,

# Multiplex Graph Networks for Multimodal Brain Network Analysis

Zhaoming Kong  Computer Science and Engineering
Lehigh University
<EMAIL_ADDRESS>Lichao Sun  Computer Science and Engineering
Lehigh University
<EMAIL_ADDRESS>Hao Peng  Liang Zhan  Cyber Science and Technology
Beihang University
<EMAIL_ADDRESS>Electrical and Computer Engineering
University of Pittsburgh
<EMAIL_ADDRESS>Yong Chen  Perelman School of Medicine
University of Pennsylvania
<EMAIL_ADDRESS>Lifang He  Computer Science and Engineering
Lehigh University
<EMAIL_ADDRESS>
###### Abstract

In this paper, we propose MGNet, a simple and effective multiplex graph
convolutional network (GCN) model for multimodal brain network analysis. The
proposed method integrates tensor representation into the multiplex GCN model
to extract the latent structures of a set of multimodal brain networks, which
allows an intuitive ’grasping’ of the common space for multimodal data.
Multimodal representations are then generated with multiplex GCNs to capture
specific graph structures. We conduct classification task on two challenging
real-world datasets (HIV and Bipolar disorder), and the proposed MGNet
demonstrates state-of-the-art performance compared to competitive benchmark
methods. Apart from objective evaluations, this study may bear special
significance upon network theory to the understanding of human connectome in
different modalities. The code is available at
https://github.com/ZhaomingKong/MGNets.

###### Index Terms:

Multiplex graph convolutional network, tensor analysis, multimodal
representation

##  I Introduction

In recent years, brain network analysis has attracted considerable interests
in the fields of neuroscience and machine learning. It plays a vital role in
understanding biologically fundamental mechanisms of brain function, such as
how the brain manages cognition, what signals the connections convey and how
these signals affect brain regions [6]. It has been found to be helpful in the
early diagnosis of several neurological disorders, such as HIV infection and
Alzheimer’s disease [7, 29]. However, databases for biomedical engineering are
significantly smaller than for other applications, which make this study very
challenging. It is a common belief that the fusion of complementary
information from multimodal data would be highly beneficial to improve the
effectiveness of brain network analysis. For example, functional magnetic
resonance imaging (fMRI) and diffusion tensor imaging (DTI) are the most
widely used modalities which can reflect morphological differences of brain
regions and yield complementary information of brain networks.
In the analysis of multimodal brain networks, various machine learning methods
have been investigated for representation learning and disease prediction from
shallow to deep models, such as multiple graph kernels [14], tensor
factorizations [3, 24, 11], and convolution neural networks [31, 37, 15].
Although significant progress has been made in this endeavor, there still
lacks a basic model that can predict cutting performance. In particular, the
brain network has sophisticated and non-linear structure, which may not be
well captured by shallow models. Meanwhile, deep learning methods may suffer
from excessive parameters, which are both difficult for training and
vulnerable to overfitting. Besides, most existing methods do not make good use
or even fail to accommodate the intrinsic graph structure of brain networks.
Recently, graph convolutional networks (GCNs) have emerged as a promising
approach to learn powerful representations for graph data and have been
successfully used for brain network analysis [21, 1, 19, 4, 22]. However, GCN
explicitly requires a known graph structure (i.e., adjacency matrix), which is
typically not available in brain networks, especially in multimodal cases. To
address this challenge, Zhang et al. [40] proposed a MVGCN method for single
modality multi-view brain network classification, where prior knowledge of
geometric coordinate is required to define a common graph structure. In
addition, the small-world model was introduced to learn a random graph during
training [41]. To the best of our knowledge, there is no general method for
multimodal brain network analysis with GCNs.
In this paper, we propose MGNet, a generic framework for multimodal brain
network analysis by integrating tensor representation into multiplex GCN
models. Different from existing methods [40, 22] that rely on prior
information such as regions of interest (ROI) for graph structure modeling,
MGNet does not require any prior knowledge. It first stacks all multimodal
brain networks across subjects and modalities into a 4D tensor, and then
employs multilinear tensor projection to leverage the structural information
of the 4D tensor to directly generate the graph structure. The main
contributions of the paper are summarized as follows:

  * •

We present a novel multimodal brain network classification framework (MGNet)
by combining tensor and GCN techniques, where a multilinear tensor projection
is introduced to take advantages of the homogeneity of multimodal data and the
nature of graphs, and a graph structure is learned by projection pursuit.
Meanwhile, an associated tensor form of GCN is proposed for combining multiple
GCN models.

  * •

Our model can be regarded as a generalization of GCN to multi-graph data with
tensor representation. It alleviates the problem of pre-determining population
graph structure, which is often ambiguous due to the intrinsic complexity of
intra-graph and cross-graph.

  * •

We examine the effectiveness of the proposed MGNet model on three challenging
real datasets (i.e., HIV, Bipolar and PPMI) for the disease prediction tasks,
and the results demonstrate that MGNet model can produce state-of-the-art
performance compared to strong baseline methods.

##  II Related Work

In this section, we briefly introduce existing research related to this work.

Tensor-based Embedding. Several tensor-based methods have been proposed to
leverage the convenience of tensor representation to characterize multimodal
graph data. Specifically, [3] proposed an iterative tensor-based method for
simultaneous graph embedding and clustering. [26] proposed a tensor-based
embedding method for classification task with some multi-view side
information. In [24], partial symmetric tensor decomposition is applied to
learn the multi-view multi-graph embeddings. Recently, [38] introduced the
$t$-product tensor factorization to handle multimodal and high-dimensional
data.
Graph Convolutional Networks. In recent years, several CNN architectures for
learning over graphs have been proposed. Specifically, [15] introduced a novel
convolution filter to leverage the topological locality of structural brain
networks. [33] extended CNN to 3D functional brain network classification and
[37] to structural brain network mining with graph reordering. More recently,
[1] explored GCN for the task of ROI identification and [19] extended GCN for
brain network pairwise-metric learning. [40] proposed a MVGCN method for
multi-view brain network classification, where prior knowledge of geometric
coordinate is required to define a common feature representation space. In
addition, [41] proposed to learn a random graph during training based on the
small-world model, and [22] designed a ROI-aware brain graph model to take
advantage of the topological and functional information of fMRI.
We notice that most of the existing methods are customized for a specific
purpose, and there still lacks a general GCN-based method for the multimodal
and cross-modal brain network analysis and classification task.

##  III Methodology

###  III-A Problem Formulation

A brain network usually leverages a graph structure to describe
interconnections between brain regions, which can be represented by a weighted
graph $G=\\{V,E,\mathbf{X}\\}$, where $V=\\{v_{i}\\}_{i=1}^{N}$ is the node
set indicating brain regions, $E=\\{e_{ij}\\}_{i,j=1}^{N}$ is the edge set
between nodes, and $\mathbf{X}=\\{x_{ij}\\}_{i,j=1}^{N}$ is the weighted
matrix where $x_{ij}$ is the corresponding edge weight. Assume that each
subject consists of $M$-modal brain networks $\\{G_{1},\cdots,G_{M}\\}$, where
each one is extracted from a specific imaging modality (or measure) such as
fMRI and DTI. These networks share the same set of nodes, i.e., using an
identical definition of brain regions, but may differ in network topology and
edge weight. Given a multimodal brain network data set of $S$ subjects with
$M$ different modalities
$\mathcal{D}=\\{(\\{G_{1s},\cdots,G_{ms},\cdots,G_{Ms}\\},y_{s})\\}_{s=1}^{S}$,
where $G_{ms}$ is the brain network data of the $m$-th modality in the $s$-th
subject and $y_{s}$ the corresponding class label, the goal of multimodal
brain network data analysis is to probe the interrelationships between
different modalities and obtain from low-level or raw relational data higher-
level descriptions of brain-behavior states to facilitate disease diagnosis or
treatment monitoring.

###  III-B Architecture of MGNet

Figure 1: The architecture of MGNet. It first conducts higher-order SVD
decomposition of tensor data to study the group-wise maps of multimodal brain
network fusion. Then multiple GCN aggregators jointly learn the
representations of brain networks by passing the information in an inter-graph
or intra-graph way. After that, the extracted features are fed to a fully
connected layer (FCN) for prediction in an end-to-end manner.

Fig. 1 provides an overview of the proposed tensor-driven multiplex graph
convolutional network (MGNet) model, which consists of two major components.
Briefly, the first part performs cross-modality bridging with tensor
decomposition, which is used to efficiently extract latent structures across
modalities and individuals. The second part is the design phase of MGNet
model, which adopts GCN to capture intrisic data graph structures and encode
relationship of different modalities. MGNet can be used in both spectral and
spatial domains, as there is an equivalence of the graph convolution process
regardless of the specific domain [2]. In this study, we consider a general
multi-layer GCN model with the following propagation rule [17]:

$\mathbf{H}^{(l+1)}=\sigma(\widetilde{\mathbf{D}}^{-\frac{1}{2}}\widetilde{\mathbf{A}}\widetilde{\mathbf{D}}^{-\frac{1}{2}}\mathbf{H}^{(l)}\mathbf{W}^{(l)})$
(1)

where $\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}_{N}$ is the adjacency
matrix of the undirected graph $G$ with self-connections $\mathbf{I}_{N}$,
$\widetilde{d}_{ii}=\sum_{j}\widetilde{a}_{ij}$ and $\mathbf{W}^{(l)}$ is a
layer-specific trainable weight matrix. $\sigma(\cdot)$ denotes an activation
function, such as the $ReLU(\cdot)=\max(0,\cdot)$, and
$\mathbf{H}^{(l)}\in\mathbb{R}^{N\times D}$ is the feature matrix of the
$l$-th layer.
In the following, we discuss each component in detail and discuss how the form
of this propagation rule can be effectively extended to multiplex models with
tensor representation, thereby enabling the formation of a multimodal learning
environment.

####  III-B1 Cross-Modality Bridging

The brain network classification problem can be reformulated as a graph
classification task, and many effective graph-centric models [30, 9] can be
utilized by treating each modality independently and specifying an aggregation
function for each graph. However, such a naive extension may lead to the loss
of multimodal information, and potentially increase the storage and
computational burden. In order to focus on correspondences between data
modalities, a set of data transformations are necessary to increase the
homogeneity of the data within and across subjects and modalities and remove
portions of the data that can be assumed not to be mutually represented.

Motivated by the fact that tensor analysis can effectively model a family of
multi-relational data and potential factor characteristics [20, 12, 18], we
take advantage of tensor representation for an effective fusions of graphs
with different modalities. Specifically, given a multimodal brain network
dataset
$\mathcal{D}=\\{(\\{G_{1s},\cdots,G_{ms},\cdots,G_{Ms}\\},y_{s})\\}_{s=1}^{S}$,
we construct a single 4D tensor $\mathcal{X}\in\mathbb{R}^{N\times N\times
M\times S}$ with weighted connectivity matrices
$\mathbf{X}_{ms}=\mathcal{X}(:,:,m,s)$ for all $M$ modalities and $S$
subjects. To explore the uniformity of multimomal brain networks and also
capture most data variation, we adopt the feature extraction method in [32],
and introduce common feature projection matrices $\mathbf{U}_{1}$ and
$\mathbf{U}_{2}$ by minimizing the following problem

$\begin{split}\min_{\mathbf{C}_{ms},\mathbf{U}_{1},\mathbf{U}_{2}}\quad&\sum_{m=1}^{M}\sum_{s=1}^{S}\|\mathbf{X}_{ms}-\mathbf{U}_{1}\mathbf{C}_{ms}\mathbf{U}_{2}^{T}\|_{F}^{2}\\\
\textrm{s.t.}&\quad\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}\quad\text{and}\quad\mathbf{U}_{2}^{T}\mathbf{U}_{2}=\mathbf{I}\end{split}$
(2)

where $\mathbf{U}_{1}$ characterizes node-level relationship, $\mathbf{U}_{2}$
is used for feature extraction and $\mathbf{C}_{ms}$ is the coefficient matrix
of $\mathbf{X}_{ms}$ obtained via
$\mathbf{C}_{ms}=\mathbf{U}_{1}^{T}\mathbf{X}_{ms}\mathbf{U}_{2}$. Projection
matrices $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ can be obtained via the higher-
order SVD (HOSVD) of tensor data $\mathcal{X}$ since Eq. (2) can be written as

$\begin{split}\min_{\mathcal{C},\mathbf{U}_{1},\mathbf{U}_{2}}\quad&\|\mathcal{X}-\mathcal{C}\times_{1}\mathbf{U}_{1}\times_{2}\mathbf{U}_{2}\|_{F}^{2}\\\
\textrm{s.t.}&\quad\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}\quad\text{and}\quad\mathbf{U}_{2}^{T}\mathbf{U}_{2}=\mathbf{I}\end{split}$
(3)

We notice that $\mathbf{U}_{1}=\mathbf{U}_{2}$ due to the symmetric property
of $\mathbf{X}_{ms}$, thus we are interested in the effectiveness of
introducing $\mathbf{U}_{1}$. Intuitively, the left projection matrix
$\mathbf{U}_{1}$ in Eq. (2) captures the global node-level relationship, and
one benefit of obtaining $\mathbf{U}_{1}$ with Eq. (2) is that it does not
require any prior knowledge such as node and graph labels, thus all data can
be utilized efficiently. Also, when new subjects are available,
$\mathbf{U}_{1}$ can be updated in an online fashion [10].

####  III-B2 Multiplex GCN Aggregator

An essential component of the propagation rule in Eq. (1) is to define graph
convolution filter in the spatial or spectral domain based on an aggregator,
e.g., normalized adjacency matrix $\widetilde{\mathbf{A}}$. Unfortunately for
brain network analysis under the multimodal environments, the common ROIs used
to calculate $\widetilde{\mathbf{A}}$ [40] are not always available in real-
world cases. To address this issue, we notice that the $i$-th row of
$\mathbf{U}_{1}$ encodes the weights for the $i$-th node in the projection
$\mathbf{U}_{1}^{T}\mathbf{X}_{ms}$. Hence, by selecting the important columns
of $\mathbf{U}_{1}$ according to the singular values [25], we could use the
truncated $\mathbf{U}_{1}$ and $K$-Nearest Neighbor (KNN) graph to define an
undirected adjacency matrix to facilitate the cross-modality learning for
multiplex GCNs. To be specific, we identify the set of nodes $N_{i}$ that are
neighbors to the node $v_{i}$ using KNN, and connect $v_{i}$ and $v_{j}$ if
$v_{i}\in N_{j}$ or if $v_{j}\in N_{i}$. More specifically, we define the
adjacency matrix $\mathbf{A}$ [40] as

${a}_{ij}=\left\\{\begin{array}[]{lr}\text{exp}(-\frac{\|\mathbf{u}_{i}-\mathbf{u}_{j}\|^{2}}{2\sigma^{2}})\quad\text{if}\,v_{i}\in
N_{j}\,\text{or}\,v_{j}\in N_{i},&\\\
0\quad\quad\quad\quad\quad\quad\quad\,\,\,\text{otherwise}.&\end{array}\right.$
(4)

where $\mathbf{u}_{i}$ is the $i$-th row of truncated $\mathbf{U}_{1}$, and
$\sigma$ is the kernel width parameter. Then we can substitute
$\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}$ with
$\widetilde{d}_{ii}=\sum_{j}\widetilde{a}_{ij}$ in Eq. (1).
For simplicity, let us consider the first layer of Eq. (1) with
$\mathbf{H}^{(0)}_{ms}=\mathbf{C}_{ms}$ as input based on Eq. (2), and let
$\hat{\mathbf{A}}=\widetilde{\mathbf{D}}^{-\frac{1}{2}}\widetilde{\mathbf{A}}\widetilde{\mathbf{D}}^{-\frac{1}{2}}$,
then in our case, the propagation rule of each input feature matrix
$\mathbf{C}_{ms}=\mathcal{C}(:,:,m,s)$ is

$\mathbf{H}^{(1)}_{ms}=\sigma(\hat{\mathbf{A}}\mathbf{C}_{ms}\mathbf{W}^{(0)}),\,m\in[1,M],\,s\in[1,S]$
(5)

which could also be written as

$\mathbf{H}^{(1)}_{ms}=\sigma(\hat{\mathbf{A}}\mathbf{U}_{1}^{T}\mathbf{X}_{ms}\mathbf{U}_{2}\mathbf{W}^{(0)})$
(6)

where both $\mathbf{U}_{2}$ and the shared weight matrix $\mathbf{W}^{(0)}$
are used for feature extraction, and $\mathbf{W}^{(0)}$ is obtained in an end-
to-end fashion, thus in practice, $\mathbf{U}_{2}$ and $\mathbf{W}^{(0)}$ can
be combined to save some computing time. According to the symmetry of
$\hat{\mathbf{A}}$, we can rewrite Eq. (5) as

$\begin{split}\mathbf{H}^{(1)}_{ms}=&~{}\sigma(\hat{\mathbf{A}}^{T}\mathbf{C}_{ms}\mathbf{W}^{(0)})\\\
=&~{}\sigma(\mathbf{C}_{ms}\times_{1}\hat{\mathbf{A}}^{T}\times_{2}\mathbf{W}^{(0)^{T}})\end{split}$
(7)

where $\times_{i}$ denotes the $i$-th mode product. Rearranging Eq. (7) in the
tensor form, the propagation rule for all graphs at the $l$-th layer is
formulated as

$\mathcal{H}^{(l+1)}=\sigma(\mathcal{H}^{(l)}\times_{1}\hat{\mathbf{A}}^{T}\times_{2}\mathbf{W}^{(l)^{T}})$
(8)

where $\mathcal{H}^{(0)}=\mathcal{C}$ and
$\mathcal{H}(:,:,m,s)=\mathbf{H}_{ms}$.

####  III-B3 Modality Pooling

For subject $s$ with $M$ modalities, the final output of Eq. (8) are $M$
feature matrices
$\\{\mathbf{H}_{1s}^{(L)},\cdots,\mathbf{H}_{ms}^{(L)},\cdots,\mathbf{H}_{Ms}^{(L)}\\}$,
where $\mathbf{H}_{ms}^{(L)}\in\mathbb{R}^{N\times D_{out}}$ corresponds to
the $m$-th modality of subject $s$ in the $L$-th layer, and $D_{out}$
represents the output feature size. To integrate information of all $M$
modalities, we add an additional modality pooling layer by introducing a 1D
trainable modality importance weight
$\boldsymbol{\alpha}=\\{\alpha_{1},\cdots,\alpha_{m},\cdots,\alpha_{M}\\}$.
The final feature embedding matrix $\mathbf{F}_{s}\in\mathbb{R}^{N\times
D_{out}}$ for the $s$-th subject is calculated as a weighted combination of
all modalities by

$\mathbf{F}_{s}=\sum_{m=1}^{M}\alpha_{m}\mathbf{H}_{ms}^{(L)}$ (9)

Notice that when all elements of $\boldsymbol{\alpha}$ are the same, then Eq.
(9) boils down to the simple average pooling strategy used by the multi-view
GCN (MVGCN) [40]. Leveraging tensor representation and the propagation rule in
Eq. (8), we can write Eq. (9) as

$\mathcal{F}_{out}=\sigma(\mathcal{H}^{(L-1)}\times_{1}\hat{\mathbf{A}}^{T}\times_{2}\mathbf{W}^{(L-1)^{T}})\times_{3}\boldsymbol{\alpha}^{T}$
(10)

where $\mathcal{F}_{out}\in\mathbb{R}^{N\times D_{out}\times S}$ is the output
feature embeddings for all $S$ subjects with
$\mathcal{F}_{out}(:,:,s)=\mathbf{F}_{s}$. Furthermore, if our model contains
only one GCN layer with $L=1$, then $\mathcal{F}_{out}$ can be obtained by

$\begin{split}\mathcal{F}_{out}&=\sigma(\mathcal{C}\times_{1}\hat{\mathbf{A}}^{T}\times_{2}\mathbf{W}^{T})\times_{3}\boldsymbol{\alpha}^{T}\\\
&=\sigma(\mathcal{X}\times_{1}(\mathbf{U}_{1}\hat{\mathbf{A}})^{T}\times_{2}(\mathbf{U}_{2}\mathbf{W})^{T})\times_{3}\boldsymbol{\alpha}^{T}\end{split}$
(11)

Our model is efficient in that it avoids directly operating on long vectors,
and contains only a few trainable parameters. Therefore, it can be
effortlessly extended to large brain network datasets.

####  III-B4 FCN Layer

Finally, a fully connected network (FCN) with softmax is applied to output
feature embeddings $\mathcal{F}$ for classification. It computes the
probability distribution over the labels:

$\displaystyle
p(y_{s}=j|\mathbf{\mathbf{f}_{s}})={\text{exp}(\mathbf{w}^{\mathrm{T}}_{j}\mathbf{f}_{s})}/{\left[\sum\nolimits_{k=1}^{K}\text{exp}(\mathbf{w}^{\mathrm{T}}_{k}\mathbf{f}_{s})\right]},$
(12)

where $\mathbf{w}_{k}$ is the weight vector of the $k$-th class, and
$\mathbf{f}_{s}$ is the vectorized output feature embedding of subject $s$
obtained from $\mathcal{F}_{out}(:,:,s)$. Algorithm 1 summarizes the main
steps of the proposed MGNet model.

Algorithm 1 Multiplex Graph Networks (MGNet)

Input: Dataset $\mathcal{X}$.
Step 1: (Cross-Modality Bridging): Calculate the first mode projection matrix
$\mathbf{U}_{1}$ of $\mathcal{X}$ according to Eq. (2).
Step 2: (Multiplex GCN Aggregator): Define the aggregated adjacency matrix
$\widetilde{\mathbf{A}}$ by Eq. (4) using the truncated $\mathbf{U}_{1}$.
Perform multi-layer GCNs according to the propagation rule in Eq. (8).
Step 3: (Modality Pooling and Prediction): Conduct modality pooling to extract
feature embeddings for input subjects, which are then fed to FCN for
classification.
Output: class label $y$.

##  IV Experiments

###  IV-A Datasets and Preprocessing

Table I lists the statistics of three real-world datasets used in our
experiments, which are briefly introduced as below, and more details of data-
preprocessing are provided in the supplementary material.
Human Immunodeficiency Virus Infection (HIV): This dataset is collected from
two modalities: functional magnetic resonance imaging (fMRI) and diffusion
tensor imaging (DTI). The original dataset is heavily unbalanced, for the sake
of exposition we randomly sampled 35 early HIV patients (positive) and 35
seronegative controls (negative). These two groups of subjects do not differ
in demographic characteristics such as age, gender, racial composition and
education level.
Bipolar Disorder (BP): This dataset is collected across fMRI and DTI
modalities, consisting of 52 bipolar I subjects in euthymia and 45 healthy
controls with matched age and gender. The resting-state fMRI data was acquired
on a 3T Siemens Trio scanner using a T2∗ echo planar imaging (EPI) gradient-
echo pulse sequence with integrated parallel acquisition technique (IPAT) and
DTI data were acquired on a Siemens 3T Trio scanner.
Parkinson’s Progression Markers Initiative (PPMI): This dataset is collected
from three modalities: Probabilistic Index of Connectivity (PICo), Hough
voting (Hough) and FSL. Similar to the HIV dataset, the original dataset is
heavily unbalanced, so we randomly sample 149 patients with Parkinson’s
disease and 149 healthy controls.

TABLE I: Details of three datasets used in the experiments.

Dataset | Class | # Samples | Sample size |  Modality
---|---|---|---|---
HIV | Health | 35 | 90 x 90 x 2 | fMRI & DTI
Patient | 35
BP | Health | 45 | 82 x 82 x 2 | fMRI & DTI
Patient | 52
PPMI | Health | 149 | 84 x 84 x 3 | PICo & Hough & FSL
Patient | 149

###  IV-B Baselines and Metrics

To demonstrate the effectiveness of our MGNet model, we compare it with the
following up-to-date strong baselines for graph and neurological disorder
analysis on the above multimodal HIV, BP and PPMI datasets.

  * •

_M2E_ [24]: It is a tensor-based method for multimodal feature fusion. We
apply it to get the embeddings of all subjects and then perform classification
with FCN.

  * •

_MIC_ [35]: It first uses the kernel matrices to form an initial tensor across
all the multiple sources. After that, CP decomposition was employed to extract
feature representation for each subject. We then perform classification using
the same settings as above.

  * •

_MPCA_ [25]: We concatenate all data information into a 4D tensor and then
apply MPCA to extract feature embeddings for each subject across modalities
and individuals. We then perform classification using the same settings as
above.

  * •

_MK-SVM_ [5]: It is a multiple kernel learning method dependent of the SVM
classifier, where the graph kernel is calculated as the weighted sum of single
modality kernels.

  * •

_3D-CNN_ [8]: For each subject, we concatenate fMRI and DTI brain networks
into a 3D tensor. Then we apply 3D-CNN for joint feature extraction and
classification in an end-to-end manner.

  * •

_GCN_ [17]: We reshape the 4D tensor data $\mathcal{X}\in\mathbb{R}^{N\times
N\times M\times S}$ into a 2D feature matrix $\mathbf{X}\in\mathbb{R}^{S\times
MN^{2}}$, where each row corresponds to the vectorized representation of 3D
multimodal data $\mathcal{X}(:,:,:,i)$, in this case, each vectorized graph
can be viewed as a node, and the GCN model can be directly applied.

  * •

_GAT_ [36]: Similar to GCN, we also vectorize the input tensor data into a 2D
feature matrix $\mathbf{X}\in\mathbb{R}^{S\times MN^{2}}$, and apply the graph
attention mechanism.

  * •

_DiffPool_ [39]: It is a hierarchical GCN method equipped with differentiable
pooling for graph classification. Since it can only handle single modal data,
we apply it to each modality independently and report the best result.

  * •

_MVGCN_ [40]: It is a multi-view GCN method, which requires prior knowledge of
common geometric coordinate information to define shared feature space.
Typically, such information is not available in multimodal data, thus we
consider to obtain the shared feature space with the average of all brain
networks across modalities and subjects, and then feed it into the MVGCN
architecture.

In order to measure the performance of all compared methods, we use Accuracy
and Area Under ROC Curve (AUC) as objective indicators of classification
effectiveness, which are two widely used evaluation metrics for disease
identification in medical fields. The larger the values, the better the
classification performance.

###  IV-C Implementation Details

For all our experiments, we use binary cross-entropy loss and smooth $L_{1}$
loss with Adam optimizer [16] to train the deep models. We emperically set the
learning rate to $0.001$ and the epoch to $50$ iterations. We vary the dropout
rate of the graph embedding layer from 0 to 0.5, and the number of MGNet
layers from $\\{1,2,3\\}$. In our proposed model, there are three major
parameters, namely the batch size, the number of the $K$ nearest neighbors
when building the KNN graph, and the output feature size $D_{out}$ in GCNs. We
apply the grid search to determine the optimal values of these three
parameters. In particular, we empirically select $D_{out}$ from
$\\{20,40,60,80,100,120\\}$, and $K$ and batch size from
$\\{2,4,6,8,10,12\\}$. In our experiments, $80\%$ of data are used for
training, $10\%$ for validation and the rest $10\%$ for testing. We also
carefully tune parameters of all compared methods according to the authors’
suggestions and report their best possible results using the same data splits
and the same 10-fold cross-validation scheme for a fair comparison. All
experiments are performed on a 8-core machine with 16GB RAM. The deep learning
backend is Tensorflow-gpu (2.2.0) with Python 3.6.

TABLE II: Comparison of different methods on HIV, BP and PPMI datasets in terms of average Accuracy and AUC scores. Model | Method | HIV | BP | PPMI
---|---|---|---|---
Accuracy | AUC | Accuracy | AUC | Accuracy | AUC
Shallow | M2E |  50.61$\pm$15.84 |  51.53$\pm$13.68 |  57.78$\pm$12.61 |  53.63$\pm$11.82 |  55.93$\pm$8.34 |  54.79$\pm$7.32
MIC |  55.63$\pm$15.28 |  56.61$\pm$13.43 |  51.21$\pm$13.78 |  50.12$\pm$16.78 |  56.03$\pm$8.26 |  55.21$\pm$6.86
MPCA |  67.24$\pm$11.56 |  66.92$\pm$12.46 |  56.92$\pm$13.33 |  56.86$\pm$13.69 |  59.66$\pm$5.64 |  59.59$\pm$5.63
MK-SVM |  65.71$\pm$12.08 |  68.89$\pm$9.61 |  60.12$\pm$10.83 |  56.78$\pm$12.86 |  58.24$\pm$7.88 |  57.36$\pm$7.62
Deep | 3D-CNN |  74.31$\pm$18.81 |  73.53$\pm$16.41 |  63.33$\pm$11.21 |  61.62$\pm$10.26 |  63.89$\pm$7.12 |  66.18$\pm$7.23
GAT |  68.58$\pm$13.51 |  67.31$\pm$14.32 |  61.31$\pm$15.04 |  59.93$\pm$13.54 |  63.45$\pm$6.93 |  63.43$\pm$7.25
GCN |  70.16$\pm$12.54 |  69.94$\pm$12.91 |  64.44$\pm$15.71 |  64.24$\pm$16.45 |  63.10$\pm$7.98 |  63.09$\pm$8.10
DiffPool |  71.42$\pm$14.78 |  71.08$\pm$15.12 |  62.22$\pm$12.83 |  62.54$\pm$13.41 |  61.12$\pm$6.79 |  61.83$\pm$6.82
MVGCN |  74.29$\pm$11.27 |  73.75$\pm$12.63 |  62.22$\pm$16.83 |  62.64$\pm$16.89 |  59.24$\pm$7.83 |  58.38$\pm$7.92
Shallow + Deep | MGNet | 81.39$\pm$13.41 | 82.08$\pm$14.81 | 67.78$\pm$12.28 | 66.31$\pm$10.24 | 66.62$\pm$7.87 | 66.96$\pm$7.98

###  IV-D Results and Discussions

To evaluate the effectiveness of our model, we conduct a series of experiments
on the task of multimodal brain network classification to answer the following
three questions: Q1: The performance of MGNet compared with other state-of-
the-art methods. Q2: The effectiveness of multiple modality learning in MGNet.
Q3: The benefits of combining tensor representation and GCN in our case.

####  IV-D1 Model Comparison

Table II lists the results of all methods on HIV, BP and PPMI datasets. Notice
that the first two datasets used in our experiments consist of less than 100
subjects, which leads to high standard deviations in the results. From Table
II, we have the following observations.
Overall, it can be seen that the proposed MGNet yields an average improvement
of 7.08 $\%$, 3.34$\%$ and 2.73$\%$ in terms of accuracy on classification
tasks compared to the strong baseline methods on HIV, BP and PPMI dataset,
respectively. Specifically, MGNet achieves more than 5$\%$ improvements
compared with other graph-based models such as MVGCN and DiffPool on all three
datasets, which demonstrates the effectiveness of the proposed embedding and
common space projection strategy when dealing with multi-modality brain
networks without ROIs. The advantage of MGNet over shallow tensor-based
methods such as M2E and MPCA demonstrates the importance of modeling graph
structures. Furthermore, compared with the straightforward GCN and GAT
implementations, the superiority of the proposed MGNet lies mainly in the
joint modeling of node- and modality-level relationship, which also preserves
the multimodal tensor structure during node embedding and feature extraction.
Furthermore, it is noticed that the 3D-CNN model produces very competitive
results on all three datsets because it benefits from the locality features of
the input data. However it obtains the graph representation flatly with
convolution and pooling layers, which may not explicitly taking multimodal
relationship and graph structure information into consideration. Besides,
instead of training a multi-layer CNN model, the proposed MGNet provides an
alternative solution with graph and tensor representation.
To sum up, by taking advantage of the tensor representation and multiplex
GCNs, MGNet is able to encode both the multimodal characteristics and capture
graph structures of the input brain networks.

####  IV-D2 Hyperparameter Analysis

(a) Modality weight $\boldsymbol{\alpha}$

(b) $\\#$ Neighbours

(c) Feature size

(d) Batch size

Figure 2: Modality weight $\boldsymbol{\alpha}$ and parameter sensitivity
analysis of MGNet on HIV, BP and PPMI datasets with varying numbers of
neighbours, features and batch size.

We investigate the effects of three important parameters in our MGNet model,
namely the number of neighbours $K$ when building the KNN graph, the
dimensions of output features $D_{out}$ produced by GCN and the batch size
used for training. According to Figs. 2(b)-(d), we notice that the performance
of MGNet are related to all three parameters, which should be carefully tuned
based on site-specific conditions.

####  IV-D3 Effectiveness of Multimodal Learning

In the proposed MGNet model, multi-modal information is integrated using the
simple and effective modality-pooling strategy. To answer Q2, we investigate
how the use of multiple modalities may improve the graph representation
learning ability and thus the classification quality. Toward this end, we use
HIV and BP datasets to compare MGNet with their fine-tuned single-modality
implementations. The classification results are listed in Table III.

TABLE III: Results of MGNet with unimodal and multimodal data.

Method | Dataset | Modality | Accuracy | AUC
---|---|---|---|---
MGNet | HIV | fMRI |  53.61$\pm$11.88 |  52.31$\pm$13.63
DTI |  75.71$\pm$15.12 |  76.08$\pm$15.61
Both | 81.39$\pm$13.41 | 82.08$\pm$14.81
BP | fMRI |  57.78$\pm$12.61 |  51.83$\pm$10.84
DTI |  64.44$\pm$13.59 |  63.34$\pm$15.91
Both | 67.78$\pm$12.28 | 66.31$\pm$10.24

From Table III, we can see that the multimodal learning strategy leads to
significant improvements on both HIV and BP datasets. Owing to the usefulness
of exploiting the complementary among multiple modalities or multiple types of
features, the multimodality learning strategy is able to effectively improve
the performance compared to the single-modality implementation. Another
interesting observation is that our model with the DTI modality produces much
better results compared to the fMRI modality, hence it is assigned a higher
weight in the modality pooling operator $\boldsymbol{\alpha}$, as illustrated
in Fig. 2 (a). Furthermore, comparing the results of Table II and Table III,
we notice that CNN produces similar performance as the single-modality
implementation of our MGNet model, which also implies the advantage of
explicitly characterizing graph structures and multimodal correlation.

####  IV-D4 Ablation Study

In the formula of MGNet (Eq. (11)), $\mathbf{U}_{1}$ captures the node-level
information while $\boldsymbol{\alpha}$ characterizes the multimodal
relationship, thus it is interesting to investigate the influence of different
projection matrices of our MGNet model. In Table IV, we compare MGNet with two
different variations: MGNet without $\mathbf{U}_{1}$ (MGNet -
$\mathbf{U}_{1}$) and MGNet with predefined average modality pooling
$\boldsymbol{\alpha}=[0.5,0.5]$ (MGNet + avg. Pooling). Comparing MGNet and
MGNet - $\mathbf{U}_{1}$, it can be seen that the introduction of the node
projection matrix $\mathbf{U}_{1}$ significantly boosts the performance of our
model.

TABLE IV: Average accuracy of three different MGNet implementations on HIV, BP
and PPMI datasets.

dataset | MGNet | MGNet+avg. Pooling | MGNet - U1
---|---|---|---
HIV | 81.39$\pm$13.41 |  80.81$\pm$12.04 |  75.71$\pm$14.33
BP | 67.78$\pm$12.28 |  66.68$\pm$10.53 |  63.33$\pm$12.52
PPMI | 66.62$\pm$7.87 |  65.17$\pm$8.14 |  62.78$\pm$7.05

To further investigate the role of $\mathbf{U}_{1}$, we visualize the
difference between $\mathbf{X}_{ms}$ and $\mathbf{U}_{1}^{T}\mathbf{X}_{ms}$
for two subjects from the HIV brain network dataset in Fig. 3. An interesting
observation is that although the two subjects belong to two different
categories, the projected data $\mathbf{U}_{1}^{T}\mathbf{X}_{ms}$ exhibit
similar patterns, where the signals of different intensity are distributed in
certain rows and columns, thus, it is also reasonable to assume that they may
share the same node-level self-similarity modeled by a uniform aggregation
function in Eq. (10).

(a) Health

(b) Patient

(c) Health

(d) Patient

Figure 3: Comparison of original and projected data from the HIV dataset. The
first row shows original data samples ($\mathbf{X}$). The second row
illustrates corresponding projected data ($\mathbf{U}_{1}^{T}\mathbf{X})$.

Interestingly, from Table IV, we also notice that using a predefined modality
pooling strategy $\boldsymbol{\alpha}$ could produce similar results. This can
be explained by the fact that the DTI modality contains much more nonzero
elements than the fMRI modality, as illustrated in Fig. 3, thus even if
average view pooling is used, the DTI modality still dominates the training
process.

####  IV-D5 Visualization

The accuracy and quality of objective evaluations may be undermined by the
very limited number of training and validation samples, also one important
task of graph-based problems is to produce satisfactory representations for
subsequent analysis purposes. Therefore, to qualitatively examine the
effectiveness of MGNet, we use t-SNE [27] to compare the graph embeddings
learned by MVGCN and our MGNet on the HIV dataset, the results are illustrated
in Fig. 4.

(a) Original

(b) GCN

(c) MVGCN

(d) MGNet

Figure 4: A tSNE visualization of the graph embeddings learned by GCN, MVGCN
and MGNet on the HIV dataset.

From Fig. 4, it can be seen that our MGNet model learns a higher quality of
graph embeddings where the graphs are well-clustered according to their
labels.
To understand the performance of MGNet from clinical perspective, we also
visualize the feature learning results on HIV and BP in Figs. 5. Left panel
shows the node embedding features obtained by tensor decomposition. The
coordinate system represents neuroanatomy and the color shows the activity
intensity of the brain region. The right panel shows the graph embedding
features which represent the factor strengths for both patients and healthy
controls. From the left panel of Fig. 5, the embedded neuroanatomy is widely
different from each other. From the right panel, the result shows that the
controls have relatively positive correlation with node embedded feature,
while the patient have relatively negative correlation. Moreover, those
neuroimaging findings generally support clinical observations of functional
impairments in attention, psychomotor speed, memory, and executive function.
In particular, regions identified in our current study are consistent with
those reported in structural and functional MRI studies of HIV associated
neurocognitive disorder, including regions within the frontal and parietal
lobes [34].

(a) HIV

(b) BP

Figure 5: Embedded features of nodes and graphs of subjects (left and right
panels respectively) from tensor decomposition and GCNs on HIV and BP
datasets.

##  V Conclusions

Multimodal brain network analysis is challenging in terms of both
effectiveness and efficiency. Existing applications of GCN models explicitly
requires a known graph structure (i.e., adjacency matrix), which is typically
not available in multimodal cases. In this paper, we have presented a generic
multiplex graph convolutional network (MGNet) for multimodal brain network
classification. It advances prior works by showing how tensor and GCN
techniques can be combined together to effectively model multimodal graph-
structured data for joint embedding and classification, without using any
prior knowledge of the data. Compared with SOTA methods, MGNet produces
superior feature embedding and classification results on three realistic
multimodal brain network datasets (i.e., HIV, Bipolar and PPMI). It is
interesting to extend the proposed model to other tensor- and graph-based
applications such as text analysis [23], dynamic graphs [28] and multi-
relational learning [13].

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