statistical signi cance of functional networks in the brain · 2 chapter 1. introduction 2013; deco...

Statistical Significance of

Functional Networks in the

Brain

Jiating Zhu

Master of Science

Artificial Intelligence

School of Informatics

University of Edinburgh

2016

Abstract

This thesis analyzed the functional networks in brain across one hundred subjects

by their functional magnetic resonance imaging (fMRI) data scanned during the

resting state. Standard pre-processing techniques have been explained and been

implemented on the data. The functional connectivity in the brain is measured

with temporal correlations among voxels selected in every cubic region. The brain

network is constructed by a selected threshold, at which the second largest con-

nected component merges to the largest connected component. We find that the

second largest component is scale-free. To explore the difference of the functional

network across subjects, k-means clustering with clustering coefficient and small-

worldnes of the second largest components are carried out. We compared the

clustering results with degree distribution parameters, and found that subjects in

different clusters have distinguishable difference in terms of the considered net-

work properties. Therefore, the proposed pipeline and method in this project

are suitable for future analysis on discriminating the functional brain of healthy

controls and Major Depression Disorder patients.

Keywords: Voxel-based functional network, Resting state fMRI, Second

largest component, K-means clustering, Depression.

iii

Acknowledgements

I would like express my sincere thanks to my supervisor, Dr Michael Herrmann,

for his support and guidance throughout my time as his student. His enthusiasm,

encouragement and teachings are priceless.

I thank Heather Sibley from Royal Edinburgh Hospital for providing me the

fMRI data and additional information for analysis. I also would like to thank

Shen Xueyi, who gave me a lot advice on understanding the brain images. I

thank Thomas Nikson for helping me logging in the computer in Royal Edinburgh

Hospital. I am grateful to the people who worked at the the same floor with me

at Royal Edinburgh Hospital. Their kindness made me feel relax and helped me

a lot.

I would like to thank Thomas Joyce. He gave me very useful feedback on my

analysis and report even very close to the deadline.

I am so grateful to the people mentioned above. I have learned a lot from

them, and their help is very important to me.

iv

Declaration

I declare that this thesis was composed by myself, that the work contained herein

is my own except where explicitly stated otherwise in the text, and that this work

has not been submitted for any other degree or professional qualification except

as specified.

(Jiating Zhu)

v

Contents

1 Introduction 1

2 Background 3

2.1 Resting state, Default mode network . . . . . . . . . . . . . . . . 3

2.2 Functional network during rest . . . . . . . . . . . . . . . . . . . 4

2.3 Data from Stratifying Resilience and Depression Longitudinally

(STRADL) project . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Resting state fMRI in Major Depression Disorder . . . . . . . . . 5

3 Methods 7

3.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Data pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Realignment . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.2 Slice timing . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.3 Coregistration . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.4 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.5 Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.6 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Brain extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Pre-processed results . . . . . . . . . . . . . . . . . . . . . 11

3.3.2 Brain representation . . . . . . . . . . . . . . . . . . . . . 13

3.4 Network extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Network properties . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 The properties of functional network connectivity . . . . . 15

3.5.2 Nodal properties . . . . . . . . . . . . . . . . . . . . . . . 18

3.5.3 Global network properties . . . . . . . . . . . . . . . . . . 18

3.5.4 Metrics selection . . . . . . . . . . . . . . . . . . . . . . . 20

vii

3.6 Network analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.6.1 Network construction . . . . . . . . . . . . . . . . . . . . . 21

3.6.2 Second largest component . . . . . . . . . . . . . . . . . . 22

3.6.3 Random graph . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6.4 Power-law degree distribution . . . . . . . . . . . . . . . . 27

3.7 Clustering evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7.1 Clustering with clustering coefficient and small-worldness . 30

3.7.2 Clustering analysis . . . . . . . . . . . . . . . . . . . . . . 30

4 Discussion 35

4.1 Pre-processing procedure . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Voxel-based network . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Correlations in the network . . . . . . . . . . . . . . . . . . . . . 36

4.5 Network properties . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6 Results interpreted from resting state . . . . . . . . . . . . . . . . 37

5 Conclusion 39

Bibliography 41

viii

Chapter 1

Introduction

When someone is performing a particular task or no particular task, we can

obtain information about the corresponding neural activity from the person’s

brain image. Relations between small regions in the brain can be inferred by

constructing functional networks from the brain image. Different networks will

be generated from a particular brain with different parameters such as number of

trials, time windows for averaging and thresholds for the activity and correlation.

Intuitively, the overall characteristics of the reconstructed networks of a particular

brain should be similar. As neurological and psychiatric disorders often share

underlying brain network pathology (Deco and Kringelbach, 2014), we assume

that subjects with similar mental health condition will have similar functional

networks.

In the last decades, it is popular to investigate the brain activity during the

rest in a wide range of mental illness (Cabral et al., 2014). The resting state

dynamics are consistent across healthy subjects, and the re-productivity is high

(Cole et al., 2010). It has also been used for classification of mental disease such

as schizophrenia patients (Arbabshirani et al., 2013).

The goal of this project is to analyze the resting state functional networks

of brains among Major Depression Disorder (MDD) patients. To study whether

common properties will be shared in networks constructed from MDD brains,

the project aims to propose an approach to analysis the networks by identifying

networks from different MDD condition subjects and healthy subjects based on

some graph properties.

Most studies in neurobiology focus on the network properties that affected

by mental disease and aging separately (Lynall et al., 2010; Arbabshirani et al.,

1

2 Chapter 1. Introduction

2013; Deco and Kringelbach, 2014; Fair et al., 2008; Cao et al., 2014), while this

project aims to offer a way to estimate networks that can tell how different the

brains are in general. That is to say, this project will consider network properties

that are important in both mental disease and aging problems. This is also a

reasonable attempt, as MDD shows only low mood, which is unlike other mental

disease that has a big difference from the healthy state.

In this project, the pipeline of functional network comparison across subjects

is: data pre-processing, brain extraction, brain network construction, network

properties calculation, and finally clustering subjects with different properties as

the predictor.

A voxel-based functional network analysis is proposed in this project to study

the depression patients and healthy controls. This approach can reveal more

information about the brain connectivity than the mainstream region-based net-

work (further discussed in section 3.6.1 and section 4.2). It is a novel attempt to

infer the difference between depression and healthy subjects from the voxel-based

functional network. This project also proposed to investigate brain with the sec-

ond largest connected component of the functional network. The consistency of

the second largest functional networks across subjects has been explored in sec-

tion 3.6.2. This is the first time (to our knowledge) to analyze the second largest

functional network in the brain under the resting state. Furthermore, complex

network properties of the second largest components are calculated for clustering

the subjects (see section 3.7). The results imply that the properties of the second

largest component are indicative of discriminating subjects with different mental

health conditions, and the method we proposed is promising for future analysis

on depression.

Chapter 2

Background

2.1 Resting state, Default mode network

When a subject is performing a task such as looking at a photo, the increased

neural activity in visual cortex increases blood flow in that region. This robust

relationship of mental activity changing reflected in changing needs of the brain

for oxygen is known well for over 100 years (Raichle et al., 2001).

Traditionally, the main focus of the brain activity analyses is task-related

which helps identifying and characterizing functionally distinct areas in the hu-

man brain. Recently, researchers become interested in the resting state of the

brain. When a person is at the resting state, he/she is not consciously perform-

ing any task, but is somehow stand by the external stimuli such as the auditory

or visual. He/she is ready to suddenly turn around to a disrupting sound or

look towards a shining point. It is reported that the patterns of brain activity at

rest are distinguishable from the ones observed in performing tasks and sleeping

(Cabral et al., 2014).

The resting state network (RSN) refers to the network of the brain regions

which are anatomically separated, but strongly functionally connected and acti-

vated during rest. Multiple RSNs can be identified by Independent Component

Analysis (ICA). As ICA decomposes the brain data at rest into a number of

components, the network for each component is an RSN module. Moussa et al.

(2012) found that the sensory/motor module, the basal ganglia module, the vi-

sual module and the Default Mode Network (DMN) module are consistent across

subjects.

Sometimes, the RSN is specifically referring to the DMN. The DMN is not ac-

3

4 Chapter 2. Background

tivated in the resting state but consistently decreases its activity during attention-

demanding and goal-directed tasks (Raichle, 2015). Unlike the DMN, other task-

positive RSNs including vision, language and basal ganglia RSNs exhibit stronger

functional connectivity during corresponding tasks.

It is natural that researchers try to associate the DMN with underlying high-

order cognitive processes, such as daydreaming, mind wondering and recalling

past memory. However, the present of the DMN in monkeys and rats (Raichle,

2015) makes it unlikely that the patterns in the DMN are caused by unconstrained

conscious cognition (Raichle, 2015; Cabral et al., 2014).

2.2 Functional network during rest

The first and most widely used technique in the field of exploring brain activity

during rest is resting state functional magnetic resonance imaging (R-fMRI),

which measures fluctuations in blood-oxygen-level dependent (BOLD) signal in

subjects at rest (Cabral et al., 2014). This technique is popular due to its ability

to measure correlations in neural activity between distant brain regions. Activity

fluctuations across the brain can be transformed to a network representation.

In such network, anatomically distinct brain areas can be represented as nodes,

which are “functionally connected” to each other if their activity correlate above

a threshold (Lord et al., 2013).

The graph theoretic approach is the most powerful and flexible method to

study R-fMRI (Power et al., 2011; Bullmore and Sporns, 2009). Cabral et al.

(2014) pointed out that some complex network properties are consistent over

time and spatial scales, such as small worldness and modularity. More interest-

ing findings are: disrupted small-world properties are reported in pathological

conditions and changes with normal aging in modularity.

An increasing number of pathological conditions also appear to be reflected in

the functional connectivity between particular regions (Power et al., 2011). Func-

tional connectivity during rest deteriorates in the progression of the Alzheimers

disease and appears a widespread decrease for schizophrenia patients (Cabral

et al., 2014).

2.3. Data from Stratifying Resilience and Depression Longitudinally (STRADL) project5

2.3 Data from Stratifying Resilience and De-

pression Longitudinally (STRADL) project

The brain image data used here were made available by the Stratifying Resilience

and Depression Longitudinally (STRADL) project (Fernandez-Pujals et al., 2015),

and is kindly offered by the University of Edinburgh’s Division of Psychiatry.

Though clinical depression is a chronic worldwide health problem affecting mil-

lions of people, little is known about what makes people vulnerable or resilient to

the condition. Rather than being one disease, clinical depression is a collection of

different disorders with one common symptom: low mood. The STRADL project

aims to identify the causes and mechanisms of clinical depression by studying

groups of people either with or without depression.

In the STRADL study, participants are asked to complete questionnaires on

their mental health and resilience. A subset of the participants is selected for

MRI brain scanning. As the image data are still growing, the mental health

information is under a non-disclosure agreement. In this project, we regard the

subjects as a group of people with unknown mental health condition.

2.4 Resting state fMRI in Major Depression Dis-

order

Wang et al. (2012) reviewed 16 resting state fMRI studies on Major Depression

Disorder and healthy control. They concluded that interactions between the DMN

and other task positive RSNs, and cortico-limbic mood regulating circuit should

play an important role in further MDD research. Three commonly used methods

in R-fMRI in MDD are region-of-interest (ROI), ICA and Regional Homogeneity

(ReHo). In a ROI analysis, temporal correlations between selected ROI regions

and other brain regions constitute the functional connectivity of the network.

ICA decomposes the whole brain into separate components and each component

depicts a functional network. ReHo maps the whole brain by comparing a given

voxel to the voxels around it in time series.

Both ROI and ICA methods mentioned above are region-based, which can

only measure the brain functional connectivity among restricted regions (further

discussed in section 4.2). ReHo, on the other hand, depending on neighbouring

voxels can not capture the functional connections between spatially distant struc-

6 Chapter 2. Background

tures. In this project, we propose a voxel-based network construction method,

which can gain both inter-regional and intra-regional connection information.

Chapter 3

Methods

3.1 Data description

In this project, data from one hundred subjects from the STRADL project will be

analyzed. The resting fMRIs are acquired when the participants were instructed

to do relax. For each subject, the resting state fMRI scan is stored in NIfTI

(Neuroimaging Informatics Technology Initiative) file format, which contains a

four dimension tensor and some other properties. Each NIfTI is about 51.1 MB.

Thus, this amounts to a total of 5.38 GB.

The first three dimensions in the tensor measure the length, width, and height

of the brain respectively. We can represent these three dimensions in y, x, z axis

correspondingly. Voxels of the brain are equivalent to the pixels in the 3D space.

The forth dimension of the tensor is the time dimension. During scanning, the

scanner moves from the bottom of the brain to the top. Thus, for each time step,

one brain volume (whole brain) activities are recorded.

For a 64 × 64 × 32 × 195 tensor from STRADL, we can say it has 195 time

steps, and 32 slices of 64× 64 2D images for each time step. Multiplying all four

dimensions in the tensor, 25,559,040 voxels in total per brain scan. Figure 3.1

shows the illustration of the 4D tensor in STRADL.

3.2 Data pre-processing

All pre-processing is done by Statistical Parametric Mapping toobox (SPM) (Ash-

burner et al., 2014). SPM is written in Matlab and can analyze fMRI imaging

data. Standard pre-processing steps - realignment, slice timing, coregistration,

7

8 Chapter 3. Methods

x

y

z

Slice

(a) One brain volume.

Brain Volume

2D space

Slic

e

Tim

e Ste

ps

(b) 4D tensor.

Figure 3.1: Data illustration

segmentation, normalization and smoothing will be reviewed in detail.

3.2.1 Realignment

In order to compare the same part of the brain across time, artefacts caused by

head movements during scanning should be removed. The realignment module

aims to make the same voxels have the same 3D positions throughout the time

steps. It minimizes the movements by two steps. Firstly, it estimates differences

between the mean image at the current time step and the one at the first time

step. Secondly, it uses the parameters estimated in the previous step to re-slice

the images at the current time point.

The estimation parameters measure translations and rotations in a 3D coor-

dinate system. An example of the estimation is shown in Figure 3.2. In SPM,

three translations are recorded in an x, y, z coordinate system, while three rota-

tions along each axis are represented by pitch, roll, and yaw.

3.2.2 Slice timing

The slice timing corrects differences in slice acquisition times. The data on ad-

jacent slices are recorded with an 1/2 Time Resolution (TR) interval in time

without time slicing. The whole brain scan is adjusted such that it can be con-

sidered to be acquired at the moment immediately after the nominal slice timing,

i.e. as if there was no time delay between the recordings of the slices.

Slice timing effects are more pronounce for long TRs (TR > 2s) and task

related fMRIs (Sladky et al., 2011). In our project, however, the TR is 1.56s

and fMRIs we study are acquired from the resting sate. Thus, we do not apply

slice timing on our data. This makes the pre-processing simple and avoids the

3.2. Data pre-processing 9

Figure 3.2: Estimation done by SPM realignment pre-processing for subject No.1.

In this example, two bigger movements around 50th and 140th time step (image)

can be observed along with some minor movements throughout out the recording

time.


risk of introducing artefacts caused by the temporal interpolation in slice timing

correction.

3.2.3 Coregistration

Coregistration matches modalities in individual subjects. For each subject, a

functional magnetic brain scan (fMRI) and a structural magnetic brain scan

(sMRI) are recorded separately. Coregistration maps fMRI and sMRI in the

same space. In Figure 3.3, at the same position (the cross point of two axises) in

a 3D space, the fMRI image shows the functional information corresponding to

the the anatomical position in the sMRI image.

(a) Mean fMRI across time. (b) Structural MRI.

Figure 3.3: fMRI and structural MRI of one subject in STRDAL after coregis-

tration.

3.2.4 Segmentation

A segmentation function maps sMRI and a sMRI template by registering grey

and white matter. It makes it possible to compare the structure of the brain

across subjects.

3.2.5 Normalisation

Brains of different subjects vary in shape and size. Normalisation brings them all

into a common anatomical space. It creates the mapping between one fMRI scan

and the sMRI template. Traditionally, normalisation depends on coregistration

3.3. Brain extraction 11

and segementation. Alternativly, a new SPM version (SPM12) (Ashburner et al.,

2014) normalizes the fMRI scan directly to the template sMRI space. After

normalisation, the size of a 4D tensor will change. In our case, a 64×64×32×195

tensor will become 95×79×79×195 tensor. This means 115,614,525 voxels need

to be processed per brain scan after normalization.

3.2.6 Smoothing

Smoothing is the last step in pre-processing. It suppresses noise and artefacts by

applying Gaussian kernel to the 3D brain scan images. Different parts of the brain

have their corresponding suitable kernels. If a brain is not smoothed, some highly

activated voxels are noise caused by the magnetic equipment during scanning. On

the other hand, if the brain is smoothed too much, some important signals will be

blurred. Danev (2016) reported that smoothing for the whole brain image made

everything appears to be more active relatively. Thus, for simplification, we do

not consider smoothing in this project.

3.3 Brain extraction

3.3.1 Pre-processed results

One hundred resting state brain scans in STRADL are pre-processed by realign-

ment and normalisation steps. Realignment for one subject can be done by SPM,

and the same to normalization. Therefore, we wrote the scripts for batch process-

ing (call the realignment function and normalization function in SPM for multiple

times) in Matlab. Realignment for 100 subjects took 3 and half hours, and roughly

the same amount of time is taken for normalization. These pre-processing steps

are standard procedures and are unavoidable, especially normalization. The head

position in the brain image varies across subjects, which makes them incompara-

ble before normalization.

Figure 3.4 shows the two normalised fMRIs and the sMRI template at the

same 3D position in the same space. Both normalised fMRIs in Figure 3.4 show

the images at the anterior commissure point as the sMRI template does. This

shows that the alignment of subjects is acceptable.


(a) Normalized fMRI for subject No.1. (b) Normalized fMRI for subject No.85.

(c) Structual MRI Template at anterior

commissure point.

Figure 3.4: Normalisation results. The cross point in the sMRI template is the

anterior commissure point. Correspondingly, at the same 3D position in the two

normalized fMRIs, they show the brain activity around the anterior commissure

point as well.

3.3. Brain extraction 13

3.3.2 Brain representation

Noise and low activity voxels are irrelevant to the analysis. Thus voxels with low

value are removed. A simply method is to ignore a given voxel if it presents all

zeros in time series. Holes in the brain images are filled up by image processing

methods - dilation and erosion (Gonzalez and Woods, 2008; Danev, 2016). The

voxel information of the brain is represented by a matrix (the values in time series

of one voxel in 3D store in one column) and an array of original 3D coordinates.

The statistical information for subject No.1 is shown in Table 3.1. As we

can see, the size of raw data reduces from 25,559,040 voxels to 115,192 voxels

after simple noise removing. After normalization, though the number of voxels

becomes bigger, it is much smaller than the normalized data before simple noise

reduction, which has 115,614,525 voxels. The negative voxel value occurs due

to the misalignment in the normalization process. The misalignment makes the

boundary of the brain to be fuzzy. We can simply set the negative values to zero,

which does not cause much difference of the voxels we get from the brain image.

In table 3.1, we are acknowledged that the mean and the standard deviation

remain close to the one without neglecting negative values.

As we only consider highly activated voxels, we set a threshold slightly higher

than the mean voxels and remove any voxels that are below the threshold. This

is implemented before dilation and erosion procedures. The last row in Table 3.1

shows that the mean becomes higher and the standard deviation becomes smaller

after high-value selection. More significantly, the size of the number of voxels is

half smaller than before.

Table 3.1: Voxel statistical information for subject No.1 with brain holes filled

and simple noise removed.

max mean min std. dev. size

raw 2613 298.08 0 456.46 115192

normalized 2728 566.12 -168 525.86 470713

normalized

(set negative to zero)2728 560 0 526.22 475604

normalized

(remove voxels below 600,

set negative to zero )

2728 1018.3 0 369.37 232432


Figure 3.5 shows the slice before brain extraction (without threshold and brain

holes are not filled). The colour in Figure 3.5 shows the activity intensity in the

brain. The higher value is in warm color while the lower value is in cold color. In

Figure 3.5a we can see that some light blue around the boundary, which is the

noise in the brain skull. After brain extraction, most of those noises are removed,

as we can see in Figure 3.5b. Figure 3.5d and Figure 3.5c show the same brain

as the one in Figure 3.5b from different views. The coronal section is the brain

section when we fix the y-axis in the brain volume. Similarly, the sagittal section

is the one with the fixed x-axis.

10 20 30 40 50 60 70

10

20

30

40

50

60

70

80

90

Horizontal section of the brain

0

500

1000

1500

2000

2500

(a) Mid-horizontal slice before brain ex-

traction.

10 20 30 40 50 60 70

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20

30

40

50

60

70

80

90

Horizontal section of the brain (after brain extraction)

0

500

1000

1500

2000

2500

(b) Mid-horizontal slice after brain ex-

traction.

10 20 30 40 50 60 70 80 90

10

20

30

40

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Mid-sagittal section of the brain

0

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1000

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2000

(c) Mid-sagittal section after brain ex-

traction.

10 20 30 40 50 60 70

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Coronal section of the brain

0

500

1000

1500

2000

(d) Mid-coronal section after brain ex-

traction.

Figure 3.5: Brain slice for subject No.1.

3.4 Network extraction

A correlation matrix Cnp represents the link between two brain areas n and p

(n, p = 1, ..., N) in a brain with N cortical regions. The connection in a functional

3.5. Network properties 15

brain network is weighted, corresponding to the correlation value between the two

brain areas. The connectivity can be binarized at a given threshold th to covert

the correlation matrix C into an adjacency matrix A (Anp = 1 if Cnp > th,

otherwise Anp = 0).

Highly correlated regions have a stronger connection, while lower correlated

regions have a weaker connection. The correlation value ranges from 0 to 1. The

execution time for a correlation matrix calculation in Matlab is slow and increases

rapidly as the size growing. For a size 11000× 11000 correlation matrix, it takes

about 3 minutes to calculate the correlation matrix (Danev, 2016). To reduce

the correlation matrix size, we can select the voxels in every 5 × 5 patch in a

horizontal direction. One voxel is selected in the center of each patch.

As we take all the vertical neighbouring voxels into account in 5×5 horizontal

patch selection procedure, a lot vertical connected pairs appear in Figure 3.6b,

due to the fact that adjacent voxels are closely affected each other.

Figure 3.7 shows the correlation matrix of the extracted network shown in

Figure 3.6.

3.5 Network properties

In this section, we are going to give the definition of the network properties that

have been mentioned in mental disease and aging problem in Cao et al. (2014)

and Lynall et al. (2010). We will implement some of them and give the analysis

results in the later sections.

The following properties we are going to review are very common and of-

ten used in the community. The property information is not only comparable

across subjects, but also further reduces the data. We can compare functional

networks across subjects with property values instead of thousands of connection

information.

3.5.1 The properties of functional network connectivity

In terms of network connectivity measurement. We are going to use the measures

defined in (Cao et al., 2014).

• network connection density


20 40 60

10

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30

40

50

60

70

80

90

0

500

1000

1500

2000

2500

(a) Extracted network shown on Mid horizontal slice

(b) Extracted network shown in 3D

Figure 3.6: Extracted network at threshold=0.96 for subject No.1.


2000 4000 6000 8000

1000

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3000

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5000

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7000

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9000

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.7: Correlation matrix for subject No.1 with 5× 5 horizontal patch. The

size of the correlation matrix is 9322× 9322.

A network G with N nodes and K edges

D(G) =2K

N(N − 1)(3.1)

• network mean connectivity strength

Str(G) =W

2K(3.2)

where W is the total weights of the network.

• network mean anatomical distance

Dis(G) =

∑1≤i,j≤N drij

2K(3.3)

where drij is the anatomical distance of edge rij.

• dynamic connectivity

A sliding window approach can be applied to adjust connectivity strength

and efficiency to time-varying brain images (Yu et al., 2015). This con-

siders the time factor, which is not common in normal network. But the

computation of such property is complex.


3.5.2 Nodal properties

• degree distribution parameter

Degree for node i is k(i) =∑

j Ai,j, where A is the binary adjacency matrix

derived from the corresponding functional connectivity (correlation) matrix

C with a threshold. The degree distribution is the counts for how many

nodes of each degree value appear in the network. The distribution can

be visualized as a curve with long tail, which can be estimated as a power

function (Eguiluz et al., 2005). In other words, P (k) ∼ k−γ, where P (k) is

the probability of a node has degree k (Bullmore and Sporns, 2009). We call

the exponent γ as the degree distribution parameter(Lynall et al., 2010).

• weighted degree centrality

rFCS(i) =1

N

∑1≤j≤N,j 6=i

wrij (3.4)

where wrij is the strength (the Pearson correlation coefficient) of edge rij.

It measures the connectivity strength of the region i to all other regions in

network (Cao et al., 2014) (Figure 3.8(B)).

• connectivity patterns among functional hubs

Regions with higher rFCSs are hubs. Correlation density among hubs and

their connections (sub-network) can be calculated by the rich club (Figure

3.8(C)) coefficient Φ (see (Cao et al., 2014) for detail).

3.5.3 Global network properties

• topological efficiency

Global efficiency

Eglob(G) =1

N(N − 1)

∑1≤i,j≤N,i 6=j

1

Lij(3.5)

where Lij is the characteristic (shortest) path length between nodes i and

j in the network G (Figure 3.8(A)). It measures the global efficiency of

information propagation in the network (Cao et al., 2014).

Local efficiency

Eloc(G) =1

N

∑1≤i≤N

Eglob(Gi) (3.6)


Figure 3.8: Illustration of network properties, figures from (Cao et al., 2014).

(A)Characteristic path length from node a to node b (red line). The characterise

feature is assumed to be functionally important. (B) Connectivity strength of

node c (average weight of the red color lines). Node c is a hub, which has a

strong connection in the network (hubs are marked as large dots in the figure).

(C)Clustered modules (shaded area). (D)Rich club organization (red dots and

lines).


where Gi denotes the sub-graph composed of the nearest neighbors of node

i. It measures the efficiency of the network when node i removed (Cao

et al., 2014).

• clustering coefficient

Clustering coefficient measures the segregation of the network. The clus-

tering coefficient for a network is the average clustering coefficient (C(i))

for the nodes in the network.

For a node i,

C(i) =δiτi

(3.7)

where δi is the number of connected triangles, τi is the number of connected

triples.

• small-worldness

σ =C/CR

Eglobal(G)R/Eglobal(G)(3.8)

where C and CR are clustering coefficients with characteristic path length L

and LR (LR > L). CR, LR and Eglobal(G)R are from a comparable random

graph. It is a “small-world” when σ > 1 (Lynall et al., 2010).

• modularity

For a given partition p of weighted network, the modularity index Q(p) can

be calculated to measure the modular structure (Figure 3.8(B)) (see (Cao

et al., 2014) for detail).

• robustness

Robustness parameter ρ is defined as the area under the curve s/n, where s

is the size of the largest connected component and n is the number of nodes

removed (Lynall et al., 2010).

3.5.4 Metrics selection

We present an overall consideration of the network by reviewing the network prop-

erties mentioned above. Some metrics are useful for weighted network, whereas

we only consider the biniarized network in our study. Some metrics measure the

components of the network, which are not helpful as we will only look a certain

3.6. Network analysis 21

component (will be discussed later). Some metrics are complex and need some

subjective parameter settings such as dynamic connectivity and connectivity pat-

terns among functional hubs. Therefore, we will only choose clustering coefficient,

small-worldness for later analysis.

3.6 Network analysis

3.6.1 Network construction

One approach to construct the whole brain network is to depict the correlations

between regions identified by ICA (Mørup et al., 2010). Another popular ap-

proach is to construct a macro-scale functional networks with nodes for brain

regions that are selected by random parcellation algorithm or anatomical defined

regions (Zalesky et al., 2010; Cao et al., 2014; Lynall et al., 2010). Both ap-

proaches focus the interactions between regions, rather than voxels, in the brain.

This is understandable that the connections between adjacent voxels are less im-

portant as they should be highly correlated and has no other implications of the

functional connectivity in a macro-scale of the brain. As discussed before, voxels

in every 5× 5 horizontal patch will cause some isolated strongly connected pairs

in vertical direction (Figure 3.6b), which is not informative.

In this project, however, we construct a voxel-based network instead of region-

based network as mentioned above. We choose voxels in every 5×5×5 cube region.

One voxel is selected in the center of each cube region. In this way, we can also

consider the selected voxels as representative voxels in their corresponding cubic

region, and neglect the interactions between neighbouring voxels. One advantage

for voxel-based networks is that they are reported to be more robust against

network fragmentation (fewer fragments with higher thresholds) compared to

region-based networks (Hayasaka and Laurienti, 2010).

Density of the network is affected by the threshold value. The higher threshold

is set, the lower density of the network will be, and more connected components

will show up. In Erdos-Renyi random graph, the sudden emergence of the “gi-

ant component” is concerned as the phase transition as the size of the largest

component increases largely (Bollobas et al., 2007). Similarly, the threshold at

the biggest second largest connected component is an interest point to look at.

At this threshold, the size of the largest connected component has an sudden

increase and the second largest component immediately merges to the largest


component below such threshold. Therefore we are going to analyse the second

largest component at the critical threshold.

The advantage to study the second largest component is that it contains less

noise than the largest connected component as it is smaller.

3.6.2 Second largest component

As we can see from Figure 3.9, it is consistent that each subject has a size drop

for the second largest component across different thresholds. Since the drop point

varies in different subjects, we can not simply select one threshold value for all

subjects. For instance, if we select 0.76, which is the threshold that cause the

highest point of node coverage in the second largest component for subject No.86,

we will not get a second largest network for subject No.96 with the same threshold

as the node coverage is around zero for subject No.96 (see Figure 3.9a and Figure

3.9b).

0.5 0.6 0.7 0.8 0.9 1

Threshold

0

20

40

60

80

100

No

de

s C

ove

rag

e

Largest Component

Second Largest Component

(a) Subject No.86

0.5 0.6 0.7 0.8 0.9 1

Threshold

0

20

40

60

80

100

No

de

s C

ove

rag

e

Largest Component


(b) Subject No.96

0.5 0.6 0.7 0.8 0.9 1

Threshold

0

50

100

150

200

Nodes C

overa

ge

Largest Component


(c) Subject’s average

Figure 3.9: The sizes of the largest component and the second largest component

change with different thresholds.


As we can see from Figure 3.10, the selected thresholds are ranging from 0.66

and 0.88 and tend to be centering at the mean 0.783 (Figure 3.10b). This shows

that the selected thresholds are meaningful as they do not vary too much across

different subjects.

0 10 20 30 40 50 60 70 80 90 100

Subject

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Th

resh

old

(a) Stem plot for selected thresholds

1

All Subjects

0.65

0.7

0.75

0.8

0.85

Thre

shold

(b) Box plot for selected thresholds

Figure 3.10: The threshold selected among subjects. The max threshold is 0.88,

the mean is 0.783, the min is 0.66 and the standard deviation is 0.045.

The whole brain network includes more noise than sub-networks such as the

second largest component. The density of the whole brain network is much smaller

than the second largest component (Figure 3.11). This is due to the fact that

some smaller clustered sub-networks (such as pairs of linked nodes) present in the

whole brain network, which will make the analysis more difficult and complicated.

An example of extracted second largest networks is shown in Figure 3.13.

Although some small noise (nodes outside the brain) appears in the right bottom

corner of the coronal section (Figure 3.12b), the overall distribution of the network

is reasonable, as it is mostly lie in the visual and motor region (middle and lateral


0 10 20 30 40 50 60 70 80 90 100

Subject

0

0.05

0.1

0.15

0.2

0.25

0.3

De

nsity

Brain Network


Figure 3.11: Network density for the whole brain network and the second largest

component across subjects

part of the brain), and it is an abstract notwork that is not clustered only in one

functional brain region (such as vision cortex).

3.6.3 Random graph

Because the fMRI derived graphs behave far from random (Mørup et al., 2010),

we can get some statistical evidence of the differences compared across individual

brains by comparing the extracted network of each subject with a random graph.

Figure 3.14 shows that fMRI derived graphs look different from a random graph

with respect to degree values.

For a Erdos–Renyi random graph G(n, p), the graph G is constructed by

connecting n nodes randomly with probability p independently.

In Figure 3.14, though the probability of the Erdos–Renyi random graph is

set to the density of the extracted second largest network, the resulting density

of the random and the extracted network is slightly different (0.028 and 0.026

respectively).

When calculating the small-worldness, the comparable random graph is an

Erdos–Renyi random graph with average node numbers and the average density

of the second largest components from the 100 subjects in STRADL.


10

20

30

40

50

60

70

X

102030405060708090

Y

0

200

400

600

800

1000

1200

1400

1600

1800

(a) The extracted second largest network in horizontal view.

10

20

30

40

50

60

70

Z

10203040506070

X

0

500

1000

1500

2000

(b) The extracted second largest network in coronal view.


10

20

30

40

50

60

70

Z

102030405060708090

Y

0

200

400

600

800

1000

1200

1400

1600

(c) The extracted second largest network in sagittal view.

(d) The extracted second largest network in 3D view.

Figure 3.13: The extracted second largest network for subject No.5. Voxels are

selected in every 5× 5× 5 cube.


0 20 40 60 80 100 120 140 160

Nodes

0

2

4

6

8

10

12

14

16

De

gre

e

Figure 3.14: Node degrees in the extracted second largest network for subject

No.1. The pink line is the extracted second largest network, the blue area rep-

resents the Erdos–Renyi random graph with the same vertices as the extracted

network and the probability is set to the density of the extracted one. The degree

distribution for the extracted second largest network and a random graph can be

seen in Figure 3.16 and Figure 3.15.

3.6.4 Power-law degree distribution

As we can see from Figure 3.16 and Figure 3.15 that the degree distribution of the

second largest functional in the brain is very different from a random graph. The

number of nodes and the density parameter of the random graph are the averages

from the 100 subjects in STRADL. We can also tell from Figure 3.17a that the

average degree distribution of the second largest network in brains has a “heavy

tail”, which obey the power law. The curve in Figure 3.17a is a fitted power

function estimated from Equation 3.10. Figure 3.17b shows the same degree

distribution in Figure 3.17a, but in a logarithmic scale.

We applied the Maximum Likelihood Estimation(MLE) for Truncated Pareto

distribution (White et al., 2008) to estimate the exponent λ ( λ = −γ).

According to White et al. (2008), the MLE λ is calculated from Equation 3.9.

lnx =−1

λ+ 1+bλ+1 ln b− aλ+1 ln a

bλ+1 − aλ+1(3.9)

where x is the degree axis in degree distribution, a ≤ x ≤ b, λ 6= 1, a ≥ 0, b ≥ 0.

The probability density function (PDF) is

f(x) = (λ+ 1)(b(λ+1) − a(λ+1))−1xλ) (3.10)


1 2 3 4 5Degree

0

20

40

60

80

100

120

140

160

180

Co

un

ts

Figure 3.15: Degree distribution of the comparable random graph.

Figure 3.16: Degree distributions of the second largest components constructed

from STRADL subjects. The label “Count/Size” (size is the total counts of the

degree in each subject) in the figure is equivalent to the probability of degree.

3.7 Clustering evaluation

In this section, we are going to cluster subjects into groups by k-means clustering

with different network properties as the predictor. Intuitively, we assume that

3.7. Clustering evaluation 29

0 50 100 150 200 250Degree

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16P

robabili

ty (

degre

e)

averagefit average

(a) Average degree distribution for 100 subjects from STRADL.

100

101

102

103

Subject

10-5

10-4

10-3

10-2

10-1

100

Pro

ba

bili

ty (

de

gre

e)

averagefit average

(b) Average degree distribution in logarithmic scale.

Figure 3.17: Average degree distribution of the second largest components con-

structed from STRADL subjects. The degree distribution parameter γ is 1.239.


there are 3 clusters among the 100 subjects from STRADL. The largest cluster

might reveal the average mental health condition in the 100 subjects. The other

two clusters might alienate from the average to two different extremes.

3.7.1 Clustering with clustering coefficient and small-worldness

Let us call the clusters with clustering coefficient as the predictor as CC clusters

(results are shown in Figure 3.18a). Similarly, we call the clusters with small-

worldness as the predictor as SW clusters (results shown in Figure 3.19c). The

number of the subjects in each cluster is reported on the right side of Table 3.2.

The small-worldness values for all subject are bigger than 1, which means all

the second largest components from the STRADL subjects are small-world. In

Figure 3.18b, the subjects that are clustered in one group with close clustering

coefficient are marked in the same shape and is consistent with the clustering

results in Figure 3.18a. It is obvious that the distribution of the CC clusters

along the small-worldness axis in Figure 3.18b is also similar to the ones along

clustering coefficient axis. The clusters at the bottom, middle and top in both

figures (Figure 3.18a and Figure 3.18b) are CC cluster2, CC cluster1 and CC

cluster3 respectively. The same observation is reported in Figure 3.19c and Figure

3.19d. Though some subjects in SW cluster2 are mixed up with subjects in SW

cluster3, the distribution of the SW clusters along clustering coefficient axis in

Figure 3.19d is roughly the same as the one in Figure 3.19c. This indicates that

the clustering results with two different predictors are roughly the same.

3.7.2 Clustering analysis

We also calculate the intersections of the subjects in the clusters of two different

clustering predictors. We find that most subjects (86%) can be clustered to the

same group with two different clustering. Table 3.2 shows the overlap of the

two clustering approaches in detail. For instance, from the 50 subjects and the

38 subjects that are clustered in CC cluster1 and SW cluster3 respectively, we

find 36 of them are grouped in both clusters. Note that CC cluster1 and SW

cluster3 are both in the middle part in Figure 3.18a and Figure 3.19c, thus they

are comparable.

Furthermore, we compare the mean degree distribution of the three clusters

(covering 86% subjects) from Table 3.2. The degree distribution parameters


0 20 40 60 80 100Subject

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Clu

ste

rin

g c

oe

ffic

ien

t

CC cluster1CC cluster2CC cluster3

(a) Clustering coefficients of the second largest components from

100 subjects in STRADL. The clustering predictor is clustering

coefficient.

0 20 40 60 80 100Subject

1

2

3

4

5

6

7

8

9

Sm

all-

wo

rld

ne

ss

CC cluster1CC cluster2CC cluster3

(b) Small-worldness values of the second largest components from

100 subjects in STRADL. The clustering predictor is clustering

coefficient.


0 20 40 60 80 100Subject

1

2

3

4

5

6

7

8

9

Sm

all-

wo

rld

ne

ss

SW cluster1SW cluster2SW cluster3

(c) Small-worldness values of the second largest components from

100 subjects in STRADL. The clustering predictor is small-

worldness.

0 20 40 60 80 100Subject

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Clu

ste

rin

g c

oe

ffic

ien

t

SW cluster1SW cluster2SW cluster3

(d) Clustering coefficients of the second largest components from

100 subjects in STRADL. The clustering predictor is small-

worldness.

Figure 3.19: K-means clustering results with clustering coefficient or small-

worldness as the predictor. All subjects have been clustered into three groups

and marked in different shapes in the figures.


Table 3.2: Comparison of the clustering results by two clustering predictors. The

subjects that are clustered in both selected source clusters (clusters shown in

Figure 3.19) are picked. For example, subjects in cluster1 are subjects in both

CC cluster1 and SW cluster3.

Subjects coverage (%) Source Subjects coverage (%)

Cluster1 36CC cluster1

SW cluster3

50

38


SW cluster2

20

34


SW cluster1

30

28

for the cluster1, cluster2 and cluster3 are 1.242, 1.695 and 0.988 respectively

(see Figure 3.20). It is interesting that the mean degree distribution parameter

increases with the increase of the clustering coefficient and small-worldness.

To summarize, the clustering results with clustering coefficient and small-

worldnes support each other, as they have high similarity. The degree distribution

parameters for the subjects overlapped in the two type of clusters also shows

distinct difference, which means the three network properties we consider are

reliable to cluster the subjects. The largest clusters are also close to the average

in all three metrics across subjects, which shows the network features of the

average mental health condition in the analyzed subject group.


100

101

102

103

Degree

10-5

10-4

10-3

10-2

10-1

100

Pro

ba

bili

ty (

de

gre

e)

averagecluster1cluster2cluster3

Figure 3.20: Degree distribution of clusters in logarithmic scale. The clusters

marked in the figure are the ones from Table 3.2. The degree distribution for

cluster1 (the largest cluster) is close to the average, while the other two clusters

deviate from the average. The degree distribution parameters for the cluster1,

cluster2 and cluster3 are 1.242, 1.695 and 0.988 respectively.

Chapter 4

Discussion

4.1 Pre-processing procedure

The choice of pre-processing technique has a big influence on the later analysis. In

section 3.2 we discussed the trade-off of implementing slice timing and smoothing,

and decide not implementing them. The pre-processing steps - realignment and

normalization are already sufficient for comparability among brains. However, in

section 3.3.2, we find that after normalisation some slight misalignments around

the boundary of the head bring negative voxel values . This suggests that the

traditional normalistion (based on coregistration and segmentation) might be a

better option. But the traditional procedure is more time consuming, as we

need to align the fMRI and sMRI for each subject before adjusting the fMRIs

to the sMRI template space. On the other hand, simply neglecting the negative

value only lose 1% of the voxels (see section 3.3.2). Therefore, although our

normalisation results is not perfect, it is acceptable.

4.2 Voxel-based network

Most functional networks in resting state fMRI study are region-based. The

most common brain parcellation is to divide the brain into 90 anatomical regions

with Automated Anatomical Labeling (AAL) template and averaging voxels of

each region (Lynall et al., 2010; Cao et al., 2014). Another similar approach is

to parcel the brain into random amount of regions (Cao et al., 2014). Func-

tional connectivity between ICA regions are also explored (Mørup et al., 2010).

Region-based functional networks are computationally easier to analyze as smaller

35

36 Chapter 4. Discussion

amount of nodes are considered. However, the region-based network is computa-

tionally expensive to be constructed, and it limits the evaluation of inter-regional

connectivity and restricts to certain regions. In contrast, voxel-based network,

which we implemented in our project, has the ability to measure inter-regional

and intra-regional connectivity. Previously, van den Heuvel et al. (2008) analyzed

voxel-based resting state functional networks among healthy subjects and found

that the networks are scale-free, but they did not make any further inference.

4.3 Dimension reduction

For computational efficiency, we reduce the dimension with a cubic selection

approach. We find that the larger the cube is, the more connected components

it will form. However, the density of the whole brain functional network will not

change with different sizes of cube (3 × 3 × 3, 4 × 4 × 4 and 5 × 5 × 5) across

different thresholds. The amount of the connected components increase as the

threshold increase, and the increasing curve is similar with different cube sizes.

Also, the observations of the sudden merge of the second largest component to

the largest component are consistent across the 3 different cube sizes we studied.

This implies that the selection of the cubic size 5× 5× 5 for dimension reduction

is feasible.

4.4 Correlations in the network

In this project, we pick correlations as the measures for interrelations of pro-

cesses. Though there are other measures such as Granger causality, independent

components and mutual information, we believe that correlations are proper mea-

surements in our big data analysis.

Mutual information is commonly used to measure the regions defined by ICA

(Dodel, 2002; Mørup et al., 2010). The mutual information for two variables is

calculated from the joint probability distribution and marginal probability distri-

bution, which are heavily depend on the observation. That is to say the choice

of bin widths and time scale for counting observations influence the accuracy of

the mutual information. Granger causality has the same disadvantage of depen-

dence on observed variables (Zhou et al., 2014). The Granger causality quantifies

the usefulness of one voxel to another voxel or a selected region in times series


(Roebroeck et al., 2005). In other words, the Granger causality identify whether

one variable in the past can predict another variable in the present. This causal-

ity information about the connectivity between voxels is less interpretable than

correlations. Therefore, Granger causality, independent components and mu-

tual information are statistically less reliable and computationally more complex,

which is not helpful for big data analysis.

4.5 Network properties

The network properties we take into consideration cover global and local fea-

tures of the network. Degree distribution parameter, clustering coefficient and

small-wordlness all can be computed to a quantity. With enough data, these

quantities can be statistically significantly estimated. Another advantage is that

these quantities are computationally accessible, which is suitable for big data

analysis.

In addition, these properties are generally used in the community, thus our

results are comparable with other previous work (such as Mørup et al. (2010);

Cao et al. (2014); Lynall et al. (2010)). Though the scale of the networks varies

in different studies, the results on the network properties are still worth for com-

parison.

4.6 Results interpreted from resting state

In this project we simply assume the highly active voxels are of great interest.

However, it is not always the case. Voxels with high values might be caused by

MRI scanner (the low-frequency fluctuations), respiratory and cardiac pulsations,

which are the main source of noise in resting state (Cordes et al., 2001). Thus, in

the future, we might implement band-pass filter to minimize such noise influence

(van den Heuvel et al., 2008).

Another assumption we made in this project is that the subjects are scanned

under the resting state. But the fact remains unknown, as the subjects might

doing some unconstrained cognitive tasks, such as mind wondering, during the

rest. As the network we extracted is not the Default Mode Network (see section

2.1), other explanation of the brain activity at rest is possible, which should be

explored in the future work.

Chapter 5

Conclusion

In this project we performed a complete analysis of fMRI brain scans across

one hundred subjects during the resting state. We discussed pros and cons of

utilizing the standard pre-processing techniques. Realignment and normalization

pre-processing steps are taken to make the subjects comparable, though they

bring some artefacts and noise.

The functional network of the brain is constructed in a voxel-based level,

which reveals inter-regional and intra-regional information. Voxels in every 5 ×5 × 5 cube are selected and a correlation matrix for these voxels is calculated.

A threshold is set to binarize the correlation matrix. We find that there is a

consistent observation of a phase transition that the second largest connected

component merges to the largest connected component. The threshold that causes

the sudden transition is selected and the second largest component is extracted

from the network constructed at such a threshold. The selected thresholds for

the subjects are range from 0.66 to 0.88.

Basic network properties are discussed and three proper metrics - degree dis-

tribution parameter, clustering coefficient and small-worldness are selected to be

analyzed for the subjects. We find that the second largest networks we generate

are scale-free, because their degree distribution follows the power-law. The de-

gree distribution for different subjects can be then compared on the characterising

exponent (degree distribution parameter). We clustered the subjects into three

groups by k-means clustering with clustering coefficient and small-worldness. The

clustering results with the two types of clustering predictors are similar in terms

of the distribution and the amount of overlapped subjects. Moreover, we find the

overlapped subjects in the three clusters shows distinguishable degree distribution

39

40 Chapter 5. Conclusion

parameters, which shows the reliability of the clustering result.

Though we do not have the mental health condition data for the studied

subjects for now, we can still conclude that the largest cluster we get reveals the

network information of the average mental health condition among the subjects.

With more information for each subject in the future, we might give more accurate

classification results on healthy controls and Major Depression Disorder patients

with the proposed network properties.

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