## 1 Introduction
Differences in structural connections of FC networks can be described quantitatively by simple enough methods such as the Jaccard index (Ji). Nevertheless, the Ji is sensitive to the density of links in each network, which can lead to suggest two networks to be symmetric only because the number of connections is high. The FC analysis with fNIRS has mostly focused on describing and understanding the cortical connections within the normal healthy brain. However, when the connectivity analysis is expanded to the diseased brain, there can be various pathological conditions that involve the alteration of neurovascular coupling mechanisms (e.g., Alzheimer's disease or stroke) and thus differences in the dynamics and in the connectivity networks of HbO\({}_{2}\) and HHb might occur. In addition, having now the capacity to monitor functional brain activity on freely moving people in the real-world through wearable fNIRS systems, bigger changes in systemic physiology might occur. Current investigations have expanded the research on more complex realistic settings which consider, for example, inter-personal interactions [21; 22]. Therefore, the analysis of data recorded in such ecologically-valid settings might require further consideration of which haemoglobin signal is the best indicator of FC due to differences in systemic interferences between the signals. The amount of in-common information between HbO\({}_{2}\) and HHb species has been estimated mainly as a complementary instead of a guiding task for the connectivity analysis. Correlations between the BOLD signal and the haemoglobin species hinted a latent incongruence between HbO\({}_{2}\) and HHb responses [23] with [PERSON] et al. [24] suggesting to address the issue prior the analysis. [PERSON] et al. [25] found different symmetric patterns across brain areas; for instance, while in the visual cortex the HbO\({}_{2}\) and HHb exhibited symmetry, in the motor cortex such symmetry was not found. These asymmetries may also occur across frequency bands [26]. More importantly, such asymmetry may substantially alter the FC analysis itself, for instance using during a independent component analysis (ICA) when analysing resting state connectivity [12] as ICA-based analysis is capable of removing certain types of noise and artefacts [27]. Despite these early studies, more studies are needed regarding the understanding of the asymmetry/symmetry between Hb species. There are very few studies that investigated the symmetry of Hb species prior to the FC analysis and none has studied the symmetry directly from the Hb connectivity networks. However, in the context of FC networks, we emphasise that any symmetry analysis metric should take into account the structural features of these networks. In this paper, our goal is to quantify differences (if any) between the FC networks retrieved from the analysis based on each Hb species, both at baseline and during task-evoked responses, in order to understand how these two Hb derived networks are related. To achieve this, we describe a new metric for the quantification of the symmetry between the FC networks of HbO\({}_{2}\) versus the HHb. In addition, we introduce FC analysis between fNIRS signals and systemic variables, allowing for the first time an analysis approach that takes into consideration systemic driven fNIRS changes. We derived a set of connectivity networks for both haemoglobin species complicated functional study in freely moving participants in an ecological situation. We select an out-of-lab situation due to the induced walk-related systemic changes whereby the observed response would be affected and therefore also the expected symmetry values. In this sense, we hypothesize that in healthy subjects during certain brain functional experiments these non-neuronal confounding factors disproportionately affect the structure of the connectivity networks causing a decrease in their symmetry.