## 3 Results and Discussion
### Network Characteristics
Centrality, as extensively documented in the electrophysiological literature, has consistently underscored the non-uniform distribution of coherence across frequencies [49]. It is well established that different systems of brain regions may exhibit varying levels of coherence at distinct frequencies [48]. The centrality patterns also demonstrate frequency-specific nuances. Specifically, at low frequencies, centrality predominantly manifests in temporal lobes, with noteworthy lateralization observed in delta and theta waves (Figure 5a,b). For alpha frequencies (Figure 5c), the coherence reveals a shift in centrality, now prominently observed in the right occipital and left parietal lobes, whereas in the case of beta and gamma waves (Figure 5d,e), the centrality is very weak and homogeneous between lobes. Centralateral coherences are observed between frontal and occipital lobes in the high-frequency bands. Figure 5: Degree centrality for (**a**) delta, (**b**) theta, (**c**) alpha, (**d**) beta, and (**e**) gamma waves at \(\sigma=0.5\). Figure 6 presents another representation of the node degrees for the lobes. It is evident that the nodes exhibit larger coherence-related connections in the low-frequency bands (Figure 6a,b). However, in the higher-frequency bands (Figure 6c-e), the connections are less pronounced. Notably, the connections in the delta band (Figure 6a) are smaller compared to those in the theta network (Figure 6b), which shows nodes with degrees of 7. A more evident alteration is observed for alpha waves (Figure 6c), where the left temporal lobe is completely disconnected, while the other lobes experience a reduction in connections. In the beta graph (Figure 6d), the right parietal lobe ceases to participate entirely. Meanwhile, for gamma waves (Figure 6e), engagement diminishes for the temporal lobes, while the rest show a degree of 1. Figure 7 illustrates the connected graph components, which are the subset of network nodes where there is a path from each node in the subset to any other node in the same subset [45]. This representation allows us to observe the formation of groups as connections begin to dissolve. Identifying connected components within an undirected graph provides insight into the network's level of connectivity and aids in the extraction of coefficient \(\kappa\) (Equation (2)). Table 2 presents the global connectivity coefficients for \(\sigma_{th}=0.5\). Figure 8 shows cycles formed in the networks for three frequency bands: delta, theta, and alpha. A cycle is a connected graph, in which each vertex has degree 2 [46]. The total number of cycles found for each graph is indicated in the figure. The theta graph (Figure 8b) displays the greatest connectivity and the most edges, resulting in a higher number of cycles within the network. In contrast, both the alpha and delta graphs (Figure 8a-c, respectively) exhibit fewer connections, with the alpha graph revealing only three cycles. Furthermore, no cycles were observed in the beta or gamma graphs. This observation is consistent with the broader literature, which suggests that slower rhythmic patterns typically exhibit a more widespread network configuration compared to faster ones [50; 51]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Frequency Band** & \(\kappa\) \\ \hline Delta & 1 \\ Theta & 1 \\ Alpha & 0.5 \\ Beta & 0.5 \\ Gamma & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Connectivity coefficients for different frequency bands. Figure 6: Node degrees for (**a**) delta, (**b**) theta, (**c**) alpha, (**d**) beta, and (**e**) gamma waves. FL—left frontal; FR—right frontal; OL—left occipital; OR—right occipital; PL—left parietal; PR—right parietal; TL—left temporal; TR—right temporal. Distances between nodes are represented as \(8\times 8\) matrices (\(Delta_{D}\), \(Theta_{D}\), \(Alpha_{D}\), \(Beta_{D}\), \(Gamma_{D}\)), showing the shortest path distances. When the nodes are not connected, the distance is infinite. The disconnection of the networks is also noticeable in these matrices.