## Materials and Methods
### Functional Connectivity Analysis  
Often, brain connectivity investigations are aimed to draw inferences from a cohort of participants at group-level. For a group-level analysis, three typical approaches are: (i) an averaged representation typically called virtual-typical-subject (VTS) [36; 37; 38]; (ii) the individual-structure (IC); (iii) the common-structure (CS). The VTS representation uses an averaged-data version of the cohort, whereby a model is learned. The IS approach considers a single model for each subject, and, after a sort of model marginalization generates a single model with coinciding patterns among them. For the CS approach a unique model is learned (as in VTS). However, the set of individual parameters are preserved for each subject. Our aim is to analyse how the contribution derived from systemic factors affects the asymmetry between HbO\({}_{2}\) and HHb connectivity networks. In order to articulate our results (see Section 3), we opt for a group-level analysis by using a VTS representation. Connectivity networks were generated for each Hb signal and each condition using a Bayesian network (BN) [39]. Bayesian networks were previously used for the investigation of brain connectivity in studies such as [36; 37; 38; 40; 41; 42; 43; 44; 45]. Formally, a BN is a pair <G(V, E), \(\theta\)> for encoding the structure or skeleton G(V, E) and the parametrization of the network \(\theta\). The structure G(V, E) is a directed acyclic graph conformed to a set of nodes (V) and links between them (E). For our purpose, the sets V and E represent the set of brain regions (channels measurements) and the functional associations between them, respectively. The BNs framework determines the connectivity between pairs of observed nodes (fNIRS channels measurements) by examining the probabilistic independence relationships between them. Statistical independence tests are carried out so that the links in a connectivity network can be established. There are many algorithms described in the literature for automatically learning of the structure G which either consider the directionality of the links or not. In our case, we have used the PC algorithm [34] (PC stands for the initials of its creators [PERSON] and [PERSON]) as we are dealing with functional connectivity (FC), and thus we only need the discovery of statistical associations between fNIRS channels regardless of the direction. Hence, we only considered the first phase of the PC algorithm that recovers the skeleton (undirected graph) of the BN. In order to do so, each processed fNIRS time series was split according to experimental conditions into blocks for baseline (BS), ongoing (OG), social prospective memory (SocPM), non-social prospective memory (NonSocPM), and ongoing contaminated (OGc). The block-splitting task was done using the start and end timestamps obtained from the analysis of video recordings. We then obtained an averaged block for every condition according to the VTS group-level analysis. Four connectivity networks are retrieved for every condition corresponding to the two-haemoglobin species (HbO\({}_{2}\), HHb) and the inclusion or not of systemic information--heart rate (HR) and breathing rate (BR)--as nodes in the network. Figure 2: Preprocessing flowchart. fNIRS measurements were acquired by using a wearable optical topography fNIRS system. Then, the haemoglobin (Hb) signals were reconstructed and preprocessed, and the systemic information was added. A set of connectivity networks were determined by the PC [34] algorithm by using the implementation in pcalg R package [35]. Finally, the symmetry between functional connectivity networks and global efficiency and modularity across networks were computed. To contextualise our findings with relevant features of the recovered FC networks, we computed the global efficiency (GE) and modularity (MOD), which are summary measures of functional integration and functional segregation in complex networks analysis, respectively. GE measures how efficiently information is exchanged over the network [46].