## 2 Materials and Methods
### Network Metrics  
In this study, since we deal with time-varying and space-dependent modulations of internal tides, we focus on the connectivity among oscillations at different depths and the connectivity structure in terms of degrees [37], clustering coefficients [38], and community structures [27; 28; 39]. We also analyze the temporal evolution of the sum of minimal spanning tree (MST) weights in order to explore the variations in the overall connectivity and structure of a dynamic network over time [40; 41]. Lastly, we examine the structural entropy quantifying the uncertainty or randomness in the organization of nodes across different communities or modules within the network [42]. The mean degree, defined as the average number of edges in the network, is obtained from the adjacency matrix \(\{A_{ij}\}\), through [37]  
\[k_{i}=\sum\
olimits_{j=1}^{N}A_{ij},\langle k\rangle=\frac{1}{N}{\sum\
olimits _{i=1}^{N}k_{i}} \tag{5}\]  
where \(k_{i}\) denotes the number of links attached to a node \(i\) and \(\langle k\rangle\) denotes mean degree of the network. Mean degree provides an idea of the overall connectivity of the network. The clustering coefficient \(C(i)\) is a measure used to quantify the degree to which nodes in a network tend to cluster together or form tightly-knit groups. It describes the local structure and organization of the network by capturing the prevalence of triangles, i.e., sets of three interconnected nodes. The local and global clustering coefficients are defined as [38]  
\[C(i)=\frac{n_{i}}{k_{i}(k_{i}-1)/2},\langle C\rangle=\frac{1}{N}{\sum\
olimits _{i}C(i)} \tag{6}\]  
where \(n_{i}\) denotes the number of real links among neighbors of a node \(i\), and \(\langle C\rangle\) is the mean clustering coefficient. Figure 4: Diagram of network construction procedure based on wavelet coherence squared (WCS). The MST is a loop-less tree that spans all the nodes in a given network, such that the sum of the edge weights is minimized. For scale-dependent wavelet-coherence matrices, we compute corresponding distance matrices [\(d_{ij}\)] via the following formula [40]:  
\[d_{ij}=\sqrt{2(1-\overline{R^{2}}_{ij})} \tag{7}\]  
where \(\overline{R^{2}}_{ij}\) denotes the averaged pair-wise WCS estimate of its corresponding matrix, and [PERSON]'s algorithm is utilized to construct MST from the established distance matrix. In this study, the total weights of MST are used to examine the structural variability of a dynamic network. Communities are groups of nodes that are densely connected to each other, with relatively fewer connections to nodes outside the group. Although there are some widely used algorithms, we adopt the Louvain algorithm based on modularity optimization [28]; this algorithm shows improvements in time consumption and optimization by using the improved modularity computation method and the network's hierarchy structure. Modularity is a metric used in network analysis to evaluate the quality of a community structure [39], by measuring the difference between the actual number of edges within communities and the number of edges that would be expected by chance if the nodes were connected randomly while preserving the nodes' degrees. Higher modularity values indicate that the network has a stronger community structure. The modularity metric, often denoted as \(Q\), is defined as [39]  
\[Q=\frac{1}{2m}\sum_{ij}\bigg{(}A_{ij}-\frac{k_{i}k_{j}}{2m}\bigg{)}\delta(c_{ i},c_{j}) \tag{8}\]  
where \(m\) is the total number of edges in the network, \(k_{i}\) and \(k_{j}\) are the degrees of nodes \(i\) and \(j\), \(c_{i}\) and \(c_{j}\) are the community assignments of nodes \(i\) and \(j\), and \(\delta(c_{i},c_{j})\) is the Kronecker delta function. Since, in this study, nodes represent depths, identified communities can provide information on vertical structural variabilities of DIT and SIT networks within a mesoscale eddy.