## 5 Robustness of the WL kernel in identifying metaclusters
### Strengths of the Proposed Framework  
* **Efficacy of Graph Kernels in Capturing Metastable Connectivity Substructures** The study systematically investigates the utility of various graph kernels, such as the Weisfeiler-Lehman (WL) kernel, Random Walk kernel, and spectral decomposition kernel, in identifying latent connectivity structures. The results confirm the superior performance of the WL kernel in both computational efficiency and accuracy, as evidenced by its ability to encode metastable brain dynamics effectively. For instance, the clustering of eigenvectors from WL kernels successfully identified distinct metastable substructures across different EEG states, as illustrated in Figures 3 and 4. These findings emphasise the robustness of graph kernels in capturing the spatio-temporal complexities of EEG connectivity. * **Integration of Koopman Operators for Linearizing Nonlinear Dynamics** By leveraging Koopman spectral operators, this framework provides a linear representation of EEG's inherently nonlinear dynamics. This enables spectral decomposition into metastable states, offering interpretable features for classification. The eigenfunctions derived from the Koopman operator reveal subtle temporal transitions in brain connectivity that are otherwise challenging to detect with traditional methods. This study is one of the first to apply Koopman-based embeddings to EEG data in combination with graph kernels, marking a significant advancement in the field. * **Robust Classification with Cost-Sensitive Learning** The integration of cost-sensitive learning (CSL) ensures that the framework effectively handles class imbalance, a common issue in EEG datasets where critical events such as seizures are underrepresented. The results demonstrate that incorporating CSL significantly improved the F1-scores for minority classes, as shown in Table 2. For example, the mPLV-WL-DT workflow achieved an F1-score of 91% with CSL compared to 88.9% without CSL, highlighting its efficacy in addressing imbalance while maintaining high classification accuracy. * **Explainability of Features** The framework provides explainable insights into the metastable substructures of brain connectivity, which are visualised through eigenfunctions and clustering structures. This addresses a critical gap in the literature, as many existing methods focus solely on accuracy metrics without delving into the interpretability of the features derived. By visualizing the connectivity substructures (Figure 3) and eigenvector clusters (Figure 4), this study enhances the transparency of the classification process and contributes to a deeper understanding of EEG connectivity dynamics. _Novelty of the Findings_  
* **First-of-Its-Kind Application of Graph Koopman Kernel Embeddings** While graph kernels and Koopman operators have been applied independently in other domains, their combined application for EEG classification is novel. This integration enables the robust estimation of latent metastable structures, bridging the gap between theoretical advancements in dynamical systems and practical EEG data analysis. * **Comprehensive Evaluation of Graph Kernels for EEG Connectivity** Unlike prior studies that utilise graph kernels in a limited scope, this work systematically evaluates the performance of multiple kernels across different workflows. The demonstrated superiority of the WL kernel in terms of runtime and accuracy offers clear guidance for future studies aiming to analyse large-scale EEG datasets efficiently. * **Improved Classification Performance** The mPLV-WL-DT workflow achieved the highest performance metrics (accuracy: 91.7%, F1-score: 91%) among all tested combinations, outperforming alternative methods such as Random Forest (RF) and Support Vector Classifier (SVC). These results underscore the effectiveness of the proposed framework in achieving reliable classification, even with challenging imbalanced datasets.