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AI Open1, 57– 81 (2020). https://doi.org/https://doi.org/10.1016/j.aiopen.2021.01.001, https://www.sciencedirect.com/science/article/pii/S2666651021000012 1 Explanation-Based Identification of Backdoored Training Graphs 1 Appendix A Backdoor Detection Results of Various Explainers As stated in our main paper, GNNExplainer fails as a method for reverse- engineering backdoor triggers. To test whether this issue is restricted to GN- NExplainer, we also explored the effectiveness of two other explainers – PGEx- plainer [25], known for its parameterized probabilistic graphical model approach in interpreting complex machine learning models, and CaptumExplainer [19], recognized for its comprehensive suite of neural network interpretability tools, including advanced algorithms like Integrated Gradients and Deconvolution. Fig. S1: An example of a backdoored sample from each dataset, after applying the mask generated by CaptumExplainer (top) and PGExplainer (bottom). 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To test whether this issue is restricted to GN- NExplainer, we also explored the effectiveness of two other explainers – PGEx- plainer [25], known for its parameterized probabilistic graphical model approach in interpreting complex machine learning models, and CaptumExplainer [19], recognized for its comprehensive suite of neural network interpretability tools, including advanced algorithms like Integrated Gradients and Deconvolution. Fig. S1: An example of a backdoored sample from each dataset, after applying the mask generated by CaptumExplainer (top) and PGExplainer (bottom). 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A deep graph convo- lutional neural network architecture for graph classification. PLOS ONE, 18, 2023. URL https://api.semantic scholar.org/CorpusID:257428249. Zügner, D., Borchert, O., Akbarnejad, A., and Günnemann, S. Adversarial attacks on graph neural networks: Perturbations and their patterns. ACM Trans. Knowl. Discov. Data , 14(5), jun 2020. ISSN 1556-4681. URL https://doi.org/10 .1145/3394520. 11 Watermarking Graph Neural Networks via Explanations for Ownership Protection A. Appendix Algorithm 1: Watermark Embedding Input: Graph 𝐺, training nodes V𝑡𝑟 , learning rate 𝜂, #watermarked subgraphs 𝑇, watermarked subgraph size 𝑠, hyperparameter 𝑟, target significance 𝛼𝑡𝑔𝑡 , watermark loss contribution bound 𝜖. Output: A trained and watermarked model, 𝑓 . Setup: Initialize 𝑓 and optimizer. With 𝛼𝑡𝑔𝑡 , 𝑇, and number of node features 𝐹 as input, compute 𝑀 using equation 11. Initialize w with values 1 and −1 uniform at random. With 𝑛𝑠𝑢𝑏 = 𝑐𝑒𝑖𝑙 (𝑠 × |V 𝑡𝑟 |), randomly sample 𝑇 sets of 𝑛𝑠𝑢𝑏 nodes from V𝑡𝑟 . These subgraphs jo","evidence_cleaned":"doi: https://doi.org/10.1016/j.aiopen.2021.01.001. URL https://www.sciencedirect.com/science/ar ticle/pii/S2666651021000012. Zhou, Y ., Huo, H., Hou, Z., and Bu, F. A deep graph convo- lutional neural network architecture for graph classification. PLOS ONE, 18, 2023. URL https://api.semantic scholar.org/CorpusID:257428249. Zügner, D., Borchert, O., Akbarnejad, A., and Günnemann, S. Adversarial attacks on graph neural networks: Perturbations and their patterns. ACM Trans. Knowl. Discov. Data, 14(5), jun 2020. ISSN 1556-4681. URL https://doi.org/10 .1145/3394520. 11 Watermarking Graph Neural Networks via Explanations for Ownership Protection A. Appendix Algorithm 1: Watermark Embedding Input: Graph 𝐺, training nodes V𝑡𝑟, learning rate 𝜂, #watermarked subgraphs 𝑇, watermarked subgraph size 𝑠, hyperparameter 𝑟, target significance 𝛼𝑡𝑔𝑡, watermark loss contribution bound 𝜖. Output: A trained and watermarked model, 𝑓 . Setup: Initialize 𝑓 and optimizer. With 𝛼𝑡𝑔𝑡, 𝑇, and number of node features 𝐹 as input, compute 𝑀 using equation 11. Initialize w with values 1 and −1 uniform at random. With 𝑛𝑠𝑢𝑏 = 𝑐𝑒𝑖𝑙 (𝑠 × |V 𝑡𝑟 |), randomly sample 𝑇 sets of 𝑛𝑠𝑢𝑏 nodes from V𝑡𝑟 . 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