## 4 Discussion
### Symmetry between HbO\({}_{2}\) and HHb Connectivity Networks  
The analysis of random networks suggested a relation between density and the symmetry in FC networks (see Figure 3). In particular, we showed that the networks derived from fNIRS recordings exhibit symmetry values between HbO\({}_{2}\) and HHb above its random counterpart. In synthetic random networks, the symmetry value is higher when the connectivity density in both networks increases. This is consistent with the experimental observation that, for highly dense diffuse optical tomography spatial configurations, both HbO\({}_{2}\) and HHb networks include a high number of functional links [11]. Such higher channels density is often associated with a higher number of links i.e., denser networks (\(\ell\approx\) L). In this case, the decision of choosing one of the two Hb species is not crucial, as we expect high symmetries between them. In case of low channel density data sets, we cannot ignore the differences in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**FC Network** & \multicolumn{2}{c}{**Density**} & \multicolumn{2}{c}{**Jaccard Symmetry**} & \multicolumn{2}{c}{**DSI Symmetry**} \\ \hline
**Condition** & **With** & **Without** & **With** & **Without** & **With** & **With** & **Without** \\  & **Systemic** & **Systemic** & **Systemic** & **Systemic** & **Systemic** & **Systemic** \\ \hline Baseline (BL) & 0.22 \(\uparrow\) & 0.20 & 0.24 \(\uparrow\) & 0.20 & 0.15 \(\uparrow\) & 0.11 \\ \hline Ongoing (OG) & 0.25 \(\uparrow\) & 0.22 & 0.26 \(\uparrow\) & 0.18 & 0.18 \(\uparrow\) & 0.09 \\ \hline Social PM (SocPM) & 0.29 \(\uparrow\) & 0.27 & 0.18 \(\downarrow\) & 0.22 & 0.09 \(\downarrow\) & 0.12 \\ \hline Non-Social PM (NonSocPM) & 0.27 \(\uparrow\) & 0.25 & 0.14 \(\uparrow\) & 0.13 & 0.06 \(\uparrow\) & 0.04 \\ \hline Ongoing Condition (OGc) & 0.29 \(\uparrow\) & 0.24 & 0.11 \(\downarrow\) & 0.14 & 0.02 \(\downarrow\) & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Density, Jaccard and DSI symmetry changes with/without systemic information from the virtual-typical-subject. Experimental connectivity networks from baseline, ongoing, social prospective memory, non-social prospective memory, and ongoing contaminated conditions. Figure 7: Measures of integration (global efficiency) and segregation (modularity) in functional networks. The size, shape, colour and padding of markers encode the amount of symmetry, experimental condition, Hb signal and the inclusion or exclusion of systemic data, respectively. the symmetry between HbO\({}_{2}\) and HHb FC networks. Most of the current fNIRS technologies are still spatially low channel density devices [2], hence likely to be accompanied with a less dense-connectivity graph. Therefore, the symmetry between HbO\({}_{2}\) and HHb FC networks must be investigated and taken into consideration to draw more correct neuroscientific conclusions based on fNIRS-derived FC measures. Regarding the fNIRS networks presented here, the social and non-social prospective memory (SocPM and NonSocPM) conditions achieved more symmetric responses (Figure 6 bigger markers) between the HbO\({}_{2}\) and HHb FC networks. In this sense, this could be related the prospective memory task itself. In fact, prospective memory involves the integration of several executive functions (e.g., planning, retrospective/working memory, cognitive flexibility and inhibitory control, attentional monitoring, etc. [50]) and requires the conscious interruption of the OG task to fulfill the delayed intention, thus being more complex than only the OG task or the rest conditions. Our data suggests that the memory-related tasks within our current functional protocol produced higher symmetric FC networks between the Hb species compared to the other tasks. In addition, the walking-related tasks were found to produce more asymmetrical FC networks between the Hb species. These results refer to the particular prospective memory protocol described here.