## Materials and Methods
### Quantification of Network Symmetry  
Networks similarity was quantified in terms of the symmetry between the connectivity matrices (graph adjacency matrices) between networks, compared using the Jaccard index (Ji) [49]. The Ji of two connectivity matrices A and B is defined as:  
\[\mathrm{Ji}(\mathrm{A},\mathrm{B})=\frac{\left|\mathrm{A}\cap\mathrm{B} \right|}{\left|\mathrm{A}\cup\mathrm{B}\right|} \tag{1}\]  
where \(\left|\bullet\right|\) indicates the cardinality of the set. The Ji yields values in the range between 0 and 1, ranging from totally dissimilar to fully identical connectivity matrices, respectively. The Ji is a convenient metric to study the symmetry of functional networks because it ignores absent connections between graph nodes. Nevertheless, the symmetry is affected by the networks density. In general, comparing two densely connected networks, functional networks will result in a large Ji, whilst comparing two sparse networks will likely result in a low Ji. This is further discussed in Section 3.1. This is a mathematical artefact with no relation to physiology which should be discounted before interpreting symmetry. To separate the mathematical effect of the Ji index from the pure physiological effect, we propose a differential symmetry index (DSI) which discounts baseline Ji values expected for random networks. Let \(D_{HbO_{2}},D_{HHb}\) be the network density for the HbO\({}_{2}\) and HHb networks respectively. A graph density is calculated according to Equation (2):  
\[D_{HbX}=\frac{2\ell}{n(n-1)} \tag{2}\]  
with \(\ell\) being the number of links, \(n\) the number of nodes, and \(X\) represents one of the Hb species. Let \(\mathrm{f}(\cdot)\) be the chosen symmetry function (Ji in our case). Then the DSI is presented in Equation (3). \[\mathrm{DSI}=\left|\mathrm{f}(D_{HbO_{2}},D_{HHb})-\hat{f}(D_{HbO_{2}},D_{HHb })\right|, \tag{3}\]  
where \(\hat{f}(\cdot)\) is the expected baseline symmetry. The expected baseline symmetry can be obtained from an analytical model describing the symmetry values produced by random connectivity networks. \(\hat{f}(D_{HbO_{2}},D_{HHb})\) can be approximated by multiple linear regression, radial basis functions approximation or interpolation methods. We choose a bilinear interpolation because of its computational simplicity. Using bilinear interpolation, \(\hat{f}(D_{HbO_{2}},D_{HHb})\) is approximated according to Equation (4):  
\[\hat{f}(D_{HbO_{2}},D_{HHb})\approx a_{0}+a_{1}D_{HbO_{2}}+a_{2}D_{HHb}+a_{3} (D_{HbO_{2}}\cdot D_{HHb}), \tag{4}\]where the model coefficients \(\mathbf{a}=(a_{0},\ldots,a_{3})\) are found by solving the linear system in Equation (5). \[\left[ \begin{array}{cccc}1&D_{HbO_{2}}^{1}&D_{HHb}^{1}&(D_{HbO_{2}}^{1}\cdot D_{ HHb}^{1})\\ 1&D_{HbO_{2}}^{2}&D_{HHb}^{2}&(D_{HbO_{2}}^{2}\cdot D_{HHb}^{2})\\ \vdots&\vdots&\vdots&\vdots\\ 1&D_{HbO_{2}}^{m}&D_{HHb}^{m}&(D_{HbO_{2}}^{m}\cdot D_{HHb}^{m})\\ \end{array} \right]\left[ \begin{array}{c}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ \end{array} \right]\left[ \begin{array}{c}f(D_{HbO_{2}}^{1},D_{HHb}^{1})\\ f(D_{HbO_{2}}^{2},D_{HHb}^{2})\\ \vdots\\ f(D_{HbO_{2}}^{m},D_{HHb}^{m})\\ \end{array} \right], \tag{5}\]  
where, \(D_{HbO_{2}}^{m},D_{HHb}^{m}\) are the connection densities of the \(m\)-th FC network from oxy-haemoglobin (HbO2) and deoxy-haemoglobin (HHb), respectively. Note that the DSI represents the residuals of a symmetry model for random networks discounting the mathematical baseline and leaving what is assumed to be the physiological contribution. The DSI has been implemented in Matlab 2017b and can be downloaded at [[https://github.com/multimodalspectroscopy/DSI.git](https://github.com/multimodalspectroscopy/DSI.git)]([https://github.com/multimodalspectroscopy/DSI.git](https://github.com/multimodalspectroscopy/DSI.git)). The proposed DSI quantifies the similarity with respect to the structure between connectivity networks and has been constructed without knowledge of the active channels.