## 4 Results and Discussions
### Modularity Analysis  
To explore the topological functional organization of the brain regions, the Louvain algorithm was computed 1000 for each network. Figure 4 shows the error bars of modularity index Q over the 1000 iterations of the algorithm for the optimum resolution value \(\gamma=1\). It is worth noting that the modularity index is related to the difference between the within-module interactions and the between-module interactions of a network, so it statistically quantifies the goodness of a network partition into modules [52]. Here, the SYNC network, exhibits a significantly higher value of Q with respect the Pearson and coherence networks for all sessions, pointing out a better hard partition into modules. This result confirms the findings of our previous work in which a modularity analysis was applied to networks defined with both the SYNC index and Pearson's correlation coefficient of task-related fMRI data [22]. In particular, both a greater Q value and a higher number of functional communities activated during the working memory task were better identified in SYNC networks, while in Pearson's correlation networks modularity patterns reflecting the working memory load were not detected during the task. The NMI values reported in Figure 5 highlight that the three partitions are quite different from each other for all the sessions. However, the NMI values between the Pearson and coherence networks are slightly higher, showing a higher degree of agreement between the partitions of these two networks. Figure 6 shows the most representative network partition across the fMRI session of the subject. Seven modules were detected in the SYNC matrix; the first module comprises areas almost becoming from the cingulate cortex, e.g., cingulate gyrus, anterior and mid cingulate. Several specialized control areas of frontal regions such as medial orbital superior frontal gyrus are included in the second module. The third module resembles both peripheral and central visual networks as it includes all the occipital lobe, the calcarine fissure and surrounding cortex. The fourth module resulted the largest as it includes all the other regions from the frontoparietal network. The fifth module overlaps with most regions of the cerebellar network; the sixth module includes: superior frontal gyrus, middle temporal gyrus, inferior temporal gyrus and cerebellar crus areas. These modules are equally distributed across the default and control networks previously described in the resting-state literature [53; 54]. The seventh is mainly composed by areas from the limbic system such as amygdala and para-hippocampal gyrus. Figure 3: Pearson correlation matrix (**a**), SYNC matrix (**b**) and coherence matrix (**c**) with the empirical probability distributions of the links of the three matrices in semi-logarithmic scale (**d**). In contrast, the Pearson and coherence network are composed by only four and three modules, respectively. The first module includes some central structures such as caudate nucleus, putamen and thalamus, the inferior frontal and orbital gyrus and several regions from cerebellar networks. The second module resulted the most specialized as it covers all the occipital lobe and several regions from cerebellar network. Most of the regions from parietal, frontal and temporal lobe are equally distributed across the first and third module. In particular, the third module comprises the cerebellar crus 1 and 2, all the areas of the superior and middle frontal lobe and orbital gyrus. The fourth module resulted the smallest one including most of the regions from cerebellum and vermis. The first module of the coherence network covers all the parietal lobe and middle frontal areas. The second module includes the areas of the occipital lobe, superior frontal, orbital and temporal regions and the cerebellar crus 1 and 2. Finally, the third module is mainly composed of the cerebellar network, inferior frontal gyrus and sub-cortical regions.