Graph Neural Networks: A Review of Methods and Applications,

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Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M.: Graph neural networks: A review of methods and applications. 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). These expla- nations were generated using the same hyperparameters as in Figure 2.

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Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M.: Graph neural networks: A review of methods and applications. 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). These expla- nations were generated using the same hyperparameters as in Figure 2

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Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. 2020. Graph neural networks: A review of methods and applications. AI Open 1 (2020), 57–81. https://doi.org/ 10.1016/j.aiopen.2021.01.001

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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 . These subgraphs jo

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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 . These subgraphs jo

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Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M., 2020. Graph neural networks: A review of methods and applications. AIOpen1,57–81. URL:https://www.sciencedirect.com/science/article/pii/S2666651021000012,doi:https: //doi.org/10.1016/j.aiopen.2021.01.001. Zesheng Liu, Maryam Rahnemoonfar:Preprint submitted to ElsevierPage 20 of 20

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Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M.: Graph neural networks: A review of methods and applications. AI Open1, 57–81 (2020) https://doi.org/10.1016/j.aiopen.2021.01.001

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Jie Zhou and Ganqu Cui and Shengding Hu and Zhengyan Zhang and Cheng Yang and Zhiyuan Liu and Lifeng Wang and Changcheng Li and Maosong Sun , keywords =. Graph neural networks: A review of methods and applications , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.aiopen.2021.01.001 , url =

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Jie Zhou and Ganqu Cui and Shengding Hu and Zhengyan Zhang and Cheng Yang and Zhiyuan Liu and Lifeng Wang and Changcheng Li and Maosong Sun, keywords =. Graph neural networks: A review of methods and applications, journal =. 2020, issn =. doi:https://doi.org/10.1016/j.aiopen.2021.01.001, url =

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J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun, “Graph neural networks: A review of methods and applications,”AI Open, vol. 1, pp. 57–81, 2020. [Online]. Available: https://doi.org/10.1016/j.aiopen.2021.01.001

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Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. 2020. Graph neural networks: A review of methods and applications.AI Open1 (2020), 57–81. https://doi.org/ 10.1016/j.aiopen.2021.01.001

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J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun. Graph neural networks: A review of methods and applications.AI Open, 1:57–81, 2020. ISSN 2666-6510. doi: https://doi.org/10.1016/j.aiopen.2021.01.001. URL https://www.sciencedirect.co m/science/article/pii/S2666651021000012

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J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, M. Sun, Graph neural networks: A review of methods and applications, AI Open 1 (2020) 57–81.doi:10.1016/j.aiopen.2021.01.001

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J. Zhouet al., “Graph Neural Networks: A Review of Methods and Applications,”AI Open1(2020) 57-81, doi:10.1016/j.aiopen.2021.01.001