (15) is zero up to floating-point precision. The ~1e-10 values in Table IV are therefore a direct consequence of the metric/output mismatch, not an empirical finding that SeeExplainer is stable.

full rationale

The load-bearing circularity is confined to the stability evaluation. The paper's own equations define stability as variation of fidelity across sparsity levels, but SeeExplainer is explicitly parameter-free and Algorithm 2 returns one subgraph with no sparsity mechanism; hence the reported near-zero stability is definitional. The fidelity comparison also lacks a stated mapping from s_k to SeeExplainer's output, so explanation size is not controlled, which is a serious confound, but the paper does not provide enough detail to prove that the fidelity advantage reduces purely by construction. The granular-ball/synergy construction is asserted rather than derived, but that is a validation gap, not a circular derivation. Self-citations to prior granular-ball work are not load-bearing for the experimental claims. Score 6 reflects one central evaluation result that is forced by construction, while the method itself is not wholly identical to its evaluation metric.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The method depends on several hand-set choices (sqrt(n) balls, two centers, split criterion, average-contribution threshold) that are not derived from theory. The central assumption that topologically defined granular balls capture task-relevant synergistic edge effects is an ad hoc premise with no supporting evidence. No new physical or conceptual entities are introduced beyond the structural graph, which is a derived representation.

free parameters (4)
  • Initial granular-ball count s = sqrt(n) = sqrt(n) (graph-dependent)
    Eq. (4) chooses the number of coarse-split centers as sqrt(n), stated to be empirical based on previous works [48],[50],[51]. It directly controls the structural graph and thus the explanation.
  • Fine-split center count (2) = 2
    Eq. (6) always picks the two highest-degree nodes in a ball as centers for BFS splitting; this is a hand-set choice affecting granularity.
  • Split criterion Q(child1)+Q(child2) > Q(parent) = Edge-density ratio comparison
    Eq. (8) uses the heuristic that a split is accepted if the sum of child edge densities exceeds the parent density. No justification beyond balance.
  • Selection threshold I(G) = Average of per-edge prediction drops
    Algorithm 2 uses I(G) from Eq. (9) as the cutoff for keeping structure nodes/edges. It is a design choice; different thresholds would change the explanation size and fidelity.
axioms (5)
  • domain assumption Global precedence cognitive rule implies coarse-to-fine granular decomposition is a useful representation.
    Invoked in Sections II-C and IV-A to motivate covering the graph with a large granular ball first; not a mathematical theorem.
  • ad hoc to paper Degree-based BFS splitting into disjoint balls yields substructures that are meaningful for the model's decision and capture synergistic edge effects.
    Core structural assumption of Section IV-A2-3; no evidence links topological degree centers to model-relevant functional groups.
  • ad hoc to paper Q(GB)=|E_GB|/|V_GB| is a valid quality measure for deciding splits.
    Eq. (7)-(8): The split condition is based on this edge-density ratio; there is no theoretical grounding.
  • ad hoc to paper Removing a structural node/edge (G\v'_i, G\e'_j) is a well-defined operation on the original graph.
    Section IV-B1 uses these expressions without defining how a structural-graph element maps to a deletion in G.
  • domain assumption The pretrained GNN's prediction differences phi(G)-phi(G\S) are meaningful additive importance scores.
    The contribution measures in Eqs. (9)-(11) and fidelity in Eqs. (12)-(14) assume prediction changes quantify importance.

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Cite this review

Pith. "Pith review of Towards Faithful Graph Explanations with Synergistic Edge Effects via Granular Balls." pith.science (2026). https://pith.science/paper/XBFTUSFQ

@misc{pith2026260721381,
  author       = {Pith},
  title        = {Pith review of: Towards Faithful Graph Explanations with Synergistic Edge Effects via Granular Balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBFTUSFQ}},
  note         = {Machine review of arXiv:2607.21381}
}
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read the original abstract

Instance-level explanations aim to reveal the rationale behind a model's decisions for a specific graph. Previous methods explain graph neural networks (GNNs) by selecting important edges to induce subgraphs, where edge importance is assessed by perturbing each edge and observing changes in the model predictions. However, they often neglect the synergistic effects among edges, which are crucial for accurately characterizing edge importance. To address this issue, we propose SeeExplainer, a parameter-free explainer to interpret GNNs. Specifically, we first introduce a granular-ball graph refinement mechanism that decomposes a graph into several disjoint granular-balls with no fixed size, and utilize them as nodes to construct a structural graph. This process can better capture the synergistic effects among edges. Then, we perturb nodes and edges in the structural graph to generate explanatory subgraphs based on their respective contributions. Experiments on several graph classification datasets of different networks show that SeeExplainer outperforms state-of-the-art baselines.

Figures

Figures reproduced from arXiv: 2607.21381 by Degang Chen, Fan Chen, Guan Wang, Jiancu Chen, Shuyin Xia.

Figure 1
Figure 1. Figure 1: The synergistic effects among edges. A substructure [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of SeeExplainer. (a) Synergistic Edge Effects Modeling Module: it constructs a structural graph by leveraging information from the original [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heatmaps of fidelity− and fidelity+ for GIN and GCN. Since fidelity is jointly determined by fidelity+ and fidelity−, we present heatmaps of the fidelity+ and fidelity− values across all datasets under different graph neural net￾works to better illustrate the effectiveness of SeeExplainer. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of time consumption between two and radical n splitting [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of cases. The first and third rows indicate Eig-Search, and the second and fourth rows represent SeeExplainer. The bold black areas are [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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Reference graph

Works this paper leans on

59 extracted references · 10 linked inside Pith

  1. [1]

    Active and semi-supervised graph neural networks for graph classification,

    Y . Xie, S. Lv, Y . Qian, C. Wen, and J. Liang, “Active and semi-supervised graph neural networks for graph classification,”IEEE Transactions on Big Data, vol. 8, no. 4, pp. 920–932, 2022

    2022

  2. [2]

    Semisuper- vised graph neural networks for graph classification,

    Y . Xie, Y . Liang, M. Gong, A. K. Qin, Y .-S. Ong, and T. He, “Semisuper- vised graph neural networks for graph classification,”IEEE Transactions on Cybernetics, vol. 53, no. 10, pp. 6222–6235, 2022

    2022

  3. [3]

    Edge classification on graphs: New directions in topological imbalance,

    X. Cheng, Y . Wang, Y . Liu, Y . Zhao, C. C. Aggarwal, and T. Derr, “Edge classification on graphs: New directions in topological imbalance,” in Proceedings of the Eighteenth ACM International Conference on Web Search and Data Mining, 2025, pp. 392–400

    2025

  4. [4]

    A simple yet effective baseline for non-attributed graph classification,

    C. Cai and Y . Wang, “A simple yet effective baseline for non-attributed graph classification,” arXiv preprint arXiv:1811.03508, 2018

    Pith/arXiv arXiv 2018

  5. [5]

    Tudataset: A collection of benchmark datasets for learning with graphs,

    C. Morris, N. M. Kriege, F. Bause, K. Kersting, P. Mutzel, and M. Neumann, “Tudataset: A collection of benchmark datasets for learning with graphs,” arXiv preprint arXiv:2007.08663, 2020

    Pith/arXiv arXiv 2007

  6. [6]

    Protein function prediction via graph kernels,

    K. M. Borgwardt, C. S. Ong, S. Schonauer, S. Vishwanathan, A. J. Smola, and H.-P. Kriegel, “Protein function prediction via graph kernels,” Bioinformatics, vol. 21, no. suppl 1, pp. i47–i56, 2005

    2005

  7. [7]

    Spline-fitting with a genetic algorithm: A method for developing classification structure activity relationships,

    J. J. Sutherland, L. A. O’brien, and D. F. Weaver, “Spline-fitting with a genetic algorithm: A method for developing classification structure activity relationships,”Journal of Chemical Information and Computer Sciences, vol. 43, no. 6, pp. 1906–1915, 2003

    1906

  8. [8]

    Towards a rigorous science of interpretable machine learning,

    F. Doshi-Velez and B. Kim, “Towards a rigorous science of interpretable machine learning,” arXiv preprint arXiv:1702.08608, 2017. 10

    Pith/arXiv arXiv 2017

  9. [9]

    Explainability in graph neural networks: A taxonomic survey,

    H. Yuan, H. Yu, S. Gui, and S. Ji, “Explainability in graph neural networks: A taxonomic survey,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 5, pp. 5782–5799, 2022

    2022

  10. [10]

    Gnnex- plainer: Generating explanations for graph neural networks,

    Z. Ying, D. Bourgeois, J. You, M. Zitnik, and J. Leskovec, “Gnnex- plainer: Generating explanations for graph neural networks,”Advances in Neural Information Processing Systems, vol. 32, 2019

    2019

  11. [11]

    Xgnn: Towards model-level explanations of graph neural networks,

    H. Yuan, J. Tang, X. Hu, and S. Ji, “Xgnn: Towards model-level explanations of graph neural networks,” inProceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2020, pp. 430–438

    2020

  12. [12]

    Gnninterpreter: A probabilistic genera- tive model-level explanation for graph neural networks,

    X. Wang and H.-W. Shen, “Gnninterpreter: A probabilistic genera- tive model-level explanation for graph neural networks,” arXiv preprint arXiv:2209.07924, 2022

    Pith/arXiv arXiv 2022

  13. [13]

    D4explainer: In-distribution explanations of graph neural network via discrete denoising diffusion,

    J. Chen, S. Wu, A. Gupta, and R. Ying, “D4explainer: In-distribution explanations of graph neural network via discrete denoising diffusion,” Advances in Neural Information Processing Systems, vol. 36, pp. 78 964–78 986, 2023

    2023

  14. [14]

    Gnnboundary: Towards explaining graph neural networks through the lens of decision boundaries,

    X. Wang and H. W. Shen, “Gnnboundary: Towards explaining graph neural networks through the lens of decision boundaries,” in TheTwelfth International Conference on Learning Representations, 2024

    2024

  15. [15]

    Graphon-explainer: Generating model- level explanations for graph neural networks using graphons,

    S. Saha and S. Bandyopadhyay, “Graphon-explainer: Generating model- level explanations for graph neural networks using graphons,”Transac- tions on Machine Learning Research, 2024

    2024

  16. [16]

    Graphlime: Local interpretable model explanations for graph neural networks,

    Q. Huang, M. Yamada, Y . Tian, D. Singh, and Y . Chang, “Graphlime: Local interpretable model explanations for graph neural networks,”IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 7, pp. 6968–6972, 2022

    2022

  17. [17]

    Eig-search: Generating edge-induced subgraphs for gnn explanation in linear time,

    S. Lu, B. Liu, K. G. Mills, J. He, and D. Niu, “Eig-search: Generating edge-induced subgraphs for gnn explanation in linear time,” inForty-first International Conference on Machine Learning, 2024

    2024

  18. [18]

    Interpreting graph neural networks for nlp with differentiable edge masking,

    M. S. Schlichtkrull, N. De Cao, and I. Titov, “Interpreting graph neural networks for nlp with differentiable edge masking,” arXiv preprint arXiv:2010.00577, 2020

    Pith/arXiv arXiv 2010

  19. [19]

    On explainability of graph neural networks via subgraph explorations,

    H. Yuan, H. Yu, J. Wang, K. Li, and S. Ji, “On explainability of graph neural networks via subgraph explorations,” inInternational Conference on Machine Learning. PMLR, 2021, pp. 12 241–12 252

    2021

  20. [20]

    Parameterized explainer for graph neural network,

    D. Luo, W. Cheng, D. Xu, W. Yu, B. Zong, H. Chen, and X. Zhang, “Parameterized explainer for graph neural network,”Advances in Neural Information Processing Systems, vol. 33, pp. 19 620–19 631, 2020

    2020

  21. [21]

    Zhang and G

    P. Zhang and G. Chartrand, Introduction to graph theory. Tata McGraw- Hill New York, 2006, vol. 2, no. 2.1

    2006

  22. [22]

    Network flows: Theory, algorithms, and applications,

    G. R. Waissi, “Network flows: Theory, algorithms, and applications,” 1994

    1994

  23. [23]

    A fast adaptive k-means with no bounds,

    S. Xia, D. Peng, D. Meng, C. Zhang, G. Wang, E. Giem, W. Wei, and Z. Chen, “A fast adaptive k-means with no bounds,”IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020

    2020

  24. [24]

    An efficient and adaptive granular-ball generation method in classification problem,

    S. Xia, X. Dai, G. Wang, X. Gao, and E. Giem, “An efficient and adaptive granular-ball generation method in classification problem,”IEEE Transactions on Neural Networks and Learning Systems, vol. 35, no. 4, pp. 5319–5331, 2024

    2024

  25. [25]

    Online graph dictionary learning

    C. Vincent-Cuaz, T. Vayer, R. Flamary, M. Corneli, and N. Courty, “Online graph dictionary learning”,International Conference on Machine Learning. PMLR, 2021, pp. 10 564–10 574

    2021

  26. [26]

    Semi-supervised classification with graph convolutional networks,

    T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” CoRR, vol. abs/1609.02907, 2016. [Online]. Available: http://arxiv.org/abs/1609.02907

    Pith/arXiv arXiv 2016

  27. [27]

    How powerful are graph neural networks?

    K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?” 2019. [Online]. Available: https://arxiv.org/abs/1810.00826

    Pith/arXiv arXiv 2019

  28. [28]

    Explainability methods for graph convolutional neural networks,

    P. E. Pope, S. Kolouri, M. Rostami, C. E. Martin, and H. Hoffmann, “Explainability methods for graph convolutional neural networks,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019

    2019

  29. [29]

    Explainability techniques for graph convolutional networks,

    F. Baldassarre and H. Azizpour, “Explainability techniques for graph convolutional networks,” arXiv preprint arXiv:1905.13686, 2019

    Pith/arXiv arXiv 1905

  30. [30]

    Degree: Decomposition based explanation for graph neural networks,

    Q. Feng, N. Liu, F. Yang, R. Tang, M. Du, and X. Hu, “Degree: Decomposition based explanation for graph neural networks,” arXiv preprint arXiv:2305.12895, 2023

    Pith/arXiv arXiv 2023

  31. [31]

    Graphsvx: Shapley value explanations for graph neural networks,

    A. Duval and F. D. Malliaros, “Graphsvx: Shapley value explanations for graph neural networks,” inJoint European Conference on Machine Learn- ing and Knowledge Discovery in Databases. Springer, 2021, pp.302–318

    2021

  32. [32]

    Dis- till n’explain: explaining graph neural networks using simple surrogates,

    T. Pereira, E. Nascimento, L. E. Resck, D. Mesquita, and A. Souza, “Dis- till n’explain: explaining graph neural networks using simple surrogates,” inInternational Conference on Artificial Intelligence and Statistics. PMLR, 2023, pp. 6199–6214

    2023

  33. [33]

    Relex: A model-agnostic relational model explainer,

    Y . Zhang, D. Defazio, and A. Ramesh, “Relex: A model-agnostic relational model explainer,” inProceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, 2021, pp. 1042–1049

    2021

  34. [34]

    Towards multi- grained explainability for graph neural networks,

    X. Wang, Y . Wu, A. Zhang, X. He, and T.-S. Chua, “Towards multi- grained explainability for graph neural networks,”Advances in Neural Information Processing Systems, vol. 34, pp. 18 446–18 458, 2021

    2021

  35. [35]

    Zorro: Valid, sparse, and stable explanations in graph neural networks,

    T. Funke, M. Khosla, M. Rathee, and A. Anand, “Zorro: Valid, sparse, and stable explanations in graph neural networks,”IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 8, pp. 8687–8698, 2022

    2022

  36. [36]

    Gstarx: Explaining graph neural networks with structure-aware cooperative games,

    S. Zhang, Y . Liu, N. Shah, and Y . Sun, “Gstarx: Explaining graph neural networks with structure-aware cooperative games,”Advances in Neural Information Processing Systems, vol. 35, pp. 19 810–19 823, 2022

    2022

  37. [37]

    Topological structure in visual perception,

    L. Chen, “Topological structure in visual perception,”Science, vol. 218, no. 4573, pp. 699–700, 1982

    1982

  38. [38]

    Dgcc: data-driven granular cognitive computing,

    G. Wang, “Dgcc: data-driven granular cognitive computing,”Granular Computing, vol. 2, no. 4, pp. 343–355, 2017

    2017

  39. [39]

    Gb-rvfl: Fusion of randomized neural network and granular ball computing,

    M. Sajid, A. Quadir, M. Tanveer, A. D. N. Initiative et al., “Gb-rvfl: Fusion of randomized neural network and granular ball computing,” Pattern Recognition, vol. 159, p. 111142, 2025

    2025

  40. [40]

    Ball kk-means: Fast adaptive clustering with no bounds,

    S. Xia, D. Peng, D. Meng, C. Zhang, G. Wang, E. Giem, W. Wei, and Z. Chen, “Ball kk-means: Fast adaptive clustering with no bounds,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 1, pp. 87–99, 2022

    2022

  41. [41]

    Gbct: efficient and adaptive clustering via granular-ball computing for complex data,

    S. Xia, B. Shi, Y . Wang, J. Xie, G. Wang, and X. Gao, “Gbct: efficient and adaptive clustering via granular-ball computing for complex data,” IEEE Transactions on Neural Networks and Learning Systems, 2025

    2025

  42. [42]

    Generation of granular-balls for clus- tering based on the principle of justifiable granularity,

    Z. Jia, Z. Zhang, and W. Pedrycz, “Generation of granular-balls for clus- tering based on the principle of justifiable granularity,”IEEE Transactions on Cybernetics, 2025

    2025

  43. [43]

    Granular-ball regeneration clustering with principle of justifiable gran- ularity,

    W. Li, L. Wei, W. Pedrycz, W. Ding, C. Zhang, T. Zhan, and S. Xia, “Granular-ball regeneration clustering with principle of justifiable gran- ularity,”IEEE Transactions on Neural Networks and Learning Systems, 2025

    2025

  44. [44]

    Aw-gbgae: an adaptive weighted graph autoencoder based on granular-balls for general data clustering,

    J. Xie, Y . Cheng, S. Xia, C. Hua, G. Wang, and X. Gao, “Aw-gbgae: an adaptive weighted graph autoencoder based on granular-balls for general data clustering,”IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025

    2025

  45. [45]

    Gbrs: A unified granular-ball learning model of pawlak rough set and neighborhood rough set,

    S. Xia, C. Wang, G. Wang, X. Gao, W. Ding, J. Yu, Y . Zhai, and Z. Chen, “Gbrs: A unified granular-ball learning model of pawlak rough set and neighborhood rough set,”IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 1, pp. 1719–1733, 2025

    2025

  46. [46]

    Constructing three-way decision with fuzzy granular-ball rough sets based on uncertainty invariance,

    J. Yang, Z. Liu, G. Wang, Q. Zhang, S. Xia, D. Wu, and Y . Liu, “Constructing three-way decision with fuzzy granular-ball rough sets based on uncertainty invariance,”IEEE Transactions on Fuzzy Systems, 2025

    2025

  47. [47]

    Fuzzy granule density-based outlier detection with multi-scale granular balls,

    C. Gao, X. Tan, J. Zhou, W. Ding, and W. Pedrycz, “Fuzzy granule density-based outlier detection with multi-scale granular balls,”IEEE Transactions on Knowledge and Data Engineering, 2025

    2025

  48. [48]

    Graph coarsening via supervised granular-ball for scalable graph neural network training,

    S. Xia, X. Ma, Z. Liu, C. Liu, S. Zhao, and G. Wang, “Graph coarsening via supervised granular-ball for scalable graph neural network training,” inProceedings of the AAAI Conference on Artificial Intelligence, 2025, pp. 12 872–12 880

    2025

  49. [49]

    D. B. West et al., Introduction to graph theory. Prentice hall Upper Saddle River, 2001, vol. 2

    2001

  50. [50]

    A new internal index based on density core for clustering validation,

    J. Xie, Z.-Y . Xiong, Q.-Z. Dai, X.-X. Wang, and Y .-F. Zhang, “A new internal index based on density core for clustering validation,”Information Sciences, vol. 506, pp. 346–365, 2020

    2020

  51. [51]

    The upper bound of the optimal number of clusters in fuzzy clustering,

    J. Yu and Q. Cheng, “The upper bound of the optimal number of clusters in fuzzy clustering,” Science in China Series:Information Sciences, vol. 44, no. 2, pp. 119–125, 2001

    2001

  52. [52]

    T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to algorithms. MIT press, 2022

    2022

  53. [53]

    Structure-activity relationship of muta- genic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity,

    A. K. Debnath, R. L. Lopez de Compadre, G. Debnath, A. J. Shusterman, and C. Hansch, “Structure-activity relationship of muta- genic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity,”Journal of Medicinal Chemistry, vol. 34, no. 2, pp. 786–797, 1991. [Online]. Available: https://doi.org/10.1021/jm00106a046

    doi:10.1021/jm00106a046 1991

  54. [54]

    Derivation and validation of tox- icophores for mutagenicity prediction,

    J. Kazius, R. McGuire, and R. Bursi, “Derivation and validation of tox- icophores for mutagenicity prediction,”Journal of Medicinal Chemistry, vol. 48, no. 1, pp. 312–320, 2005, pMID: 15634026. [Online]. Available: https://doi.org/10.1021/jm040835a

    doi:10.1021/jm040835a 2005

  55. [55]

    Comparison of descriptor spaces for chemical compound retrieval and classification,

    N. Wale, I. A. Watson, and G. Karypis, “Comparison of descriptor spaces for chemical compound retrieval and classification,”Knowledge and Information Systems, vol. 14, no. 3, pp. 347–375, 2008. 11

    2008

  56. [56]

    Fast unfolding of communities in large networks,

    V . D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre, “Fast unfolding of communities in large networks,”Journal of Statistical Mechanics: Theory and Experiment, vol. 2008, no. 10, p. P10008, 2008

    2008

  57. [57]

    Weisfeiler and leman go sparse: Towards scalable higher-order graph embeddings,

    C. Morris, G. Rattan, and P. Mutzel, “Weisfeiler and leman go sparse: Towards scalable higher-order graph embeddings,”Advances in Neural Information Processing Systems, vol. 33, pp. 21 824–21 840, 2020

    2020

  58. [58]

    A unified view of gradient-based attribution methods for deep neu- ral networks,

    M. Ancona, E. Ceolini, A. C. Oztireli, and M. H. Gross, “A unified view of gradient-based attribution methods for deep neu- ral networks,” CoRR, vol. abs/1711.06104, 2017. [Online]. Avail- able:http://arxiv.org/abs/1711.06104

    Pith/arXiv arXiv 2017

  59. [59]

    Explainability methods for graph convolutional neural networks,

    P. E. Pope, S. Kolouri, M. Rostami, C. E. Martin, and H. Hoffmann, “Explainability methods for graph convolutional neural networks,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 10 772–10 781

    2019

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.