PhD Ardalan Aarabi
email: ardalan.aarabi@u-picardie.fr

T J Huppert

Assessment of the effect of data length on the reliability of resting-state fNIRS connectivity measures and graph metrics

Keywords: data length, reliability analysis, stability analysis, Pearson's correlation, percentage bend correlation, mutual information, partial correlation, graph metrics, functional connectivity analysis, resting state fNIRS

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Functional near-infrared spectroscopy, a non-invasive optical imaging technique, uses nearinfrared light to measure variations in concentrations of oxyhemoglobin (HbO) and deoxyhemoglobin (HbR) in the cortex with a limited light penetration depth of up to 2 cm in adults [START_REF] Fukui | Monte Carlo prediction of near-infrared light propagation in realistic adult and neonatal head models[END_REF]. With a much higher temporal resolution compared to fMRI, fNIRS can be used to investigate resting-state functional brain connectivity [START_REF] White | Resting-state functional connectivity in the human brain revealed with diffuse optical tomography[END_REF][START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF][START_REF] Li | Dynamic functional connectivity revealed by resting-state functional near-infrared spectroscopy[END_REF][START_REF] Novi | Resting state connectivity patterns with nearinfrared spectroscopy data of the whole head[END_REF] by avoiding aliasing of higher frequency (cardiac or respiratory) activities, which have shown to account for a significant fraction of variance in resting-state fMRI (rsfMRI) signals in low-frequency (< 0.1 Hz) bands [START_REF] Biswal | Simultaneous assessment of flow and BOLD signals in resting-state functional connectivity maps[END_REF][START_REF] Lowe | Correlations in low-frequency BOLD fluctuations reflect cortico-cortical connections[END_REF][START_REF] Cordes | Frequencies contributing to functional connectivity in the cerebral cortex in "resting-state" data[END_REF].

Functional connectivity analysis has become a popular method to investigate connectivity patterns by analyzing temporal dependencies between fluctuations of functionally related brain regions under normal and pathological conditions using different modalities [START_REF] Rogers | Assessing functional connectivity in the human brain by fMRI[END_REF][START_REF] White | Resting-state functional connectivity in the human brain revealed with diffuse optical tomography[END_REF][START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF][START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Li | Dynamic functional connectivity revealed by resting-state functional near-infrared spectroscopy[END_REF][START_REF] Adebimpe | Functional Brain Dysfunction in Patients with Benign Childhood Epilepsy as Revealed by Graph Theory[END_REF][START_REF] Adebimpe | EEG Resting State Functional Connectivity Analysis in Children with Benign Epilepsy with Centrotemporal Spikes[END_REF][START_REF] Novi | Resting state connectivity patterns with nearinfrared spectroscopy data of the whole head[END_REF][START_REF] Barzegaran | Functional connectivity analysis in EEG source space: The choice of method[END_REF]Brookes et al., 2017). Functional connectivity analysis of rsfNIRS signals has provided promising results for identification of resting state brain networks within the sensorimotor, visual, auditory, and language regions similar to those reported in rsfMRI studies [START_REF] White | Resting-state functional connectivity in the human brain revealed with diffuse optical tomography[END_REF][START_REF] Homae | Development of global cortical networks in early infancy[END_REF][START_REF] Lu | Use of fNIRS to assess resting state functional connectivity[END_REF][START_REF] Zhang | Functional connectivity as revealed by independent component analysis of resting-state fNIRS measurements[END_REF][START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF]. Graph theoretical approaches have proven to be very useful for characterizing the functional structure of brain networks using resting state fNIRS/fMRI data [START_REF] Sporns | Organization, development and function of complex brain networks[END_REF][START_REF] Niu | Revealing topological organization of human brain functional networks with resting-state functional near infrared spectroscopy[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF][START_REF] Vico Fallani | Graph analysis of functional brain networks: practical issues in translational neuroscience[END_REF][START_REF] Andellini | Test-retest reliability of graph metrics of resting state MRI functional brain networks: A review[END_REF]. For graph theoretical analysis of brain networks, one requires to represent whole brain networks as a graph of nodes (brain regions) and edges (interactions between regions), defined based on functional associations between brain entities. To date, a few studies have attempted to characterize topological properties of wholebrain functional brain networks using graph metrics and fNIRS data [START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF][START_REF] Niu | Revealing topological organization of human brain functional networks with resting-state functional near infrared spectroscopy[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF][START_REF] Li | Dynamic functional connectivity revealed by resting-state functional near-infrared spectroscopy[END_REF]. The validation of connectivity patterns largely relies on TRT reliability of graph metrics [START_REF] Nakagawa | Repeatability for Gaussian and non-Gaussian data: a practical guide for biologists[END_REF] across different sessions and subjects.

In the past decade, several rsfMRI studies have been conducted to investigate the effect of different connectivity metrics and scan duration on the functional structure of brain networks and the reliability of graph metrics [START_REF] Smith | Network modelling methods for FMRI[END_REF][START_REF] Whitlow | Effect of resting-state functional MR imaging duration on stability of graph theory metrics of brain network connectivity[END_REF][START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF][START_REF] Liao | Functional brain hubs and their test-retest reliability: a multiband resting-state functional MRI study[END_REF][START_REF] Cao | Test-retest reliability of fMRI-based graph theoretical properties during working memory, emotion processing, and resting state[END_REF]. So far, a few studies have investigated the reliability of resting-state functional connectivity (RSFC) patterns and global and regional graph metrics using rsfNIRS data (Zhang et al., 2011;[START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF][START_REF] Blasi | Test-retest reliability of functional near infrared spectroscopy in infants[END_REF][START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF]. Overall, it is still not well understood how the methodological choices including connectivity metrics and data length can affect the topological properties of rsfNIRS-based RSFC.

Concerning connectivity metrics, most fNIRS studies of the brain's intrinsic functional connectivity have used Pearson's correlation or cross correlation to measure linear associations between brain regions [START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF][START_REF] Li | Dynamic functional connectivity revealed by resting-state functional near-infrared spectroscopy[END_REF][START_REF] Novi | Resting state connectivity patterns with nearinfrared spectroscopy data of the whole head[END_REF][START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF]. These metrics are not capable of capturing the general linear/nonlinear dependency or the direct interdependence between brain regions after ruling out the effect of other regions [START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Smith | Network modelling methods for FMRI[END_REF].

With regard to the scan length, to the best of our knowledge, there has been only one study [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF]) so far conducted to evaluate the effect of data length on the reliability and stability of a few graph metrics using rsfNIRS data. They have performed a reliability analysis on binarized brain networks constructed using whole-brain rsfNIRS data and the linear correlation analysis. Thresholding has shown to seriously affect network metrics values especially in networks with limited number of nodes [START_REF] Vico Fallani | Graph analysis of functional brain networks: practical issues in translational neuroscience[END_REF][START_REF] Garrison | The (in)stability of functional brain network measures across thresholds[END_REF][START_REF] Hahn | A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain[END_REF].

In this study, we performed a comprehensive assessment of the effect of data length on the reliability and stability of four functional connectivity metrics (PC, BC, MI and PtC) using rsfNIRS data. We further investigated the intersession reliability of twelve global and regional graph measures commonly used for characterizing functional integration and segregation of information of brain networks with different data lengths.

Materials and methods

Subjects and experimental setup

The resting state fNIRS data collected from 13 healthy adult subjects (35 ± 12 years old) were included in this study. The experimental setup and dataset have been reported in other studies [START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF][START_REF] Novi | Resting state connectivity patterns with nearinfrared spectroscopy data of the whole head[END_REF][START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. To collect the data, subjects have been instructed to remain still in a quiet dark room. From each subject, 300-sec rsfNIRS data have been collected in two separate sessions. All procedures have been approved by the Institutional Review Board at the Massachusetts General Hospital, where written informed consents were obtained from all participants. All measurements have been collected with a continuous-wave near-infrared optical system (CW5, TechEn Inc., Milford, MA) using 32 laser diodes at 2 wavelengths (690 and 830 nm), 32 avalanche photodiodes (APD) and a sampling rate of 10 Hz [START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF]. Fifty NIRS channels (source-detector pairs with a spatial separation of 3 cm) have been positioned on the head, covering up frontal, parietal, temporal and occipital lobes (Fig. 1).

Data preprocessing

The optical data were converted to relative changes in optical density as described in [START_REF] Huppert | HomER: a review of timeseries analysis methods for near-infrared spectroscopy of the brain[END_REF]. The modified Beer-Lambert law was used to calculate the concentration change of HbR and HbO based on the differences in the absorption for the two wavelengths. The concentration in HbT was calculated by summing the concentrations in HbO and HbR. The average differential path-length factor was assumed to be 6 for both wavelengths [START_REF] Duncan | Optical pathlength measurements on adult head, calf and forearm and the head of the newborn infant using phase resolved optical spectroscopy[END_REF]. The time courses of HbO, HbR and HbT signals were off-line band-pass filtered between 0.01 and 0.1 Hz using a second-order Butterworth filter to remove baseline drift and systemic physiological noise including respiration and cardiac activity. For each subject, we further removed motion artifacts by using the spatial ICA and excluded noisy channels as described in [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. To study the effect of data length on the reliability of functional connectivity measures and graph metrics, the concentration signals were segmented into overlapping epochs with lengths of 30, 60, 90 (short range), 120, 150, 180 (medium range), 210, 240 and 270 (long range) with an increment of 1 sec. To capture the overall changes in FC patterns, for each data length, 50 epochs were randomly selected based on the method largely used for nonlinearity tests using surrogate data (Schreiber and Schmit, 2000). Based on this method, the minimum number of segments used for FC analysis should be at least 39 for twosided statistical tests with a significance level of p<0.05. For the data length of 270 sec, 30 epochs were selected from the overlapping segments and the remaining 20 epochs were randomly selected through random segmentation of the 300-sec data collected in each session.

Functional connectivity analysis

We used the rsfNIRS data to examine the reliability of four FC measures across two sessions using the TRT analysis [START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF]. To construct a connectivity matrix for each data length, we used four connectivity measures including Pearson's correlation, percentagebend correlation, mutual information and partial correlation [START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Pernet | Robust correlation analyses: False positive and power validation using a new open source matlab toolbox[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF][START_REF] Kruschwitz | GraphVar: a user-friendly toolbox for comprehensive graph analyses of functional brain connectivity[END_REF][START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF]). Pearson's correlation was first used to measure the degree of linear association between the time series of fNIRS channels (X↔Y) as:

∑ ∑ ∑ = = = - - - - = = N i i N i i N i i i y y x x y y x x Y X Y X r 1 2 1 2 1 ) ( ) ( ) )( ( ) var( ). var( ) , cov( (1)
where x and y denote the average of X and Y, respectively.

As an alternative to Pearson's correlation, we used the percentage-bend correlation, which has shown to be robust to outliers associated with marginal (X and Y) distributions by down weighting a specified percentage (20% herein) of marginal observations deviating from the median [START_REF] Pernet | Robust correlation analyses: False positive and power validation using a new open source matlab toolbox[END_REF]. The percentage-bend and Pearson's correlation differ markedly when there is deviation from normality [START_REF] Wilcox | The percent-age bend correlation coefficient[END_REF].

We further used the mutual information [START_REF] Hlinka | Functional connectivity in resting-state fMRI: is linear correlation sufficient?[END_REF] to measure the general dependency (linear/nonlinear) between fNIRS channels. MI estimates the amount of information shared between two time series (X↔Y) as:

) , ( ) ( ) ( ) , ( Y X H Y H X H Y X I - + = (2)
Where H(X) and H(Y) are the entropy of X and Y, respectively, and H(X,Y) is the joint entropy measuring the uncertainty associated with both X and Y.

In general, correlation values between two channels may result from indirect relationship caused by sequential pathways. To quantify this relationship, we used the partial correlation [START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF] to prevent a false prediction of direct links between any two brain regions (fNIRS channels). The partial correlation | was computed by measuring the strength of association between any two channels (X↔Y) by controlling for the remaining channels (Z) as:

) 1 )( 1 ( 2 2 YZ XZ YZ XZ XY Z XY r r r r r r - - - = (3)
where XY r , XZ r and YZ r are the unpartialized correlation coefficients between X↔Y, X↔Z, and Y↔Z, respectively.

For each data length and epoch, four weighted 50 × 50 connectivity matrices were generated by computing each of the connectivity measures between the fNIRS channels. We computed the absolute correlation values in each weighted matrix to only consider synchronization measures scaling from unsynchonized (zero value) to synchronized (with positive/negative values). For each subject, session and FC measure, the connectivity matrices were then averaged over the fifty epochs randomly selected for each data length to create an average connectivity matrix per data length.

Network measures

We investigated the reliability of twelve global network metrics, degree, characteristic path length, global efficiency, clustering coefficient, graph radius and diameter, transitivity, betweenness, modularity, participation coefficient, vulnerability and hierarchy largely used to characterize functional integration and segregation of brain networks [START_REF] Sporns | Organization, development and function of complex brain networks[END_REF][START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. We also explored the reliability of the local properties of the constructed functional brain networks with degree, shortest path length, local efficiency, clustering coefficient, betweenness and vulnerability at nodal level [START_REF] Watts | Collective dynamics of ' small-world[END_REF][START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF].

Since thresholding might significantly affect the topological properties of brain functional networks [START_REF] Vico Fallani | Graph analysis of functional brain networks: practical issues in translational neuroscience[END_REF][START_REF] Garrison | The (in)stability of functional brain network measures across thresholds[END_REF][START_REF] Hahn | A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain[END_REF], we performed the weighted functional connectivity analysis by computing the graph theoretical measures directly from the average weighted connectivity matrices. In weighted graphs, edges are characterized with connection strengths and the entire connectivity matrix is used to compute weighted graph metrics rather than sparse binary adjacency matrices [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. Since each connectivity matrix contained N (herein 50) nodes connected by (N-1) undirected edges, the total number of edges in the graph was . Each of the graph metrics used in this study is briefly described below (see [START_REF] Watts | Collective dynamics of ' small-world[END_REF] and [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF] for more details).

Degree

Degree is a measure of a node's centrality in the network. In a weighted network, the weighted degree ( w i k , also called strength) of a node i is defined as:

∑ = = N j ij w i k 1 ω (4)
Where ωij is the connection strength between nodes i and j, and N is the total number of nodes in the network. Based on this definition, high-strength nodes can directly interact with a large number of other nodes. K w , the average degree of the network, is defined as:

∑ = = N i w i w k n K 1 1 (5)

Characteristic and shortest path length

Characteristic path length is a measure of functional integration in brain networks. The characteristic path length (L w ) for a weighted network is defined as:

∑ ∑ ∑ = ≠ = = - = = N i N i j j w ij N i w i w n d n l n L 1 , 1 1 1 1 1 (6)
Where w i l is the average shortest path length between node i and all other nodes, and w ij d is the shortest weighted distance between nodes i and j. The characteristic path length of a network is the average shortest path length between all pairs of nodes in the network. It indicates that, on average, each node can be reached from any other node along a path composed of only a few edges.

Radius, diameter and center of graph

In a graph, the eccentricity (εi) of a node is defined as the largest distance from the node to any other node. The radius (r w ) and diameter ( w ρ ) of the graph is the minimum and maximum eccentricity of any node in the graph, respectively [START_REF] Hernández | Classification of graph metrics[END_REF], defined as:

i N i w r ε min ∈ = i N i w ε ρ max ∈ = (7)
The radius and diameter are measures of how close/far nodes are to each other in a network. For a highly-connected self-centered graph, the radius and diameter are equal (Buckley, 1998).

Clustering coefficient

Clustering coefficient is a measure of local segregation in a network by measuring the degree to which nodes tend to cluster together. The average clustering coefficient ( w C ) of a weighted network is defined as:

∑ ∑ = = - = = N i w i w i w i N i w i w k k t n c n C 1 1 ) 1 ( 2 1 1 (8) 10
Where w i c is the weighted clustering coefficient for node i and w i t is the weighted geometric mean of triangles around node i, measuring the density of connections between the node's neighbors.

Transitivity

As an alternative to the clustering coefficient, transitivity measures the probability for the network to have tightly connected adjacent nodes or clusters [START_REF] Onnela | Intensity and coherence of motifs in weighted complex networks[END_REF][START_REF] Fagiolo | Clustering in complex directed networks[END_REF]. The transitivity (T w ) of a weighted network is defined based on the relative number of triangles in the graph compared to the total number of connected triples of nodes:

∑ ∑ = = - = N i w i w i N i w i w k k t T 1 1 ) 1 ( 2 (9)
Transitivity and average clustering coefficient of all nodes contain the same information about the level of segregation of the network, however, transitivity is less affected by low-degree nodes [START_REF] Chakrabarti | Graph Mining: Laws, Tools, and Case Studies, Sythesis Lectures on Data Mining and Knowlege Discovery[END_REF].

Local and global efficiency

The local efficiency of a node describes how efficiently information is propagated over the node's direct neighbors. The local efficiency of a node is computed as the inverse of the average shortest path connecting all neighbors of the node. The global efficiency describes how efficiently information is propagated through a whole network [START_REF] Latora | Efficient behavior of small-world networks[END_REF]. The global efficiency is computed as the average of the inverse of the distance matrix. A high global efficiency indicates that all pairs of nodes in a graph, on average, have short communication distances and can be reached in a few steps [START_REF] Sporns | Networks of the Brain[END_REF].

Modularity

Modularity measures the extent to which a graph can be divided into separated submodules with dense within-submodule and sparse between-submodule connections using the community structure. The modularity (Q w ) of a weighted network is defined as:

j i m m N i N j w j w i ij w l k k w l Q , 1 1 1 δ ∑∑ = =         - = (10)
Where mi and mj are the modules containing nodes i and j, respectively, l is the total number of edges in the graph, wij is the connectivity matrix, w i k and w j k are the degrees of nodes i and j, respectively, and j i m m , δ is 1 if nodes i and j belong to the same module and 0 otherwise [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF].

Betweenness centrality

Betweenness reflects a node's centrality measuring the extent to which the node is crossed by shortest paths in a graph. It measures the influence of any nodes on information flow between other nodes. The weighted betweenness centrality (

w i b ) of node i is computed as: ∑ ≠ ≠ ≠ = - - = N i j i h j h j h hj w hj w i i n n b , , 1 , ) ( ) 2 )( 1 ( 1 η η (11)
Where w hj η is the number of shortest paths between h and j, and

) (i w hj
η is the number of shortest paths between h and j that pass through i [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. A node with higher betweenness has higher control over interactions between dense subnetworks connected to it. The average betweenness (B w ) for a weighted network is computed by averaging the betweenness centralities of all nodes in the network.

Participation coefficient

Participation coefficient is a measure of the distribution of the edges of nodes or diversity of intermodular connections of nodes [START_REF] Joyce | A new measure of centrality for brain networks[END_REF][START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. The participation coefficient (Pi) of a node i is defined as

∑ =         - = M m w i w i i k m k P 1 2 ) ( 1 (12) Where ) (m k w i
denotes the number of connections between node i and other nodes within module m in a weighted network with M as the total number of modules, w i k as the degree of node i, and

w i w i k m k / ) (
as the ratio of connections that node i has within its own module [START_REF] Costa | Characterization of complex networks: A survey of measurements[END_REF][START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. The participation coefficient of a node is close to one if it has equal links to all modules of the network, and zero if its entire links remains within its own module. The average participation coefficient (P w ) of all nodes in a network measures the average diversity of all connections across modules.

Vulnerability

In a network, any damage to highly connected nodes can strongly affect the performance of the network [START_REF] Costa | Characterization of complex networks: A survey of measurements[END_REF]. The vulnerability of a node measures the drop in the network performance when the node and all its edges are removed from the network [START_REF] Costa | Characterization of complex networks: A survey of measurements[END_REF]:

E E E V i i - = (13)
where E and Ei are the global efficiency of the network before and after the removal of node i and all its edges. The vulnerability of a network is assessed [START_REF] Latora | Vulnerability and protection of critical infrastructures[END_REF] by the maximum vulnerability for all of its nodes:

i i w V V max = , N i ,..., 1 = (14)

Hierarchy

Hierarchy in a network can be quantified by investigating the power law relationship between the clustering coefficient and degree of all the nodes in the network as:

β - ≈ k C (15)
A linear regression analysis is performed between log(C) and log(k) to estimate the hierarchy (β w ) of the weighted network. A network with β w lower than 1 has limited hierarchy among all its nodes.

We computed the twelve graph metrics to quantify the topological properties of the brain networks at the global and nodal levels for each subject and data length.

Stability analysis

We used non-parametric Friedman tests (with p<0.05) using post-hoc multiple comparison tests to identify stability zones, within which the FC measures and graph metrics show stabilization with increasing/decreasing data lengths [START_REF] Fraschini | The effect of epoch length on estimated EEG functional connectivity and brain network organisation[END_REF]. To find stability zones, the whole data length range was searched for each FC measure or graph metric to identify the time interval(s), within which no significant differences were observed in correlation or graph metric values with increasing or decreasing data lengths in the interval(s). We also assessed the statistical differences between the correlation values and graph metrics derived by each of the four connectivity measures using repeated measures analysis of variance (ANOVA, with p<0.05). Post-hoc paired t-tests were also used to characterize differences between the FC measures (with p<0.05).

Test-retest reliability assessment

We further computed the Intraclass Correlation Coefficient (ICC) to assess the test-retest reliability of the FC measures and graph metrics by comparing the variability of these measures across the two sessions of the same subject to the total variation across sessions and subjects for each of the three concentration signals (HbO, HbR and HbT). To this end, a matrix was created with 13 rows (subjects) and two columns (sessions) for each FC measure and graph metric. We then used the one-way random effect model [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF][START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF] to compute ICC as follows:

w s b w b v N v v v ICC ) 1 ( - + - = (16)
where vb is the between-subject variance, vw is the within-subject variance and Ns is the number of sessions per subject. ICC varies between 0 (low reliability) and 1 (high reliability). Since ICC might have negative values indicating non-reliable estimation when the within-subject variance was larger than the between-subject variance, we set negative values of ICC to zero to avoid misinterpretation [START_REF] Rousson | Assessing intrarater, interrater and test-retest reliability of continuous measurements[END_REF]Zhang et al., 2011). All analyses were performed using custom codes written in Matlab (MATLAB R2014a, The MathWorks Inc.), the Brain Connectivity Toolbox [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF], Graphvar toolbox [START_REF] Kruschwitz | GraphVar: a user-friendly toolbox for comprehensive graph analyses of functional brain connectivity[END_REF].

Results

Stability and reliability of FC measures

Figure 2 and Table S1 show the average results of the intra-session FC analysis as a function of data length for HbO, HbR and HbT. As shown, all the FC measures showed decreases in mean values with increasing data length. A slight increase in the standard deviation was observed with increasing data length for PC and BC.

Comparing the FC measures using the repeated-measures ANOVA (p<0.05) showed no significant differences between PC and BC. MI, however, displayed significantly lower mean values in comparison with PC and BC. The mean values of PtC were much lower than those obtained for the other three measures, with a curve decreasing with data lengths longer than 150 sec, while the FC curves for PC, BC and MI decreased. Post-hoc testing using paired t-tests revealed significantly lower PC, BC, and MI mean values for HbR compared to HbO and HbT for all data lengths. For PtC, HbR exhibited lower mean values in comparison with HbO and HbT for data lengths between 60 and 150 sec. For all the four FC measures, mean values for HbT were relatively higher than those found for HbO and HbR.

Figure 2 and Table S1 also show the results of the stability analysis, illustrating intervals within which no significant changes in FC values were found between different data lengths. As shown for PC and BC, the stability zones started at data lengths of 90 sec for HbO and 150 sec for HbR. The stability zones for HbT was longer starting earlier from 60-90 sec. MI stabilized at medium-range data lengths (150-180 sec). Interestingly, PtC showed stability for data lengths shorter than 150 sec for HbO and HbR, and between 90 and 150 sec for HbT. Table 1 shows the mean Pearson correlation coefficients found between the resting-state functional connectivity (rsFC) maps computed for sessions 1 and 2 at the group level. To obtain rsFC maps, the mean connectivity matrices were first computed over all subjects for each data length and session. The correlation analysis was then performed between the rsFC maps of the two sessions for each data length. The statistical analysis revealed significantly positive correlations (p<0.05) indicating high similarity (reproducibility) between the rsFC maps for sessions 1 and 2 across all participants. Only the partial correlation showed instability with medium to long-range data lengths. Figure 3 shows the results of the TRT analysis performed on each FC measure for HbO, HbR and HbT. For each data length, boxplots show average ICC values across all channels and subjects. High reliability was found for PC and BC with mean ICC values greater than 0.8 for all data lengths and concentration signals. The repeated-measures ANOVA showed no significant differences (p>0.05) between PC and BC for all data lengths. For HbO, the reliability decreased slightly for data lengths longer than 120 sec. MI also showed high reliability (mean ICC greater than 0.75) for all data lengths, exhibiting more variability for HbO and HbT compared to HbR. The reliability of MI significantly decreased for HbO and HbT for data lengths longer than 120 sec. The reliability of MI for HbR was found to be high (mean ICC greater than 0.8) with no significant variation across data lengths longer than 60 sec. Compared to the other FC measures, PtC yielded the lowest reliability (mean ICC values between 0.6 and 0.8) for data lengths shorter than 150 sec. The reliability of PtC decreased significantly with medium to long-range data lengths with higher variations across subjects and channels.

Stability and reliability of network metrics

Figure 4 shows the boxplots of degree (k w ) and shortest path length (l w ) as a function of data length for HbO, HbR and HbT across all nodes and subjects for both sessions. The weighted graph metrics were derived by each FC measure. The results for other graph metrics studied can be found in Figure S1.

As shown, the average degree (K w ) decreased with increasing data length. The repeatedmeasures ANOVA revealed no significant differences (p>0.05) between PC and BC. MI and PtC showed significantly lower values in comparison with PC and BC. Global efficiency (

W G E ),
average clustering coefficient (C w ) and transitivity (T w ) also showed similar decreases with increasing data length (Figure S1). An inverse trend was observed for characteristic path length (L w ), graph radius (r w ) and diameter (ρ w ), modularity (Q w ), betweenness (B w ) and vulnerability (V w ) (Figure S1). The PC, BC and PtC-derived hierarchy (β w ) exhibited slight increases with increasing data length. For MI, however, the hierarchy measure displayed no significant decrease or increase for HbO and HbT, but it decreased for HbR with increasing data length. The participation coefficient values were highly variable across sessions and subjects with no specific decreasing or increasing pattern.

Figures 4 and S1 also show the results of the stability analysis for each graph metric and session. Almost for all the metrics studied, the stability zones started at short-range data lengths for HbO and HbT compared to HbR. The PC and BC-derived graph metrics stabilized at short-range data lengths in comparison with those derived by MI (Table 2). The PtC-derived metrics exhibited stability for data lengths shorter than 120 sec for HbO and HbR, and between 90 and 150 sec for HbT (Table 2). Figure 5 shows the distribution of the data lengths with which the stability zones started across all the graph metrics studied. To obtain these results, the onsets of the stability zones were determined for all the graph metrics. We then used the kernel density estimation method, a nonparametric technique, to estimate the kernel density function of the onsets over all graph metrics for each FC measure [START_REF] Rosenblatt | Remarks on Some Nonparametric Estimates of a Density Function[END_REF]. As shown, the majority of the PC and BC-derived graph metrics stabilized with a data length of 60 sec, 150 sec and 90 sec for HbO, HbR and HbT, respectively. The MI-derived graph metrics showed stability with data lengths of 150 sec or longer. In contrast, the graph metrics derived by PtC showed stability with short-range data lengths (90 sec or less) for HbO and HbR. For HbT, however, different graph metrics stabilized with data lengths varying between 60 to 150 sec. Figure 6 shows the ICC curves for the average degree (K w ) and characteristic path length (L w ) with different data lengths for HbO, HbR and HbT. As shown, K w and L w derived by PC, BC and MI showed high reliability (ICC > 0.8) for all data lengths. The lowest ICC with high variation across different data lengths were observed for the PtC-derived K w and L w .

The ICC curves for other metrics have been shown in Figure S2. Overall, K w , W G E , L w , C w , T w , r w , and ρ w were highly reliable when derived by PC, BC or MI for HbO, HbR and HbT regardless of data length. The reliability for the majority of the PtC-derived metrics was relatively moderate to high with data lengths shorter than 120 sec except for r w , and ρ w . We also found high reliability for B w , Q w , and V w for HbO, HbR and/or HbT. Among all the studied metrics, P w and β w were the least reliable metrics regardless of FC measure and concentration signal. S3. Overall, the ICC maps showed higher stability with increasing data lengths at the nodal level for the metrics computed by PC, BC and MI. The ICC maps of the metrics derived by BC and MI compared to PC were spatially more uniform across brain regions. In contrast, the reliability of the PtC-derived metrics was found to be only higher for data lengths shorter than 120 sec.

Discussion

In this study, we investigated the effect of data length on the reliability and stability of four functional connectivity measures and twelve graph theoretical metrics commonly used to study functional brain networks. We performed the test-retest reliability analysis using nine data lengths ranging from 0.5 to 4.5 min. Our results show that data length can significantly affect the results of FC analysis and the topological properties of functional brain networks derived from rsfNIRS data. The stability analysis showed that PC and BC stabilized with data lengths longer than 1-2.5 min depending on concentration signals. Compared to PC and BC, the stabilization for MI occurred with medium to long-range data lengths (longer than 2.5 min). The partial correlation showed stability only for short to medium-range data lengths (2.5 min or less) for HbO and HbR. For HbT, however, its stability occurred with data lengths between 1.5-2.5 min.

PC and BC showed high reliability with mean ICC values greater than 0.8 for all data lengths analyzed. For these two measures, the reliability was found to be higher for HbO/HbT than for HbR. The mutual information also showed high reliability (mean ICC > 0.75) for all data lengths. Interestingly, MI showed lower reliability for HbO and HbT compared to HbR for medium to long-range data lengths (> 2 min). Compared to the other three FC measures, PtC yielded the lowest reliability (with mean ICC values between 0.6 and 0.8) for short to mediumrange data lengths (shorter than 2.5 min).

Our results showed that the majority of the global and regional (nodal) graph metrics derived by PC, BC and MI stabilized with increasing data length. The PC and BC-derived graph measures showed stability with short to medium-range data lengths longer than 1, 2.5 and 1.5 min (Figure 5) for HbO, HbR and HbT, respectively. The MI-derived graph metrics required medium to long-range data length (longer than 2.5 min) to stabilize. The stability analysis on the network metrics derived by PtC revealed stability for short-range data lengths shorter than 1 min for HbO and HbR, and 2.5 min for HbT. Among all the metrics studied, the participation coefficient was found to be highly variable across sessions and subjects.

The TRT analysis showed that the reliability of the network metrics derived by PC, BC and MI could be improved by data lengths of at least 1.5 to 2.5 min, depending on FC measures and concentration signals. Compared to PC and BC, the MI-based functional connectivity analysis required medium to long-range data lengths (> 2.5 min) to achieve stable and reliable results for global and nodal metrics. The reliability of the PtC-derived network metrics differed significantly from those derived by the other three FC measures. The PtC-derived network metrics exhibited moderate to high reliability (with mean ICC >0.6) and stability only with shortrange data lengths shorter than 2 min for HbO, HbR and HbT. For the PC, BC and MI-based networks, degree, global efficiency, characteristic path length, clustering coefficient, transitivity, and graph radius and diameter exhibited high reliability (with mean ICC >0.8) with stability zones plateaued with short to medium-range data lengths longer than 1-3 min, depending on FC measures and concentration signals. For these networks, the betweenness, modularity and vulnerability measures also showed high reliability with increasing data length for HbO, HbR or HbT, but not for all of them. The participation coefficient, however, showed no specific pattern of changes or improvement with increasing data length. The hierarchy also showed variable reliability depending on data length.

Stability and reliability of FC measures

For FC analysis, we used absolute values of PC, BC, MI and PtC to compute graph metrics as it has been shown that negative correlation values can decrease TRT reliability of global network metrics [START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF] due to their higher variance in comparison with positive correlations. Negative correlations have shown to be more related with the global signal regression in resting state data [START_REF] Fox | The global signal and observed anticorrelated resting state brain networks[END_REF][START_REF] Weissenbacher | Correlations and anticorrelations in resting-state functional connectivity MRI: a quantitative comparison of preprocessing strategies[END_REF].

Our results showed no significant differences between correlation values estimated by PC and BC. This might reflect the normality of rsfNIRS data distributions as the results of the PC and BC-based connectivity analysis have shown to only differ when performed on data with non-Gaussian distributions [START_REF] Wilcox | The percent-age bend correlation coefficient[END_REF][START_REF] Rousselet | Improving standards in brain-behavior correlation analyses[END_REF]. However, we found less stability for the nodal graph metrics derived by PC in comparison with BC, more likely due to the sensitivity of PC to marginal outliers caused by noise [START_REF] Pernet | Robust correlation analyses: False positive and power validation using a new open source matlab toolbox[END_REF].

The partial correlation showed stability only for short to medium-range data lengths. In fMRI connectivity studies, Pearson's correlation and partial correlation have shown to provide good estimates of functional connections [START_REF] Smith | Network modelling methods for FMRI[END_REF]. However, the partial correlation's performance has shown to decrease significantly when the number of nodes in brain networks increases. In our study, we found significantly lower values for PtC compared to the other three measures. Its reliability also decreased significantly for long-range data lengths with higher variations across subjects and channels in comparison with the other FC measures.

As a measure of general linear/nonlinear statistical dependency between brain regions [START_REF] Fraser | Independent coordinates for strange attractors from mutual information[END_REF], MI exhibited lower values in comparison with PC and BC at the single subject and group levels and stabilized with medium to long-range data lengths. This finding is consistent with the fact that a good estimate of the true joint distribution for MI requires reasonably a large number of samples (Gao et al., 2015).

We further analyzed the correlation between the mean rsFC maps of the two sessions at the subject level [START_REF] Mcgraw | Forming inferences about some intraclass correlation coefficients[END_REF][START_REF] Zhang | Functional connectivity as revealed by independent component analysis of resting-state fNIRS measurements[END_REF]. For PC, BC and MI, we found statistically significant, strong positive correlations (greater than 0.75 with p<0.05) between the rsFC maps obtained for sessions 1 and 2 across all participants and data lengths. This finding confirms the reliability of the results of the resting-state fNIRS connectivity analysis across sessions [START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF]. [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF] have also reported statistically significant betweenrun correlations (with mean correlations r = 0.53). We only found low non-significant correlation values between the rsFC maps of the two sessions computed by PtC with long-range data lengths.

With regard to concentration signals, our post-hoc testing using paired t-tests (p<0.05) revealed significantly lower mean PC, BC and MI values for HbR compared to HbO and HbT for all data lengths (Table S1). Similarly, for PtC, HbR showed lower values for data lengths within the stability zones. For all FC measures, correlation values for HbT were higher and lower than those found for HbO and HbR, respectively.

Interestingly, the TRT analysis performed using PC, BC and MI showed higher reliability for HbR compared to HbO and HbT. The differences in ICC values between them became more significant with medium to long-range data lengths. The higher reliability found for HbR in this study is in contrast with the findings of the other studies [START_REF] Hoshi | Functional near-infrared spectroscopy: current status and future prospects[END_REF][START_REF] Lu | Use of fNIRS to assess resting state functional connectivity[END_REF]Zhang et al., 2010;Zhang et al., 2011) that have reported lower reliability for HbR, suggested to be due to the fact that HbR signals provide higher local specificity and more information on regional cerebral responses [START_REF] Kleinschmidt | Simultaneous recording of cerebral blood oxygenation changes during human brain activation by magnetic resonance imaging and near-infrared spectroscopy[END_REF][START_REF] Leff | Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies[END_REF][START_REF] Plichta | Event-related functional near-infrared spectroscopy (fNIRS): are the measurements reliable?[END_REF]. In our previous resting state analysis, we found that the spectral and frequency content of HbO and HbR signals especially within low-frequency bands were not significantly different at the global level [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. Moreover, compared to HbR, HbO and HbT signals have shown to be more affected by various systemic physiological noises including cardiac activity, fluctuations in blood pressure, and respiration [START_REF] Toronov | Near-infrared study of fluctuations in cerebral hemodynamics during rest and motor stimulation: temporal analysis and spatial mapping[END_REF][START_REF] Boas | Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy[END_REF][START_REF] Kohno | Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis[END_REF][START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. This may cause higher variability and lower reliability for correlation measures and network metrics across sessions and subjects [START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF].

In resting-state functional connectivity analysis, data length plays an important role for capturing the dynamics of brain networks as well as time varying aspects of segregation and integration of information over a specified time scale. We found higher inter-subject variability in connectivity measures with increasing data length. This might be due to variable neural and FC patterns observed during resting states over longer scanning sessions [START_REF] Greicius | Functional connectivity in the resting brain: a network analysis of the default mode hypothesis[END_REF][START_REF] Damoiseaux | Consistent resting-state networks across healthy subjects[END_REF][START_REF] Fox | The human brain is intrinsically organized into dynamic, anticorrelated functional networks[END_REF]. We also observed higher variations in channel-wise reliability as some channels showed higher reliability specially those located within frontal, temporal and posterior regions. Despite inter-channel variations in reliability, we found high reliability for FC measures at the group level, which tends to reduce variations at different optode locations (Zhang et al., 2011).

There are other factors that might explain inter-subject variations in reliability, including contribution of systemic physiological and instrumental noise and movement artifacts (Aarabi et al., 2017;[START_REF] Santosa | Characterization and correction of the false-discovery rates in resting state connectivity using functional near-infrared spectroscopy[END_REF]. In the present study, we removed high-amplitude artifacts using the spatial ICA [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF] as it has been shown that the presence of artifacts can impact the results of the reliability analysis [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF]. We also used a band-pass filter to reduce the effect of respiratory and cardiac fluctuations as well as Meyer waves (Ardalan and [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. It is obvious that very low frequency vascular changes with no direct coupling to neuronal activities cannot be removed through band-pass filtering (Ardalan and [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]). However, it has been shown that the correlation analysis within [0.01-0.1 Hz] reflect more the intrinsic properties of spontaneous neural activity [START_REF] Biswal | Functional connectivity in the motor cortex of resting human brain using echo-planar mri[END_REF][START_REF] Lowe | Functional Connectivity in Single and Multislice Echoplanar Imaging Using Resting-State Fluctuations[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF] even in the presence of systematic interferences. Further work is required to quantitatively characterize the contribution of neuronal and vascular activities to concentration signals and their effect on the result of the FC analysis.

Stability and reproducibility of network metrics

Our results (Figures 6 andS2) showed that the PC, BC and MI-derived network metrics including degree, global efficiency, characteristic path length, clustering coefficient, transitivity, graph radius and diameter were highly reliable (with mean ICC >0.8) across all data lengths and concentration signals (HbO, HbR and HbT). Our results on the PC-derived network metrics are largely consistent with the findings reported by [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF], who have studied the reliability and stability of nodal efficiency, nodal betweenness, global efficiency and clustering coefficient for different data lengths using rsfNIRS data. Similar to our results, they have found that nodal efficiency and betweenness stabilized and were reliable with data lengths of at least 1 min. However, they have shown that clustering coefficient, local and global efficiencies were only reproducible with much longer data lengths (5 min). We found high reliability for these two measures even with one-minute data segments. The discrepancies between our findings and those reported by [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF] could be due to two reasons. First, they have investigated binarized brain networks reconstructed using 46 fNIRS channels and sparsity thresholds. The thresholding may seriously affect network metrics values especially in networks with a limited number of nodes [START_REF] Vico Fallani | Graph analysis of functional brain networks: practical issues in translational neuroscience[END_REF][START_REF] Garrison | The (in)stability of functional brain network measures across thresholds[END_REF][START_REF] Hahn | A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain[END_REF]. We, however, analyzed weighted brain networks to avoid problems related to thresholding like network fragmentation. Moreover, [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF] have truncated rsfNIRS data into 30-s time bins for data lengths ranging from 1 to 10 min to assess the effect of scanning duration on functional brain connectivity and network metrics. Though it has not been clearly specified in their study, it seems that they have only analyzed one epoch per data length for each subject. This can highly affect the results of the FC analysis as connectivity patterns between different brain regions may vary over time. We, however, considered 50 randomly-selected time intervals (epochs) for each data length and subject to only take into account the overall dependency between different brain regions during resting state by ignoring epoch-specific changes in FC patterns.

Our results are also consistent with the results of [START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF], who have reported fair to excellent reliability for the majority of global metrics including clustering coefficient (HbO: 0.76; HbR: 0.78; HbT: 0.53), global efficiency (HbO: 0.76; HbR: 0.70; HbT: 0.78) and characteristic path length using 10-min HbO, HbR and HbT signals. They have also found similar results for nodal degree (0.52-0.84) and nodal efficiency (0.50-0.84). In line with our findings, they have further found higher average ICC values across all metrics for HbR in comparison with HbO and HbT and lower reliability for betweenness and modularity.

Our findings also confirm the high reliability for clustering coefficient, path length, and global efficiency reported in an fMRI study [START_REF] Telesford | Reproducibility of graph metrics in FMRI networks[END_REF]. Other fMRI studies have, however, reported low to modest reliability for most of the global network metrics due to either the sensitivity of these metrics to fluctuations in functional connections between sessions or low inter-subject or high intra-subject variance across scanning sessions [START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF][START_REF] Braun | Test-retest reliability of resting-state connectivity network characteristics using fMRI and graph theoretical measures[END_REF].

Our results showed that the global metrics derived by PC, BC or MI remained highly reliable provided that medium to large-range data lengths were used in the FC analysis. However, we found low reliability for the network metrics derived by PtC in these ranges. This finding is consistent with the results reported by [START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF]. We also found that the reliability of the PtC-derived network metrics decreased linearly with long-range data lengths, suggesting that the network analysis using PtC might lead to inappropriate topological properties, most likely due to changes in common driver components between brain regions over long-range scanning sessions [START_REF] Pereda | Nonlinear multivariate analysis of neurophysiological signals[END_REF]. This may also show that indirect connections between brain regions in a network may have variable effects on the strength and distribution of topological properties of brain networks at single-subject and regional levels. The large number of nodes could also cause poor performance for partial correlation [START_REF] Smith | Network modelling methods for FMRI[END_REF]. These findings suggest that the partial correlation should be used with caution for resting-state brain network studies [START_REF] Jalili | Constructing brain functional networks from EEG: partial and unpartial correlations[END_REF].

Among all graph metrics, modularity and hierarchy are of importance to investigate the modular and hierarchical organization (top-down relationships between regions) of brain networks. In fMRI studies, modularity has shown low to fair reliability compared to other metrics [START_REF] Braun | Test-retest reliability of resting-state connectivity network characteristics using fMRI and graph theoretical measures[END_REF]. In our study, we found low to high reliability for modularity depending on FC measures and concentration signals. We also found low modularity for rsfNIRS-based brain networks more likely due to the limited number of nodes (herein channels). This is consistent with the results indicating lower modularity for rsfNIRS-based brain networks compared to rsfMRI-based brain networks [START_REF] He | Uncovering intrinsic modular organization of spontaneous brain activity in humans[END_REF][START_REF] Niu | Revealing topological organization of human brain functional networks with resting-state functional near infrared spectroscopy[END_REF].

In line with the results reported by [START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF], we also found relatively low to moderate reliability for hierarchy. This metrics was highly dependent on data length, concentration signals and FC measures. Surprisingly, in our study only the PC and BC-derived networks showed some degree of hierarchy (values greater than 1) for HbO and HbT. For HbR, the network analysis exhibited values less than 1 and even negative, suggesting that the functional network did not exhibit a hierarchical structure [START_REF] Rodrigue | The geography of transport systems[END_REF]. The MI and PtC-derived hierarchy values were negative for all three concentration signals. These results are in contrast with other findings indicating hierarchical and modular structures for the rsfNIRSbased functional brain networks [START_REF] Niu | Revealing topological organization of human brain functional networks with resting-state functional near infrared spectroscopy[END_REF].

Stability and reproducibility of nodal metrics

In our study, we investigated the stability of nodal graph metrics with data length. Our results showed that the nodal metrics derived by PC, BC and MI reached higher stability with increasing data length. The ICC maps obtained for the BC and MI-derived nodal metrics were found to be more uniform across nodes in comparison with those computed for the PC-derived metrics.

The PC, BC and MI-derived nodal metrics also displayed higher stability and reliability in anterior and posterior regions in comparison with central regions due to higher variability in long-range intrahemispheric functional connections between frontal and parietal/occipital regions [START_REF] Li | Dynamic functional connectivity revealed by resting-state functional near-infrared spectroscopy[END_REF]. This finding is consistent with the results reported by [START_REF] Hermundstad | Structural foundations of resting-state and task-based functional connectivity in the human brain[END_REF], who have found higher FC variability between intrahemispheric frontal and posterior regions using task-derived fMRI data.

The reliability of nodal centrality metrics including degree, efficiency, betweenness, cluster coefficient, and participant coefficient has been previously investigated in an fMRI study conducted by [START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF]. They have reported poor to moderate reliability for nodal metrics including degree, which exhibited the highest TRT reliability and least variance in comparison with other metrics. We, however, found moderate to high reliability for PC, BC, and MI-derived nodal degree, efficiency, clustering coefficient, betweenness and vulnerability with mean ICC values larger than 0.65 across all hemoglobin concentration signals. Among all the nodal metrics studied in our work, nodal degree showed the highest reliability and least variation across subjects and sessions. Our results also showed that the reliability of the PC, BC, and MIderived nodal metrics was lower for HbT compared to HbO and HbR. Moreover, the reliability of the BC and MI-derived nodal metrics was found to be less variable across different data lengths in comparison with those derived by PC. The PtC-derived nodal metrics were very unreliable across different correlation measures especially with long-range data lengths.

Reliability results with regard to data length, fNIRS vs fMRI

The reliability of global network metrics has been assessed in several resting state fNIRS and fMRI studies [START_REF] Deuker | Reproducibility of graph metrics of human brain functional networks[END_REF][START_REF] Telesford | Reproducibility of graph metrics in FMRI networks[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF][START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Braun | Test-retest reliability of resting-state connectivity network characteristics using fMRI and graph theoretical measures[END_REF].

The effect of scan duration on TRT reliability of graph metrics derived from rsfMRI data has been previously addressed [START_REF] Liao | Functional brain hubs and their test-retest reliability: a multiband resting-state functional MRI study[END_REF][START_REF] Cao | Test-retest reliability of fMRI-based graph theoretical properties during working memory, emotion processing, and resting state[END_REF]. [START_REF] Liao | Functional brain hubs and their test-retest reliability: a multiband resting-state functional MRI study[END_REF] have shown that ICC values increased as the scan duration increased during the first 5-6 min and then remained unchanged with larger durations. In another fMRI study, [START_REF] Van Dijk | Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization[END_REF] have shown the stabilization of connectivity strengths with a scan duration of at least 5 min, suggested to be sufficiently long enough to produce reliable results across sessions and subjects using resting-state data [START_REF] Van Dijk | Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization[END_REF]. However, their results have been obtained within the regions of the default mode network (DMN). Similarly, [START_REF] Braun | Test-retest reliability of resting-state connectivity network characteristics using fMRI and graph theoretical measures[END_REF] also observed that decreasing the scan duration from 5 to 3 min significantly decreased the reliability of graph metrics. This is consistent with the results reported by [START_REF] Andellini | Test-retest reliability of graph metrics of resting state MRI functional brain networks: A review[END_REF], who have found a significant increase in the TRT reliability of graph metrics with data lengths longer than 5 min. [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF] have also shown that increasing data length had a significant effect on intersession reliability before reaching a plateau around 9 min. This observation is consistent with other findings, which suggest that a minimum BOLD data duration of approximately 5-7 minutes is enough for the stabilization of average correlation strengths [START_REF] Fox | The human brain is intrinsically organized into dynamic, anticorrelated functional networks[END_REF][START_REF] Van Dijk | Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization[END_REF][START_REF] Whitlow | Effect of resting-state functional MR imaging duration on stability of graph theory metrics of brain network connectivity[END_REF]. Similarly, improvement in TRT reliability of graph metrics has been reported with increasing scan length to 8-12 min for intersession rsfMRI data [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF]. Termenón et al. (2016) have also shown that a good reliability for graph metrics cannot be reached with short-duration data (5 to 10 min). In other fMRI studies, however, it has been shown that increasing the scan length from 6 to 12 min resulted only in 20% greater intrasession ICC for a subset of network connections [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF][START_REF] Anderson | Reproducibility of single-subject functional connectivity measurements[END_REF]. It has been suggested that functional brain networks become less reliable at a global level with longer scan duration because they rapidly change configurations due to cognitive representations and perceptions over longer periods of time [START_REF] Deuker | Reproducibility of graph metrics of human brain functional networks[END_REF]. Variability in intrinsic properties of functional brain networks might be also related to very low frequency fluctuations related to regional autoregulatory brain activity [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF].

The effect of data length on the reliability and stability of graph metrics has also been studied using rsfNIRS data [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF], who have shown that increasing data length can improve the results of the FC analysis. In our study, we also observed higher variability in FC values computed with short-range data lengths. However, we found that the strength of connectivity computed using fNIRS data did not change significantly with increasing data length above certain length ranges depending on FC and graph metrics. The instability in FC values computed with short-range data was more likely due to variations in functional connectivity patterns between and within brain networks [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF] as well as slow changes in hemodynamic activity caused by neuronal activity or physiological noise. These findings suggest that long-range data lengths are required to capture slow-varying network dynamics using fNIRS and fMRI signals. In studies performing FC analysis on signals with faster dynamics (like EEG and MEG signals), short-range epoch lengths can provide reliable estimates of dependencies between brain region [START_REF] Deuker | Reproducibility of graph metrics of human brain functional networks[END_REF]Matthew et al., 2011;[START_REF] Fraschini | The effect of epoch length on estimated EEG functional connectivity and brain network organisation[END_REF]. In these studies, the results of the FC analysis has also shown to be prone to noise and nuisance using short data lengths [START_REF] Fraschini | The effect of epoch length on estimated EEG functional connectivity and brain network organisation[END_REF].

Overall, these results suggest that graph metrics of rsFC are reliable enough to measure the topological properties of the brain's intrinsic functional architecture with both fNIRS and fMRI imaging techniques with certain discrepancies between the connectivity results [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF] due to differences in imaging modalities, data lengths, network and sample size [START_REF] Geng | Effect of Resting-State fNIRS Scanning Duration on Functional Brain Connectivity and Graph Theory Metrics of Brain Network[END_REF]. In fMRI studies, a band-pass filtered is usually used between (0.01-0.1 Hz) to reduce the effect of low-frequency drift and high-frequency physiological noise. However, the physiological noise like respiration and cardiac activity being aliased within (0.01-0.1 Hz) may cause high variability within and between sessions and subjects. In fNIRS studies, such a frequency band can efficiently reduce respiration and cardiac components due to the higher temporal resolution of fNIRS in comparison with fMRI [START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF]. The exact nature of these discrepancies can be investigated by performing the TRT reliability and stability analysis using concurrently collected resting-state fNIRS and fMRI data.

Technical considerations

Several technical issues might affect the accuracy of the TRT reliability analysis performed in our study. First, we used the weighted network connectivity analysis (van Wijk et al 2010). It has been suggested that compared to binarized networks, weighted networks could better characterize subtle changes in network topology [START_REF] Barrat | The architecture of complex weighted networks[END_REF]. However, the weighted approach could bias the connectivity results due to weak connections related to noise [START_REF] Ivkovic | Statistics of weighted brain networks reveal hierarchical organization and Gaussian degree distribution[END_REF]. Since non-significant connectivity values may have minor effects on the network topology (van Wijk et al., 2010), one can construct soft-thresholded networks with a continuous mapping of all correlation values to edge weights [START_REF] Schwarz | Negative edges and soft thresholding in complex network analysis of resting state functional connectivity data[END_REF].

Second, we investigated the TRT reliability of functional brain network metrics using rsfNIRS data collected with relatively short time intervals between different sessions for the same subjects. It has been reported that test-retest reliability reduces with increased time intervals between sessions [START_REF] Bennett | How reliable are the results from functional magnetic resonance imaging?[END_REF] more likely due to the fact that mental states of subjects may significantly change across different scans [START_REF] Waites | Effect of prior cognitive state on resting state networks measured with functional connectivity[END_REF][START_REF] Horovitz | Low frequency BOLD fluctuations during resting wakefulness and light sleep: a simultaneous EEG-fMRI study[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: test-retest evaluation on short-and long-term resting-state functional MRI data[END_REF]. This is why long-term scans might better elucidate real TRT reliability of global network metrics [START_REF] Liang | Effects of different correlation metrics and preprocessing factors on small-world brain functional networks: a resting-state functional MRI study[END_REF][START_REF] Niu | Test-retest reliability of graph metrics in functional brain networks: a resting-state fNIRS study[END_REF]. Moreover, we did not investigate intrasession reliability of FC measures and network metrics. The intrasession reliability analysis might better show changes in cognitive and emotional states of subjects, resulting in variations in the topological organization of functional brain networks [START_REF] Birn | The effect of scan length on the reliability of resting-state fMRI connectivity estimates[END_REF].

Third, the presence of systemic physiology in fNIRS signals especially in low frequencies can make it difficult to accurately characterize network topology. The use of denoising methods including short separation measurement [START_REF] Gagnon | Improved recovery of the hemodynamic response in diffuse optical imaging using short optode separations and state-space modeling[END_REF], independent component analysis [START_REF] Kohno | Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis[END_REF][START_REF] Aarabi | Characterization of the relative contributions from systemic physiological noise to whole-brain resting-state functional near-infrared spectroscopy data using single-channel independent component analysis[END_REF], adaptive filter method [START_REF] Zhang | Adaptive filtering to reduce global interference in non-invasive NIRS measures of brain activation: how well and when does it work?[END_REF], linear regression using auxiliary data [START_REF] Mesquita | Resting state functional connectivity of the whole head with near-infrared spectroscopy[END_REF] and principal component analysis (PCA) [START_REF] Zhang | Eigenvector-based spatial filtering for reduction of physiological interference in diffuse optical imaging[END_REF] can help to remove systemic physiology from low frequency neuronal oscillations and reduce the effect of noise on the results of FC analysis using fNIRS data. To better understand the effect of different physiological noises on brain functional organization one should perform a study in a very controlled manner under different signal to noise ratios.

Finally, we did not study the effect of longer (beyond 5 min) resting-state scanning lengths on the stability and reliability of FC measures and network metrics. Using longer scanning lengths is an efficient way to include non-overlapping segments in FC analysis for each data length. Further work is required to investigate the effect of these limitations on the topology of functional networks.

Conclusion

In this study, we investigated the effect of data length on the TRT reliability and stability of four functional connectivity measures and twelve commonly-used network metrics derived from resting-state fNIRS data. Our results show that the data length can significantly affect the results of the FC analysis and the topological properties of functional brain networks, depending on FC measures and concentration signals. Our results demonstrated higher reliability and stability for most global and nodal metrics especially degree, efficiency, clustering coefficient and path lengths with increasing data length. Our results suggest that caution should be taken when comparing results from studies on functional network organization when the FC analysis is performed with different data lengths, especially with short-range epoch lengths. Moreover, various graph metrics should be investigated and interpreted cautiously as they showed different sensitivities to data length.

Fig. 1 .

 1 Fig. 1. Whole-head optical probe used for data collection. Small bold dots show the position of the fNIRS channels used in this study.

Figure 2 .

 2 Figure 2. Boxplots showing median, interquartile range (shaded box), and 95% confidence ranges (whiskers) of absolute values for each FC measure computed over all channels and subjects for each session and concentration signal. The solid and dotted horizontal lines indicate the stability intervals, within which no significant differences (p>0.05) in connectivity values were found between different data lengths for session 1 and 2, respectively.

Figure 3 .

 3 Figure 3. Boxplots of intraclass correlation (ICC) values as a function of data length. Boxplots show median, interquartile range (shaded box), and 95% confidence limits (whiskers) computed over all subjects for HbO, HbR, and HbT.

Figure 4 .

 4 Figure 4. Boxplots for degree and shortest path length derived by each connectivity measure as a function of data length for HbO, HbR and HbT. Boxplots show median, interquartile range (shaded box), and 95% confidence limits (whiskers) computed over all nodes and subjects. The solid and dotted horizontal lines indicate the stability intervals within which no significant differences (P > 0.05) in degree or shortest path length values were found between different data lengths for session 1 and 2, respectively.

Figure 5 .

 5 Figure 5. Kernel density functions of the onsets of the stability zones found over all graph metrics for each concentration signal.

Figure 6 .

 6 Figure 6. Intraclass correlation (ICC) curves for the average degree and characteristic path length as a function of data length for Pearson's correlation (a and e), Bend correlation (b and f), Mutual information (c and g) and Partial correlation (d and h).

Figure 7

 7 Figure7shows the ICC maps for k w and l w computed using each of the four FC measures as a function of data length for HbO. The ICC maps for other nodal metrics including W L E , c w , b w and v w are shown in FigureS3. Overall, the ICC maps showed higher stability with increasing data lengths at the nodal level for the metrics computed by PC, BC and MI. The ICC maps of the metrics derived by BC and MI compared to PC were spatially more uniform across brain regions. In contrast, the reliability of the PtC-derived metrics was found to be only higher for data lengths shorter than 120 sec.

Figure 7 .

 7 Figure 7. Intraclass correlation (ICC) maps for the nodal degree and shortest path length as a function of data length for HbO.

Table 1 .

 1 Correlation results between resting-state functional connectivity (rsFC) maps computed for sessions 1 and 2 at the group level. The rsFC maps derived for sessions 1 and 2 represented mean connectivity matrices computed over all subjects for each data length and correlation metrics. Non-significant correlation values (P>0.05) are highlighted in bold face.

	Signal	measure				Data length (sec)
			30	60	90	120	150	180	210	240	270
	HbO	Pearson Correlation 0.82	0.77 0.81 0.79 0.77 0.75 0.79 0.80 0.83
		Partial Correlation	0.56	0.57 0.82 0.81 0.74 0.76 0.75 0.81 0.30
		Bend Correlation	0.82	0.77 0.80 0.77 0.75 0.73 0.78 0.79 0.82
		Mutual Correlation	0.78	0.72 0.79 0.74 0.74 0.70 0.75 0.75 0.76
	HbR	Pearson Correlation 0.86	0.81 0.79 0.84 0.85 0.84 0.84 0.83 0.84
		Partial Correlation	0.75	0.49 0.69 0.76 0.93 0.41 0.45 0.54 0.54
		Bend Correlation	0.85	0.80 0.79 0.83 0.84 0.82 0.83 0.82 0.84
		Mutual Correlation	0.67	0.82 0.86 0.86 0.88 0.84 0.84 0.83 0.86
	HbT	Pearson Correlation	0.72 0.77 0.75 0.71 0.69 0.69 0.70 0.72 0.75
		Partial Correlation	0.69	0.85 0.80 0.77 0.86 0.59 0.28 0.75 0.43
		Bend Correlation	0.72	0.77 0.75 0.70 0.69 0.70 0.72 0.74 0.78
		Mutual Correlation	0.65	0.76 0.75 0.69 0.65 0.67 0.67 0.66 0.69

Table 2 .

 2 Onsets of the stability zone for the graph metrics derived by each of the four connectivity measures at the network level. For each metric, the minimum data lengths (in sec) at the onset of the stability zones or the entire stability interval are reported.

	Graph metric	FC	HbO	HbR	HbT	Graph metric	FC	HbO	HbR	HbT
		measure					measure
	Degree	PC	> 90	> 150	> 60	Graph Radius	PC	> 90	> 120	> 60
		BC	> 60	> 150	> 90		BC	> 150	> 150	> 120
		MI	> 150	> 150	> 120		MI	> 180	> 180	> 210
		PtC	30-120	30-120	30-150		PtC	-	-	60-90
	Global efficiency	PC	> 60	> 90	> 90	Graph Diameter	PC	> 90	> 120	> 150
		BC	> 60	> 90	> 60		BC	> 120	> 120	> 120
		MI	> 150	> 150	> 150		MI	> 180	> 180	> 180
		PtC	30-120	30-120	90-150		PtC	60-150	60-120	60-150
	Clustering	PC	> 60	> 120	> 90	Vulnerability	PC	> 120	> 210	> 120
	Coefficient	BC	> 60	> 150	> 90		BC	> 120	> 180	> 120
		MI	> 150	> 180	> 150		MI	> 210	> 210	> 210
		PtC	30-120	30-120	90-150		PtC	30-90	30-80	30-90
	Transitivity	PC	> 60	> 150	> 90	Modularity	PC	> 90	> 180	> 90
		BC	> 60	> 150	> 90		BC	> 90	> 180	> 90
		MI	> 150	> 180	> 150		MI	> 180	> 240	> 180
		PtC	30-120	30-120	90-150		PtC	-	-	-
	characteristic path	PC	> 90	> 120	> 90	Participation	PC	> 30	> 30	> 90
	length	BC	> 90	> 150	> 90	Coefficient	BC	> 30	> 30	> 30
		MI	> 180	> 180	> 150		MI	> 180	> 120	> 60
		PtC	30-90	30-19	30-150		PtC	> 120	> 90	90-180
	Betweenness	PC	> 150	> 180	> 120	Hierarchy	PC	> 150	> 120	> 90
		BC	> 150	> 180	> 120		BC	> 60	> 150	> 60
		MI	> 210	> 240	> 240		MI	> 150	> 150	> 90
		PtC	30-60	30-60	30-60		PtC	120-180	150-210	150-210

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The authors thank Maria Angela Franceschini and David Boas for providing the data used in this study.

Conflicts of interest

The authors declare no conflicts of interest.