# Altered Topological Properties of Functional Brain Network Associated with
Alzheimer’s Disease

Yongcheng Yao
Department of Imaging and Interventional Radiology
The Chinese University of Hong Kong
Hong Kong
<EMAIL_ADDRESS>

###### Abstract

Functional Magnetic Resonance Imaging (fMRI) is commonly utilized to study
human brain activity, including abnormal functional properties related to
neurodegenerative diseases. This study aims to investigate the differences in
the topological properties of functional brain networks between individuals
with Alzheimer’s Disease (AD) and normal controls. A total of 590 subjects,
consisting of 175 with AD dementia and 415 age-, gender-, and handedness-
matched controls, were included. The topological properties of the brain
network were quantified using graph-theory-based analyses. The results
indicate abnormal network integration and segregation in the AD group. These
findings enhance our understanding of AD pathophysiology from a functional
brain network structure perspective and may aid in identifying AD biomarkers.
Supplementary data to aid in the validation of this research are available at
https://github.com/YongchengYAO/AD-FunctionalBrainNetwork.

_Keywords_ brain network $\cdot$ network topology $\cdot$ Alzheimer’s disease

##  1 Introduction

Alzheimer’s Disease (AD) is a chronic neurodegenerative disease prevalent in
the elderly population, characterized by cognitive decline, language problems,
memory disturbances (especially short-term memory), and disorientation. With
disease progression, severe bodily dysfunction and ultimately death can occur.
AD is the most common form of dementia, accounting for approximately half of
the cases. A rare form of AD is early-onset familial Alzheimer’s disease,
which is associated with the amyloid precursor protein and presenilin genes.
Another form of AD is sporadic AD, affecting over 15 million people worldwide,
with its cause primarily unknown. Risk factors for AD include decreased brain
size, low education level, low mental ability, head injury, and vascular-
disease-related factors [1]. The amyloid hypothesis proposes that
extracellular amyloid beta deposits cause AD [2]. The tau hypothesis suggests
that AD results from tau protein dysfunction, with neurofibrillary tangles
formed by tau protein destroying the neuron’s transport system [3].

Functional magnetic resonance imaging (fMRI) offers a non-invasive approach
for diagnosing, evaluating therapeutic interventions, and investigating the
mechanisms of AD. In brain imaging, a typical fMRI utilizes the Blood
Oxygenation Level Dependent (BOLD) contrast to indirectly reflect brain
activity through signal fluctuations. Early studies of brain function
primarily relied on task-based fMRI, where fMRI brain activity was acquired
during specific functional tasks [4, 5]. In 1995, Biswal demonstrated that
resting-state fMRI signals could depict spontaneous neuronal activity without
the need for external task experiments [6]. An increasing number of studies
have utilized resting-state fMRI to investigate brain function and disease-
related abnormalities. In recent years, resting-state fMRI has become the most
widely used neuroimaging technique in AD-related studies [7, 8, 9, 10, 11].

Graph-theory-based analysis is a prevalent method in brain function studies.
To some extent, the graph-based method is also a connectivity analysis, as the
interactions among brain regions are described in a network (a graph). In
contrast to connectivity analysis, a graph-based analysis investigates the
network’s topological properties instead of network connections. Topological
properties are quantified by various network metrics [12]. Previous studies
using functional MRI and connectivity analysis have reported disrupted
functional connectivity in AD patients. Key regions related to neural
degeneration in AD patients include the Hippocampus [8, 13], Prefrontal Cortex
[14], Parietal Lobe [15], Precuneus [16], and Posterior Cingulate Gyrus [17,
18, 16]. In a systematic review and meta-analysis of 43 studies with 1,363
subjects, regions within the default mode, salience, and limbic networks
consistently showed abnormalities in connectivity [19]. The default mode
network has been a focal point of AD studies since changes in functional
connectivity in regions of the default mode network have been observed in
preclinical AD (from very mild [20] to moderate AD [21]). These findings are
even more compelling for subjects with severe AD. It has been reported that
the map of decreased functional connectivity is similar to the amyloid
deposition map and the tau-protein deposition map [22], indicating that
amyloid deposition [23] and tau protein deposition may play a causative role
in the dysfunction of AD patients. Dysfunction in the default mode network is
a prominent feature of AD at all stages. In contrast to the disruption of
functional connections in the default mode network, increased functional
connections in the salience network have also been reported [24], which have
been validated by subsequent studies [25, 22]. Brier _et al._ [25] showed that
the pattern of aberrant functional connectivity changes with AD progression.
At the early stage of AD (with very mild symptoms), increased functional
connectivity within the salience network and decreased functional connectivity
within the default mode network, executive control network, and sensorimotor
network were observed. However, at the mild to moderate stage, decreased
functional connections were discovered in all networks. Brier and colleagues
[22] have proposed a model of disease spread to account for changes in
functional disruption in AD.

##  2 Graph-Theory-Based Analysis

###  2.1 Related Works

##### Graph Type

The analysis of a graph depends on how it is defined. A graph can be defined
as weighted or weightless (binary) based on whether a network connection has
weight or not. Additionally, a graph can be directional or bidirectional based
on its directionality. Therefore, there are four types of graphs: weighted
directional graph, weighted bidirectional graph, weightless directional graph,
and weightless bidirectional graph. A weightless graph is a special case of a
weighted graph in which all connections have the same weight of 1. Similarly,
a bidirectional graph is a unique form of a directional graph, where each
bidirectional connection can be viewed as two opposite directional
connections. In summary, a weighted and directional graph is the most common
form, and the other three types of graphs are special cases under additional
conditions.

##### Network Segregation

The ability of a network to organize nodes into clusters or modules is termed
network segregation. Clusters or modules are groups of nodes that are densely
intra-connected and sparsely inter-connected. Measures of segregation [26] are
primarily designed to identify and quantify these structures, such as the size
and number of clusters, or other metrics reflecting the properties of
connectivity within a cluster. From the perspective of the functional brain
network, functional segregation is the ability to organize the brain into
distinct functional subnetworks, each of which is responsible for a relatively
unique brain function. Therefore, the functional segregation of the human
brain is physiologically meaningful and interpretable.

##### Network Integration

The degree of interconnectivity among nodes is reflected in network
integration. Measures of integration [26] generally quantify how easily nodes
communicate with one another, with many of these measures based on the concept
of "path." In the context of the functional brain network, functional
integration can be viewed as the ability to efficiently retrieve and combine
information distributed across different functional subnetworks.

##### Network Metrics

In this study, we first constructed weightless and bidirectional graphs (brain
networks) from MRI data. Second, we quantified the networks’ topological
properties using graph-theory-based metrics. Finally, significant differences
in metrics between groups can depict the topological changes of the networks.
The graph-theory-based metrics used, as shown in Figure 1, are only for
weightless bidirectional graphs, where each edge is binary and bidirectional.
They are the same as those introduced in an previous review paper [12], which
includes elaborate explanations and definitions. Additionally, the metric
definitions for weighted and directional graphs are also listed (in Table A1
of paper [12]) and compared with their weightless and bidirectional
counterparts. It is noteworthy that metrics of different types of networks
cannot be directly compared due to their distinct definitions.

Figure 1: Illustration of Graph Theory Metrics. (1) A visualization of graph
$N$ as interconnected nodes. (2-7) Illustrations of degree / cost, clustering
coefficient, shortest path length, global efficiency, local efficiency, and
betweenness centrality.

###  2.2 Data

####  2.2.1 OASIS-3 Dataset

The subjects involved in our study were obtained from the OASIS-3 public
dataset [27], which is the latest release of the Open Access Series of Imaging
Studies (OASIS). OASIS-3 is a large longitudinal dataset that provides the
scientific community with open access not only to multi-modal neuroimaging
data but also to various clinical data and cognitive assessments. All data in
OASIS-3 are available on the OASIS Brains project website (www.oasis-
brains.org). We used the OASIS-3 dataset in this study for several reasons:

  1. 1.

Compared to other open-access datasets such as the Alzheimer’s Disease
Neuroimaging Initiative (ADNI) [28, 29, 30, 31] and the Harvard Aging Brain
Study (HABS) [32], OASIS-3 is relatively larger in terms of the number of
participants (over 1000) and MR sessions (over 2100). It is a retrospective
compilation of neuroimaging and clinical data collected across multiple
ongoing projects in the past 30 years, involving 609 cognitively normal
(normal ageing) adults and 489 subjects at various stages of cognitive
decline.

  2. 2.

It is easy to explore and download imaging and clinical data. All data in
OASIS-3 are hosted by XNAT (the extensible Neuroimaging Archive Toolkit) [33],
an informatics platform for data management and sharing.

  3. 3.

The dataset covers all common modalities of neuroimaging in the field of
ageing and cognitive decline. Since the dataset is tailored for research on
normal ageing and Alzheimer’s disease, it provides open access to over 2100 MR
sessions, including T1w, T2w, FLAIR, ASL, SWI, time-of-flight, resting-state
BOLD fMRI, and DTI. One of the advantages of OASIS-3 is the degree of data
integration, as for most sessions, the majority of modalities are available.

####  2.2.2 Inclusion Criteria

The first step of this study is to select data from a large dataset. Since
this is a between-group MRI study, the criteria are related to the parameters
of MR images and demographic information. The inclusion criteria are as
follows.

  1. 1.

Only data from one session were downloaded for each individual. Although
longitudinal data are available, we only chose one BOLD-fMRI and one T1w MRI
for each subject. The reason is that this study mainly focuses on
investigating the significant differences in topological properties of the
functional brain network between the two groups, rather than on longitudinal
changes such as disease progression. This rule ensures that the subjects
enrolled in this study are independent, which is an assumption of parametric
statistical analyses.

  2. 2.

The acquisition protocols of BOLD-fMRIs should be the same. Multi-site and
multi-scanner MR studies are challenging because of the variabilities
introduced by the differences in imaging protocols and scanners. To minimize
such variabilities, we exerted restrictions that all fMRI data should be
collected from Siemens scanners under the same imaging protocol. However, such
a restriction was not applied to T1w MRI data, because structural MRI data
were used for brain tissue segmentation to facilitate the removal of BOLD
signal artefacts, and the segmentation algorithm used in this study performs
well on T1w images with various acquisition parameters.

  3. 3.

For each individual, one BOLD-fMRI should be matched with one T1w MRI from the
same session. If there is no T1w MRI available in the same session, then the
BOLD-fMRI should be discarded, and the subject should be excluded. Since
functional MRI data have a low spatial resolution (usually on the scale of 3
mm) while T1-weighted structural MRI data have a relatively high spatial
resolution (on the scale of 1 mm), T1w MRI can better delineate the structure
of the brain, serving as a spatial reference for BOLD-fMRI. Therefore, T1w MRI
and BOLD-fMRI are usually collected in the same session. In other words, the
perfect structural reference is the one from the same scanning session.

  4. 4.

There should be no significant difference in age, gender, and handedness
between the normal control and Alzheimer’s Disease group. Significant
alterations of brain structure and function were discovered in normal ageing
adults. Gender differences and the effect of handedness were also found in
functional activation and connectivity. Thus, it is critical to ensure that
the two groups are age-, gender-, and handedness-matched.

####  2.2.3 MR Image Acquisition Parameters

Resting-state BOLD MR images were acquired using a single-shot FID EPI
sequence on a 3-Tesla scanner (Siemens, TrioTim or Biograph_mMR), with the
following parameters: TR = 2200 $ms$; TE = 27 $ms$; FA = $90^{\circ}$; slice
thickness = 4 $mm$; slice gap = 0 $mm$; number of slices (z) = 36; in-plane
resolution = 4 x 4 $mm^{2}$; in-plane matrix size (x, y) = 64 x 64; number of
time points = 164.

T1-weighted MR images were acquired using a single-shot TurboFLASH sequence on
the same 3-Tesla scanner (Siemens, TrioTim or Biograph_mMR), with the
following parameters: TR = 2400 $ms$; TE = 3 $ms$; FA = $8^{\circ}$; slice
thickness = 1 $mm$; slice gap = 0 $mm$; number of slices (z) = 176; in-plane
resolution = 1 x 1 $mm^{2}$; in-plane matrix size (y, z) = 256 x 256.

####  2.2.4 MR Image Labeling

Clinical data, including the Clinical Dementia Rating (CDR) [34] [35] and
diagnoses, can be downloaded from the "ADRC Clinical Data" field in the
OASIS-3 data browser. The CDR is a 5-point scale used to assess the severity
of dementia by characterizing 6 categories of cognitive and functional
performance relevant to Alzheimer’s Disease and related dementias. These
categories are memory, orientation, judgment and problem-solving, community
affairs, home and hobbies, and personal care. The rating for each category is
obtained through a semi-structured interview, known as the CDR Assessment
Protocol. By combining the ratings for each category, a global CDR score can
be calculated using the CDR-assignment algorithm [35]. Since clinicians base
their diagnoses not only on the global CDR score but also on other clinical
tests, we used these professional diagnoses to categorize each BOLD-fMRI into
either an AD group or a normal controls (NC) group. Specifically, we used
diagnoses from the "dx1" field to identify AD and NC data. A complete list of
unique diagnosis remarks and their corresponding labels can be found in Table
LABEL:tab:diagnoses.labels.

However, the dates of clinical assessments and MRI scanning are not the same,
so it is necessary to link the appropriate clinical diagnoses to each MR
image. To accomplish this, we first calculated the relative date of clinical
screening from the clinical data identifier. Similarly, the relative date of
MRI scanning can be calculated from the name of the MR session. For instance,
a clinical data ID "OAS30001_ClinicalData_d0339" indicates that the clinical
assessments of subject "OAS30001" were performed on the $339^{th}$ day after
their first visit. Similarly, an MR session name "OAS30001_MR_d0129" indicates
that the MRI data of subject "OAS30001" were collected on the $129^{th}$ day
after their first visit. Secondly, we linked each MRI data with its nearest
clinical results.

Table 1: Clinical Diagnoses and Labels Diagnosis | Label | Diagnosis | Label
---|---|---|---
“AD Dementia” | AD | “AD dem w/oth unusual features” | AD
“AD dem Language dysf after” | AD | “AD dem w/oth unusual features/demt on” | AD
“AD dem Language dysf prior” | AD | “AD dem/FLD prior to AD dem” | AD
“AD dem Language dysf with” | AD | “Cognitively normal” | NC
“AD dem cannot be primary” | AD | “DAT” | -
“AD dem distrubed social- after” | AD | “DAT w/depresss not contribut” | -
“AD dem distrubed social- prior” | AD | “DLBD, primary” | -
“AD dem distrubed social- with” | AD | “DLBD- primary” | -
“AD dem visuospatial, after” | AD | “DLBD- secondary” | -
“AD dem visuospatial- prior” | AD | “Dementia/PD- primary” | -
“AD dem visuospatial- with” | AD | “Frontotemporal demt. prim” | -
“AD dem w/CVD contribut” | AD | “Incipient Non-AD dem” | -
“AD dem w/CVD not contrib” | AD | “Incipient demt PTP” | -
“AD dem w/Frontal lobe/demt at onset” | AD | “No dementia” | -
“AD dem w/PDI after AD dem contribut” | AD | “Non AD dem- Other primary” | -
“AD dem w/PDI after AD dem not contrib” | AD | “ProAph w/o dement” | -
“AD dem w/depresss contribut” | AD | “Unc: impair reversible” | -
“AD dem w/depresss not contribut” | AD | “Unc: ques. Impairment” | -
“AD dem w/depresss, not contribut” | AD | “Vascular Demt primary” | -
“AD dem w/depresss- contribut” | AD | “Vascular Demt- primary” | -
“AD dem w/depresss- not contribut” | AD | “Vascular Demt- secondary” | -
“AD dem w/oth (list B) contribut” | AD | “uncertain possible NON AD dem” | -
“AD dem w/oth (list B) not contrib” | AD | “uncertain dementia” | -
“AD dem w/oth unusual feat/subs demt” | AD | “uncertain- possible NON AD dem” | -

####  2.2.5 Demographic Information

A total of 590 subjects are involved in the current study, including 175
subjects with AD dementia and 415 normal controls. As shown in Table 2, there
is no significant difference on age ($t=1.5125$, $p>0.05$), gender
($\chi^{2}=2.1782$, $p>0.05$), and handness ($\chi^{2}=0.3926$, $p>0.05$)
between the two groups.

Table 2: Demographic Information | AD Group | NC Group | Statistics | P-value
---|---|---|---|---
Gender (Male / Female) | $95/80$ | $196/219$ | $2.1782^{a}$ | $0.1400$
Age (Mean $\pm$ SD) | $75.11\pm 7.67$ | $74.12\pm 6.12$ | $1.5125^{b}$ | $0.1316$
Handness (Left / Right / NA) | $18/154/3$ | $34/370/11$ | $0.3926^{a}$ | $0.5309$
a: Welch’s two-sample t-test
b: Pearson’s chi-squared test with Yates’ continuity correction
NA: Not applicable (indicating missing data)
Figure 2: Processing Pipeline of Graph-theory-based Analysis. T1-weighted MR
images are normalized into the MNI-152 space, and then segmented into grey
matter (GM), white matter (WM), and cerebrospinal fluid (CSF). For BOLD fMRI,
realignment (head motion estimation and correction), slice timing correction,
outlier scans detection, and normalization are applied. Various regressors are
used to remove confounding effects: (1) Principal components (PCs) from WM and
CSF, (2) head motion parameters, (3) scrubbing variables, and (4) linear
regressor. Finally, band-pass filtering ($0.01-0.1$) is applied.

###  2.3 MR Images Processing

MRI data were processed using the SPM12 and functional connectivity toolbox
(CONN 18.b) [36]. The processing pipeline (Figure 2) used in this study is
mainly based on methods from SPM12. However, some other methods are applied in
place of the ones from SPM12. For instance, a components-based method with
anatomical noise ROI, aCompCor [37], is adopted to reduce the effects of noise
from white matter (WM) and cerebrospinal fluid (CSF), instead of the global
signal regression used in SPM12.

####  2.3.1 Normalization and Segmentation of T1-weighted MRI

Normalization of raw MR images into a common space is necessary for inter-
subject comparison, and the MNI-152 space is the most widely used standard
space. The MNI-152 space is defined by the prior tissue probability map (TPM)
generated from structural MR images of 152 subjects. Specifically, first, all
structural images are registered together; then, each voxel is assigned to one
tissue type, resulting in individual tissue masks for each brain tissue;
finally, the binary masks are averaged across all subjects to generate the
tissue probability map. Therefore, various prior tissue probability maps (for
each brain tissue), as a whole, represent the average shape and position of
the brain in an MR image, which is the definition of a "standard space". The
tissue probability map used is provided by SPM12 (Figure 3), referred to as
the TPM atlas.

For T1-weighted MR images, the first step applied is spatial normalization,
which wraps the MRI volumes in individual spaces into the standard MNI-152
space. The spatial normalization consists of two steps: (1) estimating a non-
linear deformation field that best overlays the TPM on the individual
T1-weighted image; (2) wrapping the raw image with the inverse deformation
estimated in step (1).

For segmentation, individual brain tissue probability maps for grey matter
(GM), white matter (WM), and cerebrospinal fluid (CSF) are estimated. Then, a
threshold is applied to each tissue probability map to generate a binary mask,
which is further refined by morphology erosion. The eroded WM and CSF masks
can serve as noise ROI masks, while the eroded GM mask is useful in
restricting the subsequent analyses within the grey matter area.

Figure 3: Tissue Probability Map. Defined in the MNI-152 space, the tissue
probability map gives the prior probability for a voxel of being a specific
brain tissue, including (1) grey matter, (2) white matter, (3) cerebrospinal
fluid, (4) skull, (5) scalp, and (6) non-brain area. The values of these maps
range $[0,1]$. For one voxel, the probabilities sum up to 1.

####  2.3.2 Head Motion Estimation and Correlation

Head motion can introduce systematic co-variation across voxels, which
increases the estimations of functional connections. More importantly, the
distance-dependent signal modulation effect is reported [38] [39] [40] in
functional connectivity studies. It indicates that the variance added by
motion artefact is similar in nearby voxels, resulting in a stronger short-
distance correlation. Additionally, due to the application of head-motion-
reduction processing methods, the observed long-distance correlation would
often be decreased [41]. Even a small head movement would cause signal
disruption in BOLD fMRI [41] and add spurious variance that can increase or
decrease the observed functional connections. Thus, the estimation of head
motion and the removal of head-motion-induced effects are crucial.

Among various artefacts, head motion is unique since it can be estimated from
MR images via the realignment process (unlike, for example, the cardiac and
respiratory artefacts that requires external recordings). Over the years,
effort has been made to measure head motion, and considerable progress has
been achieved. The simplest method is rigid body estimation, which measures
the translation and rotation in the x, y, and z axes, producing 6 motion
parameters. In 1996, an expansion of this motion estimation method was
proposed by Karl Friston [42]. The expansion takes the form
[$RR^{2}R_{t-1}R_{t-1}^{2}\dots R_{t-k}R_{t-k}^{2}$], where R denotes the 6
rigid body estimates, $R^{2}$ denotes their squares, and $R_{t-k}$ represents
6 parameters of the $k^{th}$ preceding time points. For instance,
[$RR^{2}R_{t-1}R_{t-1}^{2}$] stands for a 24-parameter estimate, and
[$RR^{2}R_{t-1}R_{t-1}^{2}R_{t-3}R_{t-3}^{2}$] refers to a 36-parameter
estimate.

In this study, the 6-parameter rigid body realignment method is adopted, which
is the strategy followed in other studies [38] [39] [40]. These 6 parameters
and their first-order derivatives, 12 parameters taking the form
[$RR^{\prime}$] in total, are included in a linear regression model to regress
out the head-motion-related variance in the following artefacts removal step.
Although subsequent studies [43] have pointed out the inadequacy of these
parameters, research by Jonathan Power and colleagues [44] shows that more
motion parameters cannot capture all head-movement-induced variance, and
similar results were observed when 12, 24, and 36 motion parameters are
included as regressors.

In the pre-processing pipeline of BOLD MR images, realignment is always the
first step. Realignment is actually the head motion estimation and correction.
It registers all other MR volumes in time series to a reference volume and
exports the head motion parameters. The reference volume can be the volume at
a specific time point, or the mean image averaged across time. In this study,
the first volume is chosen as the reference. B-spline interpolation is used in
the registration process. Head-motion-related variance can be partially
reduced by realignment.

####  2.3.3 Slice Timing Correction

During the acquisition of a BOLD MR image with EPI sequence, a 3-D volume is
actually a stack of 2-D slices collected one at a time. Therefore, for an fMRI
volume at a time point, the voxel activations of each slice are not at the
same time point. However, the ideal situation is that we can observe the
activation of the whole brain at the same time. To resolve the conflict, slice
timing correction can be used to interpolate slices to a reference slice. It
has been shown to be an effective solution that can reliably increase
sensitivity and effect power. The implementation details are as follows: (1)
the acquisition time for each slice is read from the BIDS sidecar that comes
with each NIfTI file; (2) all slices in a volume are interpolated to the slice
acquired in the middle of time.

####  2.3.4 Outlier Scans Detection

The Artifact Detection Tools (ART) are utilized for the automatic detection of
volumes with severe head motion and global mean intensity change. The volumes
that exceed the specified threshold are considered outlier scans and labelled
by scrubbing variables. The ART-based outlier detection function is
conveniently integrated into the CONN toolbox. The "intermediate settings,"
which treat the $97^{th}$ percentiles in the normative sample as the
threshold, are selected. At the end of this step, a set of scrubbing variables
is stored for further use as regressors in the general linear model.

####  2.3.5 Normalization of BOLD MRI

In the CONN toolbox, spatial normalization of BOLD MR images can be
implemented using two methods: direct and indirect normalization.

##### Direct normalization

It involves normalizing the T1-weighted MR images and BOLD MR images
separately. It is implemented in the same way as spatial normalization of T1w
MR images: (1) first, a nonlinear deformation field that wraps the TPM atlas
to the individual BOLD MR image is estimated; (2) then, the inverse
transformation is applied to normalize the BOLD MR image into MNI-152 space.

##### Indirect normalization

It involves two steps: (1) first, the BOLD MR image is co-registered to the
T1-weighted MR image, generating a transformation matrix $T_{1}$; (2) then,
the obtained transformation matrix $T_{1}$ is multiplied by the transformation
matrix $T_{2}$, which wraps the T1-weighted image to MNI-152 space. Finally,
the BOLD image can be transformed into the standard space with the matrix
$T_{2}T_{1}$.

As a step in the default pre-processing pipeline of the CONN toolbox, the
direct normalization scheme is applied.

####  2.3.6 Spatial Smoothing of BOLD MRI

Spatial smoothing with a Gaussian kernel is typically the final step in a
spatial processing pipeline for a BOLD MR image. However, it is not utilized
in this study because the current study only involves ROI-based graph
analyses. That is, the final processed and de-noised signal is the mean time
series of an ROI. Spatial smoothing prior to averaging would not significantly
alter the resulting mean signal. We only consider spatial smoothing to be a
necessary image processing step when it is followed by voxel-based analyses.
This approach is also used by the CONN toolbox.

####  2.3.7 Artefacts Removal

The removal of artefacts is a critical step in BOLD signal processing, aimed
at mitigating or eliminating the confounding effects of non-neuronal
oscillations caused by head movement, cardiac pulsation, respiratory motion,
and other systematic noises. Without this step, it is challenging for
researchers to determine whether the findings are genuine or simply driven by
artefacts. A general linear model (GLM) is used for artefacts removal, with
mean BOLD signals extracted from ROIs defined by a prior brain atlas, and a
variety of variables defined as regressors of the GLM (as shown in Figure 2).

In the previous realignment step, the nuisance head motion effect can be
partially mitigated by interpolation. Further, this confounding effect is
reduced by regressing out 12 head motion parameters (including 3 translation
parameters, 3 rotation parameters, and their first-order derivatives) and
scrubbing variables. One effect of head motion is increased short-distance
correlations and decreased long-distance correlations. Interestingly, it has
been reported that scrubbing high-motion frames from fMRI data can decrease
short-term correlations and increase long-term correlations [38], indicating
that the variance originating from head motion can be explained by scrubbing
variables.

To remove other nuisance effects, the aCompCor method [37] is utilized, which
is a component-based method with anatomical noise ROIs. Specifically, WM and
CSF masks are used to define the WM and CSF areas as noise ROIs. Then, five
principal components (PCs) for each noise ROI are calculated via principal
component analysis. Lastly, the five PCs from WM and five PCs from CSF are
entered into the linear model as regressors. Additionally, the linear trend is
removed by adding a linear regressor into the GLM. Finally, the residual time
series are band-pass filtered at [$0.01-0.1$] Hz to retain neuroactivity-
related intrinsic signal fluctuations.

###  2.4 Functional Brain Network Construction

####  2.4.1 Nodes Definition

In functional brain network analyses, nodes are typically defined as brain
regions that are structurally or functionally distinct. However, delineating
the boundaries of brain regions is a complex and continually evolving research
field. Over the past few decades, numerous atlases have been developed,
including the Automated Anatomical Labeling (AAL) atlas [45, 46], the 7- and
17-Network Parcellation [47], the 400-, 600-, 800-, and 1000-area parcellation
[48], the Atlas of Intrinsic Connectivity of Homotopic Areas (AICHA) [49], the
Human Connectome Project Multi-Modal Parcellation (HCP-MMP) [50], the Cortical
Area Parcellation [51] released by a group in Washington University, and the
Human Brainnetome Atlas [52]. In the current study, we use a customized atlas
named the “Harvard-Oxford-AAL” atlas, which includes 91 cortical ROIs and 15
subcortical ROIs from the Harvard-Oxford Atlas distributed with the FMRIB
Software Library (FSL), as well as 26 cerebellum ROIs from the AAL atlas. An
illustration of all 132 ROIs can be found in Figure 4, and Table
LABEL:tab:atlas.info provides additional details. Figure 5 displays the
contours of cortical regions.

Figure 4: Brain Atlas. It is a customized brain regions parcellation scheme, combining (1) 91 cortical regions (from the Harvard-Oxford Atlas), (2) 15 subcortical structures (from the Harvard-Oxford Atlas), and (3) 26 cerebellum regions (from the AAL Atlas). In the 3-D rendering of subcortical and cerebellum regions, spatial smoothing is applied for better visualization. (A: anterior; P: posterior; L: left; R: right; S: superior) Figure 5: Contours of Cortical Regions. The contours of 91 cortical regions defined in the Harvard-Oxford Atlas. Number is the ID for each region (Table LABEL:tab:atlas.info). Table 3: ID, Names, and Abbreviations of Brain Regions ID | Abbrev. | Brain Region
---|---|---
1 | FP r | Frontal Pole Right
2 | FP l | Frontal Pole Left
3 | IC r | Insular Cortex Right
4 | IC l | Insular Cortex Left
5 | SFG r | Superior Frontal Gyrus Right
6 | SFG l | Superior Frontal Gyrus Left
7 | MidFG r | Middle Frontal Gyrus Right
8 | MidFG l | Middle Frontal Gyrus Left
9 | IFG tri r | Inferior Frontal Gyrus, pars triangularis Right
10 | IFG tri l | Inferior Frontal Gyrus, pars triangularis Left
11 | IFG oper r | Inferior Frontal Gyrus, pars opercularis Right
12 | IFG oper l | Inferior Frontal Gyrus, pars opercularis Left
13 | PreCG r | Precentral Gyrus Right
14 | PreCG l | Precentral Gyrus Left
15 | TP r | Temporal Pole Right
16 | TP l | Temporal Pole Left
17 | aSTG r | Superior Temporal Gyrus, anterior division Right
18 | aSTG l | Superior Temporal Gyrus, anterior division Left
19 | pSTG r | Superior Temporal Gyrus, posterior division Right
20 | pSTG l | Superior Temporal Gyrus, posterior division Left
21 | aMTG r | Middle Temporal Gyrus, anterior division Right
22 | aMTG l | Middle Temporal Gyrus, anterior division Left
23 | pMTG r | Middle Temporal Gyrus, posterior division Right
24 | pMTG l | Middle Temporal Gyrus, posterior division Left
25 | toMTG r | Middle Temporal Gyrus, temporooccipital part Right
26 | toMTG l | Middle Temporal Gyrus, temporooccipital part Left
27 | aITG r | Inferior Temporal Gyrus, anterior division Right
28 | aITG l | Inferior Temporal Gyrus, anterior division Left
29 | pITG r | Inferior Temporal Gyrus, posterior division Right
30 | pITG l | Inferior Temporal Gyrus, posterior division Left
31 | toITG r | Inferior Temporal Gyrus, temporooccipital part Right
32 | toITG l | Inferior Temporal Gyrus, temporooccipital part Left
33 | PostCG r | Postcentral Gyrus Right
34 | PostCG l | Postcentral Gyrus Left
35 | SPL r | Superior Parietal Lobule Right
36 | SPL l | Superior Parietal Lobule Left
37 | aSMG r | Supramarginal Gyrus, anterior division Right
38 | aSMG l | Supramarginal Gyrus, anterior division Left
39 | pSMG r | Supramarginal Gyrus, posterior division Right
40 | pSMG l | Supramarginal Gyrus, posterior division Left
41 | AG r | Angular Gyrus Right
42 | AG l | Angular Gyrus Left
43 | sLOC r | Lateral Occipital Cortex, superior division Right
44 | sLOC l | Lateral Occipital Cortex, superior division Left
45 | iLOC r | Lateral Occipital Cortex, inferior division Right
46 | iLOC l | Lateral Occipital Cortex, inferior division Left
47 | ICC r | Intracalcarine Cortex Right
48 | ICC l | Intracalcarine Cortex Left
49 | MedFC | Frontal Medial Cortex
50 | SMA r | Supplementary Motor Cortex Right
51 | SMA L | Supplementary Motor Cortex Left
52 | SubCalC | Subcallosal Cortex
53 | PaCiG r | Paracingulate Gyrus Right
54 | PaCiG l | Paracingulate Gyrus Left
55 | AC | Cingulate Gyrus, anterior division
56 | PC | Cingulate Gyrus, posterior division
57 | Precuneous | Precuneous Cortex
58 | Cuneal r | Cuneal Cortex Right
59 | Cuneal l | Cuneal Cortex Left
60 | FOrb r | Frontal Orbital Cortex Right
61 | FOrb l | Frontal Orbital Cortex Left
62 | aPaHC r | Parahippocampal Gyrus, anterior division Right
63 | aPaHC l | Parahippocampal Gyrus, anterior division Left
64 | pPaHC r | Parahippocampal Gyrus, posterior division Right
65 | pPaHC l | Parahippocampal Gyrus, posterior division Left
66 | LG r | Lingual Gyrus Right
67 | LG l | Lingual Gyrus Left
68 | aTFusC r | Temporal Fusiform Cortex, anterior division Right
69 | aTFusC l | Temporal Fusiform Cortex, anterior division Left
70 | pTFusC r | Temporal Fusiform Cortex, posterior division Right
71 | pTFusC l | Temporal Fusiform Cortex, posterior division Left
72 | TOFusC r | Temporal Occipital Fusiform Cortex Right
73 | TOFusC l | Temporal Occipital Fusiform Cortex Left
74 | OFusG r | Occipital Fusiform Gyrus Right
75 | OFusG l | Occipital Fusiform Gyrus Left
76 | FO r | Frontal Operculum Cortex Right
77 | FO l | Frontal Operculum Cortex Left
78 | CO r | Central Opercular Cortex Right
79 | CO l | Central Opercular Cortex Left
80 | PO r | Parietal Operculum Cortex Right
81 | PO l | Parietal Operculum Cortex Left
82 | PP r | Planum Polare Right
83 | PP l | Planum Polare Left
84 | HG r | Heschl’s Gyrus Right
85 | HG l | Heschl’s Gyrus Left
86 | PT r | Planum Temporale Right
87 | PT l | Planum Temporale Left
88 | SCC r | Supracalcarine Cortex Right
89 | SCC l | Supracalcarine Cortex Left
90 | OP r | Occipital Pole Right
91 | OP l | Occipital Pole Left
92 | Thalamus r | Thalamus Right
93 | Thalamus l | Thalamus Left
94 | Caudate r | Caudate Right
95 | Caudate l | Caudate Left
96 | Putamen r | Putamen Right
97 | Putamen l | Putamen Left
98 | Pallidum r | Pallidum Right
99 | Pallidum l | Pallidum Left
100 | Hippocampus r | Hippocampus Right
101 | Hippocampus l | Hippocampus Left
102 | Amygdala r | Amygdala Right
103 | Amygdala l | Amygdala Left
104 | Accumbens r | Accumbens Right
105 | Accumbens l | Accumbens Left
106 | Brain-Stem | Brain-Stem
107 | Cereb1 l | Cerebellum Crus1 Left
108 | Cereb1 r | Cerebellum Crus1 Right
109 | Cereb2 l | Cerebellum Crus2 Left
110 | Cereb2 r | Cerebellum Crus2 Right
111 | Cereb3 l | Cerebellum 3 Left
112 | Cereb3 r | Cerebellum 3 Right
113 | Cereb45 l | Cerebellum 4 5 Left
114 | Cereb45 r | Cerebellum 4 5 Right
115 | Cereb6 l | Cerebellum 6 Left
116 | Cereb6 r | Cerebellum 6 Right
117 | Cereb7 l | Cerebellum 7b Left
118 | Cereb7 r | Cerebellum 7b Right
119 | Cereb8 l | Cerebellum 8 Left
120 | Cereb8 r | Cerebellum 8 Right
121 | Cereb9 l | Cerebellum 9 Left
122 | Cereb9 r | Cerebellum 9 Right
123 | Cereb10 l | Cerebellum 10 Left
124 | Cereb10 r | Cerebellum 10 Right
125 | Ver12 | Vermis 1 & Vermis 2
126 | Ver3 | Vermis 3
127 | Ver45 | Vermis 4 & Vermis 5
128 | Ver6 | Vermis 6
129 | Ver7 | Vermis 7
130 | Ver8 | Vermis 8
131 | Ver9 | Vermis 9
132 | Ver10 | Vermis 10

####  2.4.2 Edges Definition

In weightless bidirectional graph analysis, defining the edges involves two
steps: (1) determining the connectivity measure, and (2) applying a threshold
to create weightless connections.

In functional brain network analysis using BOLD fMRI data, there are two major
types of connectivity measures: correlation measures and regression measures.
Correlation measures utilize a correlation coefficient to quantify the
statistical relationship between two BOLD signals, such as Pearson’s linear
correlation coefficient, intra-class correlation, or rank-based correlation
coefficients like Spearman’s rank correlation coefficient and Kendall tau rank
correlation coefficient. Various correlation measures have their own
characteristics and usability, but they all assume values ranging from $-1$ to
$+1$, where $\pm 1$ indicates the strongest relationship, and 0 represents the
weakest relationship. Regression measures, on the other hand, use a regression
coefficient to measure the relationship between variables. Regression measures
offer more flexibility, such as when analyzing the relationship between two
variables using bivariate regression measures or controlling for all other
factors using multivariate regression measures.

In this study, the functional connectivity between two ROIs is defined using
Pearson’s linear correlation coefficient, which is further subjected to Fisher
z-transformation, a standard step in functional connectivity analyses for
hypothesis testing of the population correlation coefficient. To construct a
weightless graph from these weighted connections, thresholding is necessary.
Various thresholding methods exist, categorized into absolute and relative
thresholding methods. Absolute thresholding applies a fixed threshold to
connectivity measures, while relative thresholding retains a specific
proportion of connections within a graph. In this study, relative thresholding
is used, retaining only the top $15\%$ connections.

In fMRI connectivity analysis, negative connections (anti-correlations) can be
observed using linear correlation coefficients. Whether these negative
connections are spurious or physiologically meaningful is an open question. To
investigate their effects on the results, three parallel experiments were
conducted: exploring the properties of the negative functional network by
retaining only anti-correlations, exploring the properties of the positive
functional network by eliminating all anti-correlations and exploring the
properties of the mixed functional network using absolute correlation values.

###  2.5 Graph-Theory-Based Network Properties

####  2.5.1 Degree and Cost

##### Nodal Degree and Cost

Two ways to quantify the centrality of each node in a graph are measuring the
number and proportion of edges directly linked to that node – that is,
calculating the degree and cost respectively. Degree and cost are similarly
defined and related, both of which are meant to characterize the local
connectivity of each node. Here, local connectivity depicts how densely one
node is connected to all other nodes in the same graph. If one node is densely
connected to others, then its degree or cost will be high. On the contrary, if
one node is sparsely connected, its degree or cost will be low.

  * •

_Nodal degree_ represents the number of edges directly connected to a node and
is defined as

$K_{i}=\sum_{i\neq j\in N}a_{ij},$ (1)

where $K_{i}$ is the degree of node $i$, $N$ stands for the set of all nodes
in a graph, and $a$ is a binary value (0 for non-connection, 1 for connection)
indicating the connectivity, in other words, the connection status.

  * •

_Nodal cost_ represents the proportion of direct edges among all possible
connections and is defined as

$C_{i}=\frac{1}{n-1}\sum_{i\neq j\in N}a_{ij},$ (2)

where $C_{i}$ is the cost of node $i$, $N$ is the set of all nodes, $a_{ij}$
represents connectivity, and $n$ is the number of nodes in the graph.

Actually, it is obvious that the equation $C_{i}=\frac{K_{i}}{n-1}$ holds. The
major reason why cost is defined in addition to degree is that the variation
derived from the size of the graph needs to be corrected when the cross-graph
comparison is performed. For example, directly comparing the degree of a node
in a 90-nodes-graph and that in a 264-nodes-graph is meaningless since the
former is most likely smaller than the latter. Yet, the cost of a node can be
used for direct comparison in this scenario. When degree and cost refer to the
individual properties of a node, we usually call it nodal degree and nodal
cost, or the degree of a node and the cost of a node.

##### Mean Degree and Cost

From the nodal property, we can calculate the global property of the whole
graph – the degree or cost of a graph, which is simply the average of nodal
degree or nodal cost. The degree and cost of a graph are also called mean
degree and mean cost. While nodal degree and cost are metrics for measuring
nodal centrality, mean degree and cost are measures of network integration,
with larger values denoting a higher level of integration, and vice versa.

  * •

_Mean degree_ is defined as

$K=\frac{1}{n}\sum_{i\in N}K_{i}=\frac{1}{n}\sum_{i\in N}\sum_{i\neq j\in
N}a_{ij}.$ (3)

  * •

_Mean cost_ is defined as

$C=\frac{1}{n}\sum_{i\in N}C_{i}=\frac{1}{n}\sum_{i\in N}\frac{\sum_{i\neq
j\in N}a_{ij}}{n-1}.$ (4)

####  2.5.2 Shortest Path Length

##### Average Shortest Path Length

In a binary bidirectional graph, all edges (_i.e._ , connections or links) are
weightless and bidirectional. Two nodes within a network can be directly or
indirectly connected, with many possible paths between them. The shortest path
length between two nodes is defined as the minimum distance of all possible
paths connecting them. If no path exists between the two nodes, the shortest
path length is defined as $+\infty$. The average shortest path length of all
paths emitted from a node quantitatively describes the centrality of that node
and can be used as a measure of integration. It is calculated as

$L_{i}=\frac{1}{n-1}\sum_{i\neq j\in N}d_{ij},$ (5)

where $L_{i}$ is the average shortest path length of node $i$, $d_{ij}$ is the
shortest path length between node $i$ and $j$, $N$ is the set of all nodes,
and $n$ is the number of nodes in the graph. Like nodal degree and cost, the
average shortest path length is a measure of nodal centrality, with a smaller
value indicating a higher level of centrality. However, the definition of
infinite distance means that if a node is disconnected from node $i$, the
average shortest path length of node $i$ becomes infinite. This is referred to
as "the infinity property of the average shortest path length" in this
article. It should be noted that this is a drawback of the definition of the
average shortest path length because any disconnected node will render this
metric infinite.

##### Characteristic Path Length

The average shortest path length is a measure of nodal property in a graph,
and its corresponding global property is called the characteristic path length
of a graph [53]. It is defined as

$L=\frac{1}{n}\sum_{i\in N},$ (6) $L_{i}=\frac{1}{n}\sum_{i\in
N}\frac{\sum_{i\neq j\in N}d_{ij}}{n-1},$ (7)

which is simply the mean of the average shortest path length across all nodes
in a graph. Like mean degree and cost, the characteristic path length is a
measure of network integration. The smaller its value, the more integrated the
network is. An integrated network implies that, on average, each node has
short paths to all other nodes, meaning that the nodes are closely linked to
one another. However, due to the infinity property of the average shortest
path length, any isolated node in the graph will also make the characteristic
path length infinite. Therefore, its application is restricted to connected
graphs. To overcome this limitation, researchers have defined metrics not
based on the shortest path length, such as global efficiency.

####  2.5.3 Clustering Coefficient

##### Nodal Clustering Coefficient

In a graph, the neighbours of node $i$ are defined as all nodes that directly
connect to node $i$. In the analysis of small-world networks, the clustering
coefficient of node $i$ is defined by Watts and colleagues [53] as the
fraction of edges among all the possible edges in the subgraph of its
neighbours. It is calculated as

$CC_{i}=\frac{\sum_{p,q\in N{i}}a_{pq}}{K_{i}\left(K_{i}-1\right)},$ (8)

where $CC_{i}$ denotes the clustering coefficient of node $i$, $N_{i}$ denotes
the subgraph of its neighbours, $a_{pq}$ indicates connectedness, and $K_{i}$
is the degree of node $i$. As a measure of nodal property, the clustering
coefficient of a node is also referred to as the nodal clustering coefficient.
It characterizes the cliquishness or clustering property of a node.

##### Mean Clustering Coefficient

Similarly, the clustering coefficient of a graph is defined as the average of
all nodal clustering coefficients:

$CC=\frac{1}{n}\sum_{i\in N}CC_{i}=\frac{1}{n}\sum_{i\in N}\frac{\sum_{p,q\in
N_{i}}a_{pq}}{K_{i}\left(K_{i}-1\right)}.$ (9)

It is sometimes referred to as the mean clustering coefficient. Unlike mean
degree, mean cost, and characteristic path length, the mean clustering
coefficient is used to measure the segregation of a network. It reflects, on
average, the extent of local connectivity around a node.

####  2.5.4 Efficiency

The existence of the infinity property means that the rational application of
the average shortest path length or characteristic path length is subject to
the connectivity of the graph, which requires that all nodes in the graph are
connected. To address this limitation, Latora and colleagues [54] proposed the
concept of efficiency to describe the behaviour of a network. As its name
suggests, efficiency is meant to quantify how efficient the communication or
information exchange between nodes is. Different papers use various
terminologies to refer to the concept of efficiency in graph-theory-based
metrics, which can lead to confusion. In this article, we provide a brief
explanation of the terms used. First, there are two classes of metrics - one
for nodal property and the other for the overall property of the whole graph.
Some may use "local" and "global" to indicate nodal and graph-level
properties. However, "local" and "global" have specific meanings in the
definitions of efficiency - local efficiency and global efficiency are two
different metrics. Therefore, when discussing efficiency, the terms "local"
and "global" solely indicate the types of efficiency, but not the scope.
Second, to clarify the scope of a metric, we state it clearly in the name. For
example, "global efficiency of a node" indicates that it is a nodal property,
and the measure here is "global efficiency". Another example is that "local
efficiency of a graph" refers to graph-level local efficiency. We avoid using
terms like "nodal local efficiency" or "nodal global efficiency" to prevent
confusion.

##### Global Efficiency of a Node

Efficiency is closely related to the shortest path length by definition. Let
us assume that the transfer of information in a network is parallel, i.e., the
signal from a node is sent concurrently through all edges. As a result, the
efficiency of communication between node $i$ and $j$ is determined by the
shortest path between them. The efficiency is then defined as the inverse of
the shortest path length: $\epsilon_{ij}=\frac{1}{d_{ij}}$, where
$\epsilon_{ij}$ is the efficiency of information transfer and $d_{ij}$ is the
shortest path length. The global efficiency of node $i$ is defined as

$E_{i,glob}=\frac{1}{n-1}\sum_{i\neq j\in N}\epsilon_{ij}.$ (10)

With this definition, the global efficiency of a node can be meaningfully
calculated on a disconnected graph, where isolated nodes exist. For example,
when node $i$ and $j$ are disconnected, $d_{ij}=+\infty$ and
$\epsilon_{ij}=0$, and therefore, $E_{i,glob}$ or $E_{j,glob}$ is not affected
by the lack of connection.

##### Global Efficiency of a Graph

We can calculate the global efficiency of a graph, which is a measure of the
whole network, by averaging the global efficiency of all nodes. The global
efficiency of a node is the reciprocal of the average shortest path length
between the node and all other nodes in the network. If a graph is
disconnected, the global efficiency is still valid as it is not affected by
disconnection. Specifically, the global efficiency of a graph is defined as
follows:

$E_{glob}=\frac{1}{n}\sum_{i\in N}E_{i,glob}=\frac{1}{n}\sum_{i\in
N}\frac{\sum_{i\neq j\in N}\epsilon_{ij}}{n-1}=\frac{1}{n}\sum_{i\in
N}\frac{\sum_{i\neq j\in N}d_{ij}^{-1}}{n-1}.$ (11)

The global efficiency and characteristic path length are both based on the
shortest path length between nodes and serve as evaluation indices of network
integration. A highly integrated network has a high global efficiency or a
short characteristic path length. The difference between global efficiency and
characteristic path length can be understood through an analogy provided by
the original paper [54]. Global efficiency is like a parallel system where
information is sent simultaneously through all paths, while the characteristic
path length is like a sequential system where information is transferred
through one path at a time. To further illustrate the analogy, we can define
efficiency as the number of packets transferred in a time unit. In a parallel
system (left column in Figure 6), packets are sent concurrently through all
paths, and the shortest path length determines the efficiency. In a sequential
system (right column in Figure 6), packets are transferred through one path at
a time, and the efficiency is the number of packets successfully transferred
through an imaginary "average path" in a time unit. Long paths decrease the
efficiency by lengthening the "average path".

Figure 6: Parallel and Sequential System. In a parallel system (left column),
the delivery of information from node $i$ to node $j$ occurs concurrently
through every path, making the shortest path length the bottleneck of system
efficiency. In a sequential system (right column), the transmission of
information can be considered as sending a packet through one random path at a
time. Therefore, the system efficiency is determined by the imaginary “average
path”. A parallel system is not affected by long paths while a sequential
system is.

##### Local Efficiency of a Node

The global efficiency of a node measures its communication with all other
nodes in the graph, while the local efficiency quantifies the global
efficiency of a subgraph consisting only of its neighbours. Specifically, the
local efficiency of node $i$ is defined as the global efficiency of its
subgraph of neighbours:

$E_{i,local}=\frac{1}{K_{i}}\sum_{p\in N_{i}}\frac{\sum_{p\neq q\in
N_{i}}\epsilon_{pq}}{K_{i}-1}=\frac{\sum_{p\neq q\in
N_{i}}d_{pq}^{-1}}{K_{i}\left(K_{i}-1\right)},$ (12)

where $K_{i}$ is the degree of node $i$, $N_{i}$ is the subgraph of its
neighbours, $\epsilon_{pq}$ represents the efficiency of the shortest path
between node $p$ and $q$, and $d_{pq}$ is the shortest path length.

##### Local Efficiency of a Graph

The local efficiency of a graph is the average local efficiency of all
subgraphs, which is the mean of the local efficiencies of all nodes:

$E_{local}=\frac{1}{n}\sum_{i\in N}E_{i,local}=\frac{1}{n}\sum_{i\in
N}\frac{\sum_{p\neq q\in N_{i}}d_{pq}^{-1}}{K_{i}\left(K_{i}-1\right)},$ (13)

where $n$ is the number of nodes in the graph, $N$ is the set of all nodes,
$N_{i}$ is the subgraph of neighbours of node $i$, $K_{i}$ is the degree of
node $i$, and $d_{pq}$ is the shortest path length between nodes $p$ and $q$.
Similar to the clustering coefficient of a graph, the local efficiency of a
graph measures network segregation. As each node is not included in its own
subgraph, the local efficiency of a graph reveals the fault tolerance of the
network by showing how efficiently communication within the network is
maintained when a node is removed, on average.

####  2.5.5 Betweenness

##### Nodal Betweenness

Betweenness is a measure of centrality, similar to nodal degree and cost, that
captures the importance of nodes in controlling information flow. It
quantifies the proportion of all shortest paths in a graph that pass through a
given node:

$B_{i}=\frac{1}{\left(n-1\right)\left(n-2\right)}\sum_{i\neq p\neq q\in
N}\frac{\rho_{pq}^{\left(i\right)}}{\rho_{pq}},$ (14)

where $B_{i}$ is the betweenness of node $i$, $n$ is the number of nodes in
the graph, $\rho_{pq}^{\left(i\right)}$ represents the number of shortest
paths between nodes $p$ and $q$ that pass through node $i$, and $\rho_{pq}$
refers to the total number of shortest paths between nodes $p$ and $q$. Nodal
betweenness is an important metric for identifying connector hubs in a
network. Nodes with high betweenness centrality can be thought of as the
busiest ports in real-world networks. Removing such nodes can significantly
decrease the global efficiency of the network by breaking down many short
paths.

##### Mean Betweenness

Mean betweenness refers to the betweenness of a graph. It is defined as the
average nodal betweenness:

$B=\frac{1}{n}\sum_{i\in
N}B_{i}=\frac{1}{n\left(n-1\right)\left(n-2\right)}\sum_{i\neq p\neq q\in
N}\frac{\rho_{pq}^{\left(i\right)}}{\rho_{pq}}.$ (15)

###  2.6 Statistical Analysis

Initially, six nodal metrics, comprising degree (or cost), shortest path
length, clustering coefficient, global efficiency, local efficiency, and
betweenness, are computed for each region of interest (ROI) delineated in a
brain atlas. Subsequently, two-tailed two-sample Student’s t-tests are
utilized to compare each nodal metric between two groups, namely AD and NC.
Finally, the False Discovery Rate (FDR) correction is executed to rectify for
numerous comparisons across 132 ROIs. Outcomes with FDR-corrected p-value less
than 0.05 are deemed statistically significant.

###  2.7 Results

####  2.7.1 Quality Assessment for Image Processing

The initial phase in deriving dependable conclusions is rational and high-
quality image processing. Quality evaluation and control of image processing
are therefore essential. The quality assessment encompasses the manual, semi-
automatic, or automatic inspection of the quality of preprocessed images.
Quality control, on the other hand, refers to the actions taken in response to
quality assessment. In certain situations, researchers may need to modify
parameters and rerun specific experiments to obtain better image processing
outcomes. In the most extreme cases, the exclusion of subjects from subsequent
analyses may be necessary.

##### Segmentation

Figure 7 presents a T1-weighted MR image segmentation illustration. Precise
brain tissue probability maps (sub-figures 2-4 and 6-8 in Figure 7) can be
produced through this segmentation process. The combination of segmentation
and normalization also generates tissue masks in standard space (sub-figures
10-12 in Figure 7).

Figure 7: Segmentation of T1-weighted MRI. (1) Raw T1-weighted MR image in
native space. (2-4) CSF, GM, WM probability maps in native space. (5)
Normalized T1-weighted image in MNI-152 space. (6-8) Normalized CSF, GM, WM
probability maps in MNI-152 space. (9) Skull-stripped normalized T1 image.
(10-12) Normalized and eroded CSF, GM, WM masks.

##### Normalization

The assessment of normalization quality can be accomplished through visual
inspection, wherein the contour of the template is superimposed on the
normalized image. Figure 8 displays the normalized T1-weighted image, CSF
probability map, GM probability map, and WM probability map, which closely
align with the template contour. Figure 9 presents the BOLD fMRI in both
standard space (Figure 9 (1)) and native space (Figure 9 (2)). It demonstrates
that direct normalization of the BOLD MR image, which has relatively low
spatial resolution, is satisfactory.

Figure 8: Normalization of T1-weighted MRI. (1) The normalized T1-weighted MR
image. (2) The normalized CSF probability map. (3) The normalized GM
probability map. (4) The normalized WM probability map. All are overlapped
with the contour (in yellow) of template. Figure 9: Normalization of BOLD
fMRI. (1) The normalized BOLD fMRI overlapped with the contour (in yellow) of
template. (2) The BOLD fMRI in native space overlapped with the contour (in
yellow) of template.

####  2.7.2 Artefacts Removal

Figure 10 and Figure 11 depict the BOLD signals before and after the removal
of artefacts, respectively. The BOLD signals of all voxels within the GM mask
are stacked vertically in time series boxes. It is evident that certain global
signal variances (e.g., the striped pattern in blue box A in Figure 10) have
been eliminated (blue box B in Figure 10). Alongside the BOLD signal
visualization, three variables are plotted: (1) the global BOLD signal changes
in z-values, (2) the integrated head motion estimator, calculated from 6
movement parameters, and (3) the scrubbing variable exported from the Artifact
Detection Toolbox (ART). For the global signal changes, the mean BOLD signal
(the global mean) across all GM voxels is first computed and then converted to
z-values. The integrated head motion estimator from ART is a composite motion
measure that estimates the maximum voxel displacement resulting from the
combined effect of translation and rotation displacement measures. The
scrubbing variable is a frame identifier pointing out the outlier scans.

Note that two types of global signal variances are highlighted in Figure 10
and Figure 11. In each red box, we can see that sudden global signal changes
always occur with severe head motion, particularly in box 6 and 7 (Figure 11),
where MR scans are identified as outliers at the same time point. Therefore,
we can conclude that the red boxes (boxes 4 and 5 in Figure 10, boxes 6-9 in
Figure 11) indicate global signal changes potentially related to subject
motion. The purple boxes (boxes 1 and 2 in Figure 10, box 3 in Figure 11)
highlight global signal changes that may originate from sources other than
head movement.

Figure 10: BOLD Signal Before and After Artefacts Removal (1). The BOLD
signals (before and after artefacts removal) of 2 subjects are shown in time
series boxes, along with which are the BOLD global signal (GS) changes in
z-values, the head motion estimator, and the scrubbing variable (“outlier”).
Blue boxes (boxes A & B) highlights the removal of global signal changes shown
as stripes pattern. Purple boxes (boxes 1 & 2) point out the appearances of
global signal changes that maybe originate from sources other than head
movement. Red boxes (boxes 4 & 5) indicate the occurrences of global signal
changes that are potentially related to the subject motion. Figure 11: BOLD
Signal Before and After Artefacts Removal (2). The BOLD signals (before and
after artefacts removal) of another 2 subjects are shown in time series boxes,
along with which are the BOLD global signal (GS) changes in z-values, the head
motion estimator, and the scrubbing variable (“outlier”). Purple boxes (box 3)
point out the appearances of global signal changes that maybe originate from
sources other than head movement. Red boxes (box 6-9) indicate the occurrences
of global signal changes that are potentially related to the subject motion.

The impact of artefacts removal on the distribution of functional connections
(FCs) can be observed from Figure 12. First, GM voxels are segmented into 1000
clusters, defining 1000 nodes. Second, mean BOLD signals are computed from
each cluster. Third, Pearson’s linear correlation coefficient ($r$) is used to
measure FC. Finally, the distribution of FCs is plotted. Similar effects can
be observed in four subjects. The FCs distribution before artefacts removal
exhibits varying degrees of positive skewness, indicating the presence of a
systematic bias towards positive FCs. However, after removing artefacts, it
becomes approximately normally distributed with a mean close to zero.

Figure 12: Distribution of FCs Before and After Artefacts Removal. The
distribution of functional connections (FCs) of 4 subjects are shown. Before
artefacts removal, the distributions are right-skewed and sometimes non-bell-
shaped. After artefacts removal, the FCs distributions become normal with
approximate zero-mean and bell-shape.

Furthermore, Figure 13 provides additional information on FCs by incorporating
distance-related data. The definition of FCs in Figure 13 is the same as in
Figure 12. To draw the distance-FCs maps in Figure 13, an extra step is
required: calculating the distance (in $mm$) between two nodes. As shown in
the first row of Figure 13, each scattered dot represents the mean FC at a
specific distance, and the shaded area indicates the standard deviation.
Compared to the FCs distributions in Figure 12, the FCs are shifted towards
positive values before removing artefacts. After artefacts removal, most of
the mean FCs are near zero, except for short-distance FCs. The distance-FCs
maps for four subjects (Figure 13) demonstrate similar distance-related FCs
properties before and after artefacts removal. They also indicate that
removing artefacts can reduce the bias towards positive values in mediate and
long distance FCs. This effect is confirmed in Figure 14, where the distance-
FCs maps for all subjects are superimposed on top of each other. It shows
that, after removing artefacts, FCs are less divergent and more centred around
zero at mediate to long distances, while short-distance FCs are stronger than
their mediate and long-distance counterparts.

Figure 13: Distance-FCs Map Before and After Artefacts Removal. The figure
shows the functional connectivity plotted against distance (in $mm$). Each
scattering dot represents the mean functional connection (FC) at a specific
distance, and the shadow area indicates the standard deviation. Similar to the
findings in Figure 12, FCs are shifted to the positive side before artefacts
removal. After artefacts removal, most of the mean FCs are near zero except
the short-distance ones. Figure 14: Distance-FCs Map Before and After
Artefacts Removal (All Subjects). The distributions and distance-related
properties of FCs before and after artefacts removal are shown. It suggests
that FCs are less divergent and more zero-centred in mediate to long distances
after artefacts removal. Short-distance FCs are stronger compared with their
long-distance counterparts.

###  2.8 Abnormal Functional Brain Network Properties in AD

As described in the method section, three types of networks are constructed to
explore the effects of negative functional connections (anti-correlations).
The positive network is created by retaining only the top $15\%$ positive FCs.
The negative network is constructed by retaining the top $15\%$ negative FCs.
In the mixed network, the anti-correlations are first replaced by their
absolute values and then thresholded in the same manner as the positive
network.

####  2.8.1 Abnormalities in Positive Network

Table 4 illustrates significant between-group differences (FDR-p $<$ $0.05$)
found in six nodal metrics, namely local efficiency, clustering coefficient,
degree, cost, average shortest path length, and global efficiency. The results
consistently indicate a singular direction of abnormality. A summary of the
statistically significant findings is presented below.

Table 4: Abnormal Graph Metrics in AD (Results from Positive Network) Metrics | Region |  T Statistics (AD - NC) | p-value |  FDR-p
---|---|---|---|---
Local Efficiency | PT l | -4.59 | 0.000005 | 0.000701 [t]
pMTG r | -3.70 | 0.000234 | 0.014016
PT r | -3.62 | 0.000319 | 0.014016
CO r | -3.28 | 0.001113 | 0.036481
aSTG r | -3.21 | 0.001382 | 0.036481
HG r | -3.12 | 0.001882 | 0.041105
Cuneal l | -3.08 | 0.002180 | 0.041105
CO l | -2.99 | 0.002913 | 0.048057 [b]
Clustering Coefficient | PT l | -4.26 | 0.000024 | 0.003168
Average Shortest Path Length | Cereb3 r | -3.96 | 0.000083 | 0.010937 [t]
Ver8 | -3.69 | 0.000246 | 0.012355
Cereb3 l | -3.65 | 0.000281 | 0.012355 [b]
Global Efficiency | Cereb3 r | 4.53 | 0.000007 | 0.000948 [t]
Cereb3 l | 3.64 | 0.000293 | 0.018702
Cereb6 r | 3.54 | 0.000425 | 0.018702
Cereb45 r | 3.32 | 0.000965 | 0.026026
Ver45 | 3.31 | 0.000986 | 0.026026
Ver3 | 3.24 | 0.001274 | 0.028018
Ver8 | 3.12 | 0.001900 | 0.035825
Cereb8 r | 3.06 | 0.002285 | 0.037695 [b]
Degree / Cost | Cereb3 r | 4.39 | 0.000013 | 0.001744 [t]
Cereb6 r | 3.47 | 0.000550 | 0.036294 [b]

  * •

Compared with the NC group, decreased local efficiencies are discovered in the
bilateral Central Opercular Cortex (CO), the bilateral Planum Temporale (PT),
the left Cuneal, the right Heschl’s Gyrus (HG), the right posterior Middle
Temporal Gyrus (pMTG), and the right anterior Superior Temporal Gyrus (aSTG)
(Figure 15) in AD group.

  * •

Compared with the NC group, the decreased clustering coefficient is observed
in the left Planum Temporale (PT) (Figure 15) in the AD group.

  * •

Compared with the NC group, increased degrees and costs are only found in the
cerebellum, including the area 3 and area 6 of the right Cerebellum (Cereb3 &
Cereb6) (Figure 16) in the AD group.

  * •

Compared with the NC group, decreased average shortest path lengths are
observed in area 3 of the bilateral Cerebellum (Cereb3) and area 8 of the
Vermis (Ver 8) (Figure 16) in the AD group.

  * •

Compared with the NC group, increased global efficiencies are solely found in
the Cerebellum, including area 3 of the bilateral Cerebellum (Cereb3), the
area 4 & 5, area 6 and area 8 of the right Cerebellum (Cereb45, Cereb6 and
Cereb8), and the area 3, 4 & 5, and 8 of the Vermis (Ver3, Ver45, and Ver8)
(Figure 16) in the AD group.

Figure 15: Abnormal Graph Metrics in Cortical Cortices and Subcortical Regions
(Results from Positive Network). l: left; r: right; aSTG: anterior Superior
Temporal Gyrus; PT: Planum Temporale; pMTG: posterior Middle Temporal Gyrus;
CO: Central Opercular Cortex; Cuneal: Cuneal Cortex; HG: Heschl’s Gyrus.
Figure 16: Abnormal Graph Metrics in Cerebellum (Results from Positive
Network). l: left; r: right; Cereb: Cerebellum; Ver: Vermis.

####  2.8.2 Abnormalities in Mixed Network

As shown in Table 5, similar but less group-differences (FDR-p $<$ $0.05$) are
observed. Below is a summary of the results.

Table 5: Abnormal Graph Metrics in AD (Results from Mixed Network) Metrics | Region |  T Statistics (AD - NC) | p-value |  FDR-p
---|---|---|---|---
Local Efficiency | pMTG r | -4.88 | 0.000001 | 0.000179 [t]
CO r | -4.32 | 0.000018 | 0.001188
PT l | -3.68 | 0.000250 | 0.010996
aSTG l | -3.49 | 0.000519 | 0.017134
pMTG l | -3.15 | 0.001714 | 0.045251 [b]
Clustering Coefficient | pMTG r | -3.97 | 0.000081 | 0.010721 [t]
CO r | -3.67 | 0.000265 | 0.017475
PT l | -3.45 | 0.000601 | 0.026451 [b]
Global Efficiency | Cereb3 l | 3.69 | 0.000242 | 0.031955

  * •

Compared with the NC group, decreased local efficiencies are discovered in the
AD group in the bilateral posterior Middle Temporal Gyrus (pMTG), the left
Planum Temporale (PT), the left anterior Superior Temporal Gyrus (aSTG), and
the right Central Opercular Cortex (CO) (Figure 17).

  * •

Compared with the NC group, decreased clustering coefficients are observed in
the AD group in the left Planum Temporale (PT), the right Central Opercular
Cortex (CO), and the right posterior Middle Temporal Gyrus (pMTG) (Figure 17).

  * •

Compared with the NC group, increased global efficiency is only found in the
AD group in area 3 of the left Cerebellum (Cereb3) (Figure 18).

Figure 17: Abnormal Graph Metrics in Cortical Cortices and Subcortical Regions
(Results from Mixed Network). l: left; r: right; aSTG: anterior Superior
Temporal Gyrus; PT: Planum Temporale; pMTG: posterior Middle Temporal Gyrus;
CO: Central Opercular Cortex. Figure 18: Abnormal Graph Metrics in Cerebellum
(Results from Mixed Network). l: left; r: right; Cereb: Cerebellum.

###  2.9 Discussion

####  2.9.1 Removal of Global Effects

##### Global Signal

In functional neuroimaging, the global signal is defined as the mean time
series averaged across all brain voxels, including those in gray matter (GM),
white matter (WM), and cerebrospinal fluid (CSF). GM is typically the region
of interest (ROI) in brain fMRI studies because the dense neurons allow for
indirect reflection of neural activity through fluctuations in the BOLD
signal. However, the BOLD contrast is an indirect measure resulting from
complex interactions between mainly three factors: cerebral metabolic rate of
oxygen (CMRO2), blood volume, and blood flow. Thus, any factor that disrupts
the balance between these three parameters can alter the BOLD signal. For
example, head motion [44], cardiac and respiratory cycles [55], arterial CO2
concentration, cerebral autoregulation, blood pressure, and vasomotion [56]
can introduce non-neuronal variances to all BOLD time series. Consequently,
the global effects of artefacts can cause two BOLD signals to be more
statistically correlated, and the distribution of correlations between BOLD
signals is heavily skewed towards positive values. Many non-neuronal confounds
can contribute to the global signal. However, it has also been demonstrated
that the global signal can be tightly linked to neural activity [57].

##### Global Signal Regression

Global Signal Regression (GSR) is a method proposed to remove the confounding
effects of non-neuronal sources [58, 59]. GSR employs linear regression for
each voxel to remove variances explained by the global signal, and the
resulting residuals can be considered as the de-noised signal. One of the most
significant and widely discussed effects of GSR is the emergence of more anti-
correlations. The mathematical details of the algebraic operation of GSR and
the impact of mandating anti-correlations have been published [60, 61]. It is
explained that mathematically, the seed-to-voxel regression coefficients
($\beta$) have a mean of exactly zero, and the distribution of Pearson’s
linear correlation coefficients is approximately zero-centred with GSR. Murphy
and colleagues [60] were the first to mathematically demonstrate that GSR
forces approximately half of the correlations to become anti-correlations and
concluded that the anti-correlations between task-negative regions (i.e., DMN
regions) and task-positive regions are most likely an outcome of GSR.
Conversely, other researchers [61, 62] support the idea that the anti-
correlations after GSR have biological origins. To date, whether GSR should be
applied remains an open question, and contradictory recommendations have been
made [63]. Without a "gold standard," it is difficult to determine whether
anti-correlation is merely an artefact of the pre-processing strategy or
physiologically meaningful.

##### Drawbacks of GSR

On the one hand, it has been reported that GSR can not only eliminate non-
neuronal artefacts but also enhance the specificity of positive correlations
and their correspondence to structural connectivity [61]. On the other hand,
the drawbacks of GSR have been discussed. GSR can introduce spurious negative
correlations between brain regions [60, 64, 65] and alter the group comparison
of inter-regional correlations [65]. Murphy and colleagues [60] used simulated
data to demonstrate that GSR is not effective in removing nuisance confounds,
and the locations of anti-correlated areas depend on the relative phases of
the global signal and seed voxel time series. Saad and colleagues [65] used an
illustrative model to show that GSR distorts the inter-regional correlations
in a way that depends on the true correlation structure of the entire network
and regional size distribution. Anderson and colleagues [64] also demonstrated
that anti-correlations after GSR can be introduced even in completely
uncorrelated networks.

##### Alternatives to GSR

Since GSR adds uncertainty and interpretive complexity to scientific findings,
various alternatives to GSR have been explored and proposed. All of these
alternatives share the same motivation as GSR, which is to remove the common
non-neuronal BOLD signal fluctuations and reveal the true functional
connectivity. One type of technique involves the inclusion of physiological
data. Physiological signals must be recorded simultaneously with fMRI data
acquisition. When these physiological data are not available, there are still
methods to remove global effects. For example, the aCompCor technique [37]
uses anatomical masks to increase spatial specificity and principal components
instead of mean signal to eliminate confounding artefacts. More complex
methods, such as the Random Subspace Method for Functional Connectivity
(RSMFC) and the Affine Parameterization of Physiological Large-Scale Error
Correction (APPLECOOR), are summarized in the review paper by Kevin Murphy and
Michael Fox [63].

##### Reasons for Using aCompCor

The current study is a functional network study based on graph theory that
aims to investigate topological reconfiguration. Anti-correlations are treated
differently in three network construction strategies: (1) only positive
correlations are used for network construction in the analysis of positive
networks; (2) similarly, only anti-correlations are used for negative
networks; and (3) the absolute values of positive and negative correlations
are used for mixed networks. By comparing the results of these three
experiments, the effects of anti-correlations can be investigated.
Furthermore, it has been reported that both local and global topological
properties, quantified by local and global graph metrics, have higher
reliability when GSR is not applied compared to when it is applied [66].

####  2.9.2 Comparison of Findings in Positive and Mixed Network

For each network metric, consistent results are observed for both positive and
mixed networks. Here, the consistency is reflected not only by the same
direction of alteration, but not also by the rationality of network metrics
alterations. A summary of the consistency of findings from the positive and
mixed network are as followed.

##### The finding are not contradictory to each others.

Results (Table 4 and Table 5) show that only decreased local efficiency,
decreased clustering coefficient, decreased average shortest path length,
increased global efficiency, and increased degree (or cost) are found.

  * •

Decreased local efficiency and decreased clustering coefficient are
consistent. By definition, the local efficiency of node $i$ is the global
efficiency of the subgraph ($N_{i}$) of its neighbouring nodes. The clustering
coefficient of node $i$ is defined as the proportion of edges (connections)
among all the possible edges in the subgraph ($N_{i}$). Importantly, the
global efficiency is directly and inversely related to the shortest path
length: it is defined as the harmonic mean of the shortest path lengths. The
logical connection between local efficiency and clustering efficient can be
seen in the following example. If the neighbours of node $i$ are densely
interconnected which is equivalent to a high clustering coefficient for node
$i$, the shortest path length for each pair of nodes in $N_{i}$ would be low
which would lead to high global efficiency for subgraph $N_{i}$. And high
global efficiency for subgraph $N_{i}$ is equal to high local efficiency for
node $i$, by definition. Therefore, the clustering coefficient and local
efficiency are positively correlated. For example, decreased local efficiency
and decreased clustering coefficient are found in left Planum Temporale (PT)
(Table 4 & Table 5).

  * •

Decreased average shortest path length and increased global efficiency are
consistent. As explained above, the global efficiency of node $i$ is defined
as the harmonic mean of all shortest path length between node $i$ and all
other nodes in the same graph. Thus, the shortest path length and global
efficiency are negatively correlated.

  * •

Increased degree (or cost), increased global efficiency and decreased average
shortest path length are consistent. Although degree and cost are not directly
related to global efficiency in their definitions, they are more likely to be
positively correlated to the average shortest path length and global
efficiency. Supposed node $i$ has a high degree which means node $i$ is
directly connected with many nodes, a great deal of shortest paths emitted
from node $i$ will have length 1, decreasing the average shortest path length
for node $i$. For example, decreased average shortest path length, increased
global efficiency, and increased degree and cost are found in area 3 of the
right Cerebellum (Cereb3) (Table 4).

  * •

Global efficiency quantifies network integration, while local efficiency
quantifies network segregation. Though the altered global efficiencies and
local efficiencies found in the AD group are in opposite direction, the author
holds the opinion that they depict distinct aspects of network properties, and
are not strongly correlated.

##### The altered network properties found in the mixed network is consistent
with that of the positive network, but less in the amount.

  * •

Decreased local efficiencies are found in the left Planum Temporale (PT),
right posterior Middle Temporal Gyrus (pMTG), and right Central Opercular
Cortex (CO) in both positive and mixed network (Table 4 & Table 5).

  * •

Decreased clustering coefficient is found in left Planum Temporale (PT) in the
both positive and mixed network (Table 4 & Table 5).

  * •

Increased global efficiency is found in left area 3 of right Cerebellum
(Cereb3) in the both positive and mixed network (Table 4 & Table 5).

  * •

Altered metrics found only in the positive network include: decreased local
efficiency (in right PT, right aSTG, right HG, left Cuneal, and left CO),
decreased average shortest path length (in bilateral Cereb3 and Ver8),
increased global efficiency (in right Cereb3, right Cereb6, right Cereb45,
Ver45, Ver3, Ver8, and right Cereb8), and increased degree and cost (in right
Cereb3 and right Cereb6) (Table 4).

  * •

Altered metrics found only in the mixed network include decreased local
efficiency (in left aSTG and left pMTG), decreased clustering coefficient (in
right pMTG and right CO) (Table 5).

##### Compared with NC, weaker functional segregation is discovered in the AD
group in only cortical regions.

Decreased local efficiencies and clustering coefficients, which are two
measures of network segregation, are only observed in cortical cortices, but
not in any region of subcortical structures and cerebellum.

##### Compared with NC, stronger functional integration is discovered in the
AD group in only regions of subcortical structures and cerebellum.

Decreased average shortest path length, increased global efficiency, and
increased degree, which are measures of network integration, are only observed
in subcortical structures and cerebellum.

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