## 2 Materials and Methods
### Complex Network Methodology
#### 2.4.2 Network Metrics  
The IMF network constructed from an ensemble of same-mode IMFs with a specific peaked timescale is first used to examine the spatial pattern of scale-dependent cross-correlations of SSTA variability among regularly distributed lattice points (nodes in the network topology) in the EJS and, second, is utilized for delimiting the spatial influencing extent of any dominant forcing corresponding to the respective IMF. The spatial distribution of degrees (connectivity) of nodes is used for the former goal, and the second goal is achieved via the modularity-based community structure [34; 39; 40]. A degree, defined as the number of edges (or links) attached to a single node, is obtained from an undirected adjacency matrix \(\{A_{ij}\}\) via the following formula:  
\[k_{i}=\sum_{j=1}^{N}A_{ij},\;k=\frac{1}{N}\sum_{i=1}^{N}k_{i} \tag{7}\]  
where \(k_{i}\) denotes the number of links attached to a node \(i\), with \(k\) denoting the mean degree of the network. Since a mean degree usually provides information on the global connectivity of a network, we use the degree of individual nodes instead of the mean degree for examining the spatial pattern of connectivity of a network. In addition, in order to explore the connectivity structure of the network, we use the network metric of assortativity [41], referring to the nodes' tendency in a network for nodes with similar degrees to connect together. Generally, networks are classified as assortative, disassortative, and neutral; assortative networks tend to make a cluster of hubs (nodes with a high degree), while disassortative networks have a dispersed structure of hubs, if any hubs. In this study, we used the link-based assortativity coefficient, \(r\), quantifying the level of assortativity in a network [41]; it ranges from \(-1\) to \(1\). For a network consisting of N nodes linked by M edges with a degree probability of \(p_{k}\), the assortativity coefficient \(r\) is defined as follows:  
\[r=\frac{1}{\sigma_{q}^{2}}\sum_{jk}jk\Big{(}e_{jk}-q_{j}q_{k}\Big{)} \tag{8}\]  
where \(e_{jk}\) denotes the joint probability for the two nodes of a randomly chosen edge having \(j+1\) degrees and \(k+1\) degrees, respectively [41; 42], and \(q_{k}\), defined as \(q_{k}=\frac{(k+1)p_{k+1}}{\sum_{j}jp_{j}}\), denotes the normalized probability for one end of the randomly chosen link having \(k+1\) degrees, along with the variance \(\sigma_{q}^{2}=\sum_{k}k^{2}q_{k}-\left[\sum_{k}kq_{k}\right]^{2}\). And its interpretation is as follows:  
1. \(r>0\) indicates an assortative network, characterized by a big clustered structure, robustness by multiple hubs, and stability in the network's structure [41]. 2. \(r<0\) indicates a disassortative network, characterized by a negative correlation between adjacent two nodes. 3. \(r\approx 0\) implies that there is no correlation between two nodes of a network. It should be noted that all IMF networks in this study are assortative. Lastly, identifying communities in a modulated geophysical network is crucial for understanding the region-specific topological structure; it can reveal the spatial extent of any forcing affecting the variability of a node's property, like SSTA variability. A community is a group consisting of nodes that have dense connections with nodes inside the group but relatively fewer connections with nodes outside the group. Among a lot of widely used algorithms, we use the Louvain algorithm based on modularity optimization [40].