## 3 Methods
### Analysis of fMRI Synchronization-Based Connectivity
#### 3.2.2 Identification of Hub Nodes  
Two graph metrics have been used to characterize the topological organization of the synchronization-based functional network, Pearson correlation network and coherence network. More specifically, the participation coefficient and within-module degree z score were computed to assess the regional importance of the brain nodes. For a community partition,  
* the participation coefficient \(PC_{i}\) is defined as: \[PC_{i}=1-\sum_{i=1}^{N_{M}}\left(\frac{K_{is}}{K_{i}}\right)^{2},\] (8) where \(K_{i}\) is the sum of the weighted links attached to \(i\), \(K_{is}\) is the sum of weighted links from \(i\) to community \(s\) and \(N_{M}\) is the total number of modules. Thus, the participation coefficient quantifies  
Figure 2: Flowchart of the modularity analysis. The Louvain algorithm is applied for \(M=1000\) iterations to each functional connectivity matrix for different values of \(\gamma\). \(\gamma_{opt}\) is the resolution value corresponding to the maximum \(Q\) value averaged over the \(M\) partitions. This procedure returns a set of different partitions corresponding to the optimal \(\gamma\) value. Hence, a consensus algorithm is applied to identify the most representative final community partition that is used to identify both the functional modules and the hub nodes of the matrix.  
how the links of a node are distributed across modules. A node's participation coefficient is maximum if it has an equal sum of edge weights to each module in the network. A node's participation coefficient is 0 if all its links are within a single community.
* the within-module degree z score for each node is computed as: \[z_{i}=\frac{k_{i}-\bar{k}_{s_{i}}}{\sigma_{k_{s_{i}}}},\] (9) where \(k_{i}\) is the number of links of node i to other nodes in its community \(s_{i}\), \(\bar{k}_{s_{i}}\) is the average of \(k\) over all of the nodes in \(s_{i}\) and \(\sigma_{k_{s_{i}}}\) is the standard deviation of \(k\) in \(s_{i}\). Thus, the within-module degree z score measures how well-connected node \(i\) is to other nodes in the community relative to other nodes in the community.  
These graph metrics are often used to capture the centrality of nodes in brain networks and to map regions that act as hubs [48]. Here, we used a cartographic system mapping each node in a plane through the ordered pair of values \((PC_{i},z_{i})\) to identify the connective hubs, i.e., those nodes with higher connection strength and high centrality values [49; 50; 51]. This cartographic system allows identification through a statistical threshold level, different roles of the nodes in a network. Hub nodes are central nodes, so they are characterized by high \(z\) values. In addition, hubs can be divided into three different categories: provincial hubs, i.e., hub nodes with the vast majority of links within their module (lower values of \(PC_{i}\)); connector hubs with many links to most of the other modules and kinless hubs with links homogeneously distributed among all modules (highest \(PC\) values). We used the 25 th and 75 th percentile of both metrics to divide the plane \((PC,z)\) into nine regions to identify the hubs of the networks. To compare the topologies of the three functional networks across all the sections of the subject, we included into the final three sets only the hubs that appear with a specific role in at least six sessions of each functional connectivity network.