## 4 Discussion  
This study proposes and validates a novel framework that integrates graph kernel-based Koopman embeddings with cost-sensitive learning for robust EEG classification, addressing several critical gaps in the literature. By focusing on estimating metastable connectivity substructures and their classification, this work demonstrates the potential of combining advanced graph theoretical methods, dynamical system modelling, and machine learning techniques for EEG data analysis. Considering the performance results from Table 1, Figure 6, and Table 2, the workflow combining mPLV connectivity measure,  
Figure 6: Performance comparison regarding accuracy, AUC, precision, recall, and F1-score with and without cost-sensitive learning for different workflows comprising different combinations of connectivity metrics, graph kernels, and classifiers: Plot (**a**) shows the combinations with mPLV; Plot. (**b**) shows combinations with correlation coefficient. WL graph Koopman kernel, and Decision Tree outperformed the alternative combinations, particularly considering the accuracy (91.7%) and F1-score (88.9%). Considering Table 2, employing cost-sensitive learning improved the F1-score for the mPLV-WL-DT workflow to 91% compared to 88.9% without cost-sensitive learning. _Interpretation and Implications of Metacluster Substructures for Discriminating Spatial Temporal Patterns_  
The analysis of metacluster substructures, as represented by the clusters derived from the Gram matrices generated using Weisfeiler-Lehman (WL) kernel-based graph embeddings, provides a novel way to investigate the latent spatio-temporal patterns in EEG data associated with preictal and ictal states in the epileptiform data for example. These clusters encode metastable connectivity substructures in the brain, which is critical for distinguishing between these neurological states. **1. How coherent subclusters represent inter-channel synchronisation--**The Gram matrix, denoted as \(K\), is an \(N\times N\) matrix where each element \(K\)(i,j) represents the similarity between two EEG connectivity graphs at time points i and j. The intensity of entries in the Gram matrix is directly related to how structurally similar the underlying EEG connectivity patterns are. High-intensity (Bright) subclusters indicate substantial graph similarity over time points, meaning that EEG connectivity patterns remain stable over that period. This suggests sustained high inter-channel synchronisation, where multiple EEG channels exhibit coordinated activity. Low-intensity (dark) regions represent low similarity between graphs at different time points, indicating transient or shifting connectivity structures. This suggests low or desynchronised inter-channel activity, where brain regions interact dynamically with varying connectivity strengths. **2. The role of the WL Kernel in encoding synchronisation patterns--**The WL kernel enhances EEG graph representations by incorporating hierarchical neighbourhood aggregation, which refines how local and global graph structures are captured. This means EEG graphs that maintain stable inter-channel synchronisation exhibit consistent neighbourhood structures across time, leading to high kernel similarity values and thus forming high-intensity subclusters in the Gram matrix. On the other hand, EEG graphs that undergo connectivity transitions (e.g., preictal-to-ictal shifts) show decreasing similarity, reflected in fragmented, lower-intensity patterns in the Gram matrix. **3. Emergence of synchronisation patterns in different EEG states--**The structure of coherent subclusters in Gram matrices can provide valuable insights into neural synchronisation dynamics in different EEG states:  
**A. Seizure dynamics (preictal vs. ictal)--**During the (preictal) period leading up to a seizure, brain regions often exhibit progressively increasing synchronisation in specific regions before the full onset of the seizure.