{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T04:41:40Z","timestamp":1760330500758,"version":"build-2065373602"},"publisher-location":"Cham","reference-count":46,"publisher":"Springer Nature Switzerland","isbn-type":[{"type":"print","value":"9783032083296"},{"type":"electronic","value":"9783032083302"}],"license":[{"start":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:00:00Z","timestamp":1760400000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:00:00Z","timestamp":1760400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Recent advancements in graph neural networks (GNNs) have significantly enhanced the performance of AI systems in tasks such as community detection, user friendship prediction, and drug discovery. However, the opaque nature of these models undermines user trust, especially in sensitive domains like health and finance. Graph Counterfactual Explanation (GCE) methods aim to mitigate this issue by providing insights into model predictions and suggesting user actions for alternative outcomes. Yet, GCEs produced by different methods often vary in quality, diversity, and alignment with the original model\u2019s predictions. This work introduces an ensemble-based approach designed to address these inconsistencies by leveraging multiple GCE methods. Our approach comprises two main strategies: <jats:italic>Selection<\/jats:italic>, employing multi-criteria optimization to choose the optimal base explanation for each case, and <jats:italic>Aggregation<\/jats:italic>, combining multiple explanations to form a more robust overall explanation. We propose three selection strategies and six aggregation strategies. Our experimental evaluation demonstrates that these ensemble methods, particularly <jats:italic>Ideal-Point Multi-Criteria Selection<\/jats:italic>, consistently outperform individual GCE methods across diverse datasets in terms of quality, thereby significantly improving the interpretability of GNNs.<\/jats:p>","DOI":"10.1007\/978-3-032-08330-2_9","type":"book-chapter","created":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T03:11:13Z","timestamp":1760325073000},"page":"177-201","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exploring Ensemble Strategies for\u00a0Graph Counterfactual Explanations"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0491-3515","authenticated-orcid":false,"given":"Mario Alfonso","family":"Prado-Romero","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2991-2279","authenticated-orcid":false,"given":"Bardh","family":"Prenkaj","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2092-0213","authenticated-orcid":false,"given":"Giovanni","family":"Stilo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,14]]},"reference":[{"key":"9_CR1","doi-asserted-by":"crossref","unstructured":"Abrate, C., Bonchi, F.: Counterfactual graphs for explainable classification of brain networks. In: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 2495\u20132504 (2021)","DOI":"10.1145\/3447548.3467154"},{"issue":"6","key":"9_CR2","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1038\/nrg2102","volume":"8","author":"U Alon","year":"2007","unstructured":"Alon, U.: Network motifs: theory and experimental approaches. Nat. Rev. Genet. 8(6), 450\u2013461 (2007)","journal-title":"Nat. Rev. Genet."},{"key":"9_CR3","unstructured":"Artelt, A., Vrachimis, S., Eliades, D., Polycarpou, M., Hammer, B.: One explanation to rule them all\u2013ensemble consistent explanations. arXiv preprint arXiv:2205.08974 (2022)"},{"key":"9_CR4","first-page":"5644","volume":"34","author":"M Bajaj","year":"2021","unstructured":"Bajaj, M., et al.: Robust counterfactual explanations on graph neural networks. Adv. Neural. Inf. Process. Syst. 34, 5644\u20135655 (2021)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"9_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-88908-3","volume-title":"Multiobjective Optimization","year":"2008","unstructured":"Branke, J., Deb, K., Miettinen, K., S\u0142owi\u0144ski, R. (eds.): Multiobjective Optimization. LNCS, vol. 5252. Springer, Heidelberg (2008). https:\/\/doi.org\/10.1007\/978-3-540-88908-3"},{"issue":"2","key":"9_CR6","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1023\/A:1018054314350","volume":"24","author":"L Breiman","year":"1996","unstructured":"Breiman, L.: Bagging predictors. Mach. Learn. 24(2), 123\u2013140 (1996)","journal-title":"Mach. Learn."},{"key":"9_CR7","doi-asserted-by":"crossref","unstructured":"Bunke, H.: Recent developments in graph matching. In: Proceedings 15th International Conference on Pattern Recognition, ICPR-2000, vol.\u00a02, pp. 117\u2013124. IEEE (2000)","DOI":"10.1109\/ICPR.2000.906030"},{"key":"9_CR8","doi-asserted-by":"crossref","unstructured":"Campagner, A., Ciucci, D., Cabitza, F.: Aggregation models in ensemble learning: a large-scale comparison. Inf. Fusion (2022)","DOI":"10.1016\/j.inffus.2022.09.015"},{"key":"9_CR9","unstructured":"Commission, E.: On artificial intelligence\u2013a European approach to excellence and trust (2020)"},{"key":"9_CR10","unstructured":"Dutta, S., Long, J., Mishra, S., Tilli, C., Magazzeni, D.: Robust counterfactual explanations for tree-based ensembles. In: Chaudhuri, K., Jegelka, S., Song, L., Szepesvari, C., Niu, G., Sabato, S. (eds.) Proceedings of the 39th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol.\u00a0162, pp. 5742\u20135756. PMLR, 17\u201323 July 2022"},{"key":"9_CR11","unstructured":"Faber, L., Moghaddam, A.K., Wattenhofer, R.: Contrastive graph neural network explanation. In: Proceedings of 37th Graph Representation Learning and Beyond Workshop at ICML 2020, p.\u00a028. International Conference on Machine Learning (2020)"},{"key":"9_CR12","doi-asserted-by":"crossref","unstructured":"Guidotti, R.: Counterfactual explanations and how to find them: literature review and benchmarking. Data Mining and Knowledge Discovery, pp. 1\u201355 (2022)","DOI":"10.1007\/s10618-022-00831-6"},{"key":"9_CR13","doi-asserted-by":"crossref","unstructured":"Guidotti, R., Ruggieri, S.: Ensemble of counterfactual explainers. In: International Conference on Discovery Science, pp. 358\u2013368. Springer (2021)","DOI":"10.1007\/978-3-030-88942-5_28"},{"key":"9_CR14","doi-asserted-by":"crossref","unstructured":"Huang, Z., Kosan, M., Medya, S., Ranu, S., Singh, A.: Global counterfactual explainer for graph neural networks. In: Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining, pp. 141\u2013149 (2023)","DOI":"10.1145\/3539597.3570376"},{"issue":"1","key":"9_CR15","doi-asserted-by":"publisher","first-page":"8360","DOI":"10.1038\/s41598-022-12201-9","volume":"12","author":"K Jha","year":"2022","unstructured":"Jha, K., Saha, S., Singh, H.: Prediction of protein-protein interaction using graph neural networks. Sci. Rep. 12(1), 8360 (2022)","journal-title":"Sci. Rep."},{"key":"9_CR16","unstructured":"Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, 24\u201326 April 2017, Conference Track Proceedings. OpenReview.net (2017), https:\/\/openreview.net\/forum?id=SJU4ayYgl"},{"key":"9_CR17","doi-asserted-by":"crossref","unstructured":"Liu, Y., Chen, C., Liu, Y., Zhang, X., Xie, S.: Multi-objective explanations of gnn predictions. In: 2021 IEEE International Conference on Data Mining (ICDM), pp. 409\u2013418. IEEE (2021)","DOI":"10.1109\/ICDM51629.2021.00052"},{"key":"9_CR18","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s10044-012-0284-8","volume":"16","author":"L Livi","year":"2013","unstructured":"Livi, L., Rizzi, A.: The graph matching problem. Pattern Anal. Appl. 16, 253\u2013283 (2013)","journal-title":"Pattern Anal. Appl."},{"key":"9_CR19","unstructured":"Ma, J., Guo, R., Mishra, S., Zhang, A., Li, J.: CLEAR: generative counterfactual explanations on graphs. In: Oh, A.H., Agarwal, A., Belgrave, D., Cho, K. (eds.) Advances in Neural Information Processing Systems (2022), https:\/\/openreview.net\/forum?id=YR-s5leIvh"},{"key":"9_CR20","unstructured":"McAleese, S., Keane, M.: A comparative analysis of counterfactual explanation methods for text classifiers. arXiv preprint arXiv:2411.02643 (2024)"},{"key":"9_CR21","doi-asserted-by":"crossref","unstructured":"Musch, S., Borrelli, M., Kerrigan, C.: The eu ai act as global artificial intelligence regulation. Available at SSRN 4549261 (2023)","DOI":"10.2139\/ssrn.4549261"},{"key":"9_CR22","doi-asserted-by":"crossref","unstructured":"Nguyen, T.M., Quinn, T.P., Nguyen, T., Tran, T.: Explaining black box drug target prediction through model agnostic counterfactual samples. IEEE\/ACM Trans. Comput. Biol. Bioinf. (2022)","DOI":"10.1109\/TCBB.2022.3190266"},{"key":"9_CR23","doi-asserted-by":"crossref","unstructured":"Numeroso, D., Bacciu, D.: Meg: Generating molecular counterfactual explanations for deep graph networks. In: 2021 International Joint Conference on Neural Networks (IJCNN), pp.\u00a01\u20138. IEEE (2021)","DOI":"10.1109\/IJCNN52387.2021.9534266"},{"key":"9_CR24","doi-asserted-by":"crossref","unstructured":"Piaggesi, S., Bodria, F., Guidotti, R., Giannotti, F., Pedreschi, D.: Counterfactual and prototypical explanations for tabular data via interpretable latent space. IEEE Access (2024)","DOI":"10.1109\/ACCESS.2024.3496114"},{"key":"9_CR25","doi-asserted-by":"crossref","unstructured":"Prado-Romero, M.A., Prenkaj, B., Stilo, G.: Developing and evaluating graph counterfactual explanation with gretel. In: Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining, pp. 1180\u20131183 (2023)","DOI":"10.1145\/3539597.3573026"},{"key":"9_CR26","unstructured":"Prado-Romero, M.A., Prenkaj, B., Stilo, G.: Revisiting countergan for counterfactual explainability of graphs. In: Maughan, K., Liu, R., Burns, T.F. (eds.) The First Tiny Papers Track at ICLR 2023, Tiny Papers @ ICLR 2023, Kigali, Rwanda, 5 May 2023. OpenReview.net (2023)"},{"issue":"19","key":"9_CR27","first-page":"21518","volume":"38","author":"MA Prado-Romero","year":"2024","unstructured":"Prado-Romero, M.A., Prenkaj, B., Stilo, G.: Robust stochastic graph generator for counterfactual explanations. Proc. AAAI Conf. Artif. Intell. 38(19), 21518\u201321526 (2024)","journal-title":"Proc. AAAI Conf. Artif. Intell."},{"key":"9_CR28","doi-asserted-by":"crossref","unstructured":"Prado-Romero, M.A., Prenkaj, B., Stilo, G.: Are generative-based graph counterfactual explainers worth it? In: Meo, R., Silvestri, F. (eds.) Machine Learning and Principles and Practice of Knowledge Discovery in Databases, pp. 152\u2013170. Springer, Cham (2025)","DOI":"10.1007\/978-3-031-74633-8_10"},{"key":"9_CR29","doi-asserted-by":"crossref","unstructured":"Prado-Romero, M.A., Prenkaj, B., Stilo, G., Giannotti, F.: A survey on graph counterfactual explanations: definitions, methods, evaluation, and research challenges. ACM Comput. Surv. 56(7) (2024)","DOI":"10.1145\/3618105"},{"key":"9_CR30","doi-asserted-by":"crossref","unstructured":"Prado-Romero, M.A., Stilo, G.: Gretel: graph counterfactual explanation evaluation framework. In: Proceedings of the 31st ACM International Conference on Information and Knowledge Management, ACM (2022)","DOI":"10.1145\/3511808.3557608"},{"issue":"4","key":"9_CR31","doi-asserted-by":"publisher","first-page":"e1249","DOI":"10.1002\/widm.1249","volume":"8","author":"O Sagi","year":"2018","unstructured":"Sagi, O., Rokach, L.: Ensemble learning: a survey. Wiley Interdisc. Rev. Data Min. Knowl. Disc. 8(4), e1249 (2018)","journal-title":"Wiley Interdisc. Rev. Data Min. Knowl. Disc."},{"issue":"1","key":"9_CR32","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1109\/TNN.2008.2005605","volume":"20","author":"F Scarselli","year":"2008","unstructured":"Scarselli, F., Gori, M., Tsoi, A., Hagenbuchner, M., Monfardini, G.: The graph neural network model. IEEE Trans. Neural Networks 20(1), 61\u201380 (2008)","journal-title":"IEEE Trans. Neural Networks"},{"issue":"2","key":"9_CR33","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1023\/A:1022648800760","volume":"5","author":"RE Schapire","year":"1990","unstructured":"Schapire, R.E.: The strength of weak learnability. Mach. Learn. 5(2), 197\u2013227 (1990)","journal-title":"Mach. Learn."},{"key":"9_CR34","unstructured":"Skulimowski, A.: Applicability of ideal points in multicriteria decision-making. In: Proceedings of the 9th International Conference on Multiple Criteria Decision-Making, Fairfax, USA. pp.\u00a05\u20138 (1990)"},{"issue":"1","key":"9_CR35","doi-asserted-by":"publisher","first-page":"119","DOI":"10.61822\/amcs-2024-0009","volume":"34","author":"I Stepka","year":"2024","unstructured":"Stepka, I., Lango, M., Stefanowski, J.: A multi-criteria approach for selecting an explanation from the set of counterfactuals produced by an ensemble of explainers. Int. J. Appl. Math. Comput. Sci. 34(1), 119\u2013133 (2024)","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"9_CR36","unstructured":"Steuer, R.: Multiple Criteria optimization: theory, computation, and application. (WILEY SERIES IN PROBABILITY AND MATHEMATICAL STATISTICS), Wiley (1986), https:\/\/books.google.it\/books?id=0H9jQgAACAAJ"},{"key":"9_CR37","doi-asserted-by":"crossref","unstructured":"Tan, J., Geng, S., Fu, Z., Ge, Y., Xu, S., Li, Y., Zhang, Y.: Learning and evaluating graph neural network explanations based on counterfactual and factual reasoning. In: Proceedings of the ACM Web Conference 2022, WWW 2022, pp. 1018\u20131027. ACM, New York, NY, USA (2022)","DOI":"10.1145\/3485447.3511948"},{"issue":"13","key":"9_CR38","doi-asserted-by":"publisher","first-page":"3697","DOI":"10.1039\/D1SC05259D","volume":"13","author":"GP Wellawatte","year":"2022","unstructured":"Wellawatte, G.P., Seshadri, A., White, A.D.: Model agnostic generation of counterfactual explanations for molecules. Chem. Sci. 13(13), 3697\u20133705 (2022)","journal-title":"Chem. Sci."},{"key":"9_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ddtec.2020.11.009","volume":"37","author":"O Wieder","year":"2020","unstructured":"Wieder, O., et al.: A compact review of molecular property prediction with graph neural networks. Drug Discov. Today Technol. 37, 1\u201312 (2020)","journal-title":"Drug Discov. Today Technol."},{"issue":"2","key":"9_CR40","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/S0893-6080(05)80023-1","volume":"5","author":"DH Wolpert","year":"1992","unstructured":"Wolpert, D.H.: Stacked generalization. Neural Netw. 5(2), 241\u2013259 (1992)","journal-title":"Neural Netw."},{"key":"9_CR41","doi-asserted-by":"crossref","unstructured":"Wu, S., Tang, Y., Zhu, Y., Wang, L., Xie, X., Tan, T.: Session-based recommendation with graph neural networks. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol.\u00a033, pp. 346\u2013353 (2019)","DOI":"10.1609\/aaai.v33i01.3301346"},{"key":"9_CR42","doi-asserted-by":"crossref","unstructured":"Wu, X., et al.: Clare: a semi-supervised community detection algorithm. In: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 2059\u20132069 (2022)","DOI":"10.1145\/3534678.3539370"},{"key":"9_CR43","unstructured":"Xu, K., Hu, W., Leskovec, J., Jegelka, S.: How powerful are graph neural networks? In: 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, 6\u20139 May 2019. OpenReview.net (2019), https:\/\/openreview.net\/forum?id=ryGs6iA5Km"},{"key":"9_CR44","unstructured":"Ying, Z., Bourgeois, D., You, J., Zitnik, M., Leskovec, J.: Gnnexplainer: generating explanations for graph neural networks. In: Advances in Neural Information Processing Systems, vol. 32 (2019)"},{"key":"9_CR45","doi-asserted-by":"crossref","unstructured":"Yuan, H., Yu, H., Gui, S., Ji, S.: Explainability in graph neural networks: a taxonomic survey. IEEE Trans. Pattern Anal. Mach. Intell. (2022)","DOI":"10.1109\/TPAMI.2022.3204236"},{"key":"9_CR46","doi-asserted-by":"crossref","unstructured":"Zemni, M., Chen, M., Zablocki, E., Ben-Younes, H., P\u00e9rez, P., Cord, M.: Octet: object-aware counterfactual explanations. In: Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 15062\u201315071, June 2023","DOI":"10.1109\/CVPR52729.2023.01446"}],"container-title":["Communications in Computer and Information Science","Explainable Artificial Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-08330-2_9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T04:02:58Z","timestamp":1760328178000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-08330-2_9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,14]]},"ISBN":["9783032083296","9783032083302"],"references-count":46,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-08330-2_9","relation":{},"ISSN":["1865-0929","1865-0937"],"issn-type":[{"type":"print","value":"1865-0929"},{"type":"electronic","value":"1865-0937"}],"subject":[],"published":{"date-parts":[[2025,10,14]]},"assertion":[{"value":"14 October 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Disclosure of Interests"}},{"value":"xAI","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"World Conference on Explainable Artificial Intelligence","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Istanbul","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"T\u00fcrkiye","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"9 July 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"11 July 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"3","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"xai2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/xaiworldconference.com\/2025\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}