## 3 Results  
### Network Symmetry Analysis  
Understanding brain networks must be done with respect to random networks. For this reason, we simulated synthetic random networks as follows: pairs of random networks representative of the Hb species were generated starting with one random link (\(\ell_{1}=\ell_{2}=1\)) each one. Because our experimental dataset has 16 channels we fixed the random networks size to sixteen nodes (\(n=16\)). Henceforth, we continue increasing their density by randomly adding one link at time until both networks are fully connected (\(\ell_{1}=\ell_{2}=\) L with L = (\(n\times(n-1)\))/2). For each pair of edges cardinality \(<\)\(\ell_{1}\), \(\ell_{2}\)\(>\) we repeated the process 50 times. Then, the connection densities and their associated Ji were calculated for every scenario. Finally, the mean Ji across the 50 repetitions was calculated. Figure 3 shows the relationship between the density of networks and their average random Ji symmetry. Also, the symmetries between the HbO2 and HHb experimental networks retrieved from the six participants with systemic information is projected over the expected symmetry space in Figure 3. The experimental functional connectivity (FC) networks exhibit low densities but with comparatively higher symmetry than would be expected from random behaviour. Then, we can expect small absolute changes in both symmetry and density values when systemic variables are added. The symmetry of experimental FC networks (M = 0.2, SD = 0.07) significantly differ from the symmetry of the random networks pairs (M = 0.09, SD = 0.02) (\(t\)-test: \(t\)(509) = 26.39, \(p<\) 0.0001 assuming equal variances; C.I. 95% for the Difference (0.11,0.13)) (see Figure 3 blue rectangle). Focusing on the symmetry values due to physiological changes, we apply the DSI to the FC networks derived from Hb species as per Equation (3). First, from the Ji values of the random FC networks pairs, the \(\mathbf{a}\) coefficients of the bilinear interpolation in Equation (4) to approximate \(\hat{f}(\cdot)\) were computed (\(\mathbf{a}\) = (0.0768, \(-\)0.0484, \(-\)0.0485, 1.0512)). The mean Ji based symmetry from pairs of random networks and the approximated bilinear model are presented in Figure 4.  
Figure 5a shows the separation of the Ji based networks symmetry between the experimental and the expected baseline random values described by the bilinear model. Figure 5b shows the DSI values for fNIRS network. The distribution of the DSI values (residual differences between observed symmetry and the expected baseline counterpart) suggests that most fNIRS FC networks exhibit symmetry below DSI \(<\) 0.2. This suggests an empirical threshold for deciding when to proceed with connectivity analysis across both Hb; networks pairs with DSI \(<\) 0.2 could be considered as asymmetric, and the FC analysis including the two Hb-derived networks is encouraged. Figure 4: Symmetry space from random networks. Surfaces obtained from the Ji values (**a**) and the approximation from the bilinear model (**b**). The graphical comparison and the differences between both surfaces (**c**). Higher order interpolation models can yield a better approximation with increasing degree of complexity. Figure 5: (**a**) Projection of the symmetry values of RNIRS FC networks obtained by Jaccard index (blue dots) over the random baseline approximated by a bilinear model (green surface); (**b**) differential symmetry index (DSI) values grouped by condition and its distribution (left margin histogram). A threshold (grey dashed line) derived from the distribution of DSI values is also shown. Figure 3: Jaccard symmetry response from experimental and random connectivity networks pairs varying density. The x-axis represents HbO\({}_{2}\) density and the y-axis represents HHb density. Color encodes the extent of symmetry. The background symmetry field corresponds to values expected for random synthetic networks from disconnected to complete connected ([0\(-\)1]) in both networks.