## 1 Introduction  
Diffuse near-infrared light can be used to interrogate brain haemodynamics and oxygenation non-invasively [1; 2]. By measuring the light attenuation changes of the reflected light from the head, functional near infrared spectroscopy (fNIRS) quantifies the relative changes in the brain tissue concentrations of oxygenated (\(\Delta\)[HbO\({}_{2}\)]) and deoxygenated (\(\Delta\)[HHb]) haemoglobin in response to neuronal activation in the cerebral cortex. The bivariate nature of the paired haemoglobin reconstruction conveys rich information about the brain haemodynamics and oxygenation but increases the complexity of the analysis. Brain connectivity expresses patterns of co-activity among brain regions and it is often investigated following two major types of relations, (i) associative or simply functional connectivity (FC); (ii) causal referred to as effective connectivity [3; 4]. It is accepted that fNIRS is sensitive to the haemodynamic response subsequent to the neuronal activity. Ignoring the transient inverse response phase, the neuronal activity is inferred from the concomitant increase in HbO\({}_{2}\) and decrease in HHb [1; 5], consistent with our current understanding of the neurovascular coupling [6; 7]. Most approaches for the analysis of connectivity yield a graph's binary adjacency matrix characterizing the connectivity network. These approaches have two inherent shortcomings: (i) the binarization depends on a threshold that severely affects the graph density with important implications for interpretation [8]; (ii) a step to decide which haemoglobin (Hb) signal is to be used for subsequent analysis or whether analysing both Hb species is needed. Many researchers have studied different aspects of FC with fNIRS [9; 10; 11; 12; 13; 14; 15]; however, the analysis of connectivity maps is often done only using the fNIRS HbO\({}_{2}\) time courses. It is important to understand whether during the analysis both Hb species time courses are relevant, or if either provide analogous information and hence univariate analysis of connectivity may sufficient. In fNIRS connectivity analysis, an important decision is to opt for one of the haemoglobin species to generate the connectivity network of interest, whether the HbO\({}_{2}\) or HHb. A part of the scientific community favours the connectivity networks derived from the HbO\({}_{2}\) signal due to its higher signal-to-noise (SNR) ratio. Another part favours an analysis based on the HHb signal arguing its higher specificity to activity and higher robustness to physiological contaminations [16]. The rest of the community opts either for considering two univariate analysis or a true bivariate analysis [17]. A critical evaluation has not yet been performed on whether the connectivity network revealed by either haemoglobin species are equivalent - and thus a univariate analysis is enough--or not--and thus bivariate analysis should be employed considering both Hb species. This issue has not been investigated yet, as it might not be well understood what kind of information on brain connectivity is shared between the two Hb connectivity networks and how to quantify such information. Considering that HbO\({}_{2}\) and HHb often exhibit highly anticorrelated patterns during brain activation, one might expect a strong match between the connectivity graphs generated from either signal. However, this is not always the case. For instance, classical reconstruction methods based on inverting the modified Beer-Lambert law introduce cross-talk between the two signals species [6], further artifactually supporting the univariate argument of equivalent results. In addition, these fNIRS haemodynamic signals have notable differences in their contribution from factors such as systemic physiology [18; 19; 20]. When two brain regions are co-active, this may conceal any substantial differences in the HbO\({}_{2}\) and HHb network structures.