IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 1
Graph Learning: A Survey
Feng Xia, Senior Member, IEEE, Ke Sun, Shuo Yu, Member, IEEE, Abdul Aziz, Liangtian Wan, Member, IEEE,
Shirui Pan, and Huan Liu, Fellow, IEEE
Abstract —Graphs are widely used as a popular representation
of the network structure of connected data. Graph data can
be found in a broad spectrum of application domains such
as social systems, ecosystems, biological networks, knowledge
graphs, and information systems. With the continuous penetra-
tion of artificial intelligence technologies, graph learning (i.e.,
machine learning on graphs) is gaining attention from both
researchers and practitioners. Graph learning proves effective for
many tasks, such as classification, link prediction, and matching.
Generally, graph learning methods extract relevant features of
graphs by taking advantage of machine learning algorithms.
In this survey, we present a comprehensive overview on the
state-of-the-art of graph learning. Special attention is paid to
four categories of existing graph learning methods, including
graph signal processing, matrix factorization, random walk,
and deep learning. Major models and algorithms under these
categories are reviewed respectively. We examine graph learning
applications in areas such as text, images, science, knowledge
graphs, and combinatorial optimization. In addition, we discuss
several promising research directions in this field.
Index Terms —Graph learning, graph data, machine learning,
deep learning, graph neural networks, network representation
learning, network embedding.
IMPACT STATEMENT
Real-world intelligent systems generally rely on machine
learning algorithms handling data of various types. Despite
their ubiquity, graph data have imposed unprecedented chal-
lenges to machine learning due to their inherent complexity.
Unlike text, audio and images, graph data are embedded
in an irregular domain, making some essential operations
of existing machine learning algorithms inapplicable. Many
graph learning models and algorithms have been developed
to tackle these challenges. This paper presents a systematic
review of the state-of-the-art graph learning approaches as
well as their potential applications. The paper serves mul-
tiple purposes. First, it acts as a quick reference to graph
learning for researchers and practitioners in different areas
such as social computing, information retrieval, computer
vision, bioinformatics, economics, and e-commence. Second,
it presents insights into open areas of research in the field.
Third, it aims to stimulate new research ideas and more
interests in graph learning.
I. I NTRODUCTION
F. Xia is with School of Engineering, IT and Physical Sciences, Federation
University Australia, Ballarat, VIC 3353, Australia
K. Sun, S. Yu, A. Aziz, and L. Wan are with School of Software, Dalian
University of Technology, Dalian 116620, China.
S. Pan is with Faculty of Information Technology, Monash University,
Melbourne, VIC 3800, Australia.
H. Liu is with School of Computing, Informatics, and Decision Systems
Engineering, Arizona State University, Tempe, AZ 85281, USA.
Corresponding author: Feng Xia; e-mail: f.xia@ieee.orgGRAPHS, also referred to as networks, can be extracted
from various real-world relations among abundant enti-
ties. Some common graphs have been widely used to formulate
different relationships, such as social networks, biological
networks, patent networks, traffic networks, citation networks,
and communication networks [1]–[3]. A graph is often defined
by two sets, i.e., vertex set and edge set. Vertices represent en-
tities in graph, whereas edges represent relationships between
those entities. Graph learning has attracted considerable atten-
tion because of its wide applications in the real world, such as
data mining and knowledge discovery. Graph learning meth-
ods have gained increasing popularity for capturing complex
relationships, as graphs exploit essential and relevant relations
among vertices [4], [5]. For example, in microblog networks,
the spread trajectory of rumors can be tracked by detecting
information cascades. In biological networks, new treatments
for difficult diseases can be discovered by inferring protein
interactions. In traffic networks, human mobility patterns can
be predicted by analyzing the co-occurrence phenomenon with
different timestamps [6]. Efficient analysis of these networks
massively depends on the way how networks are represented.
A. What is Graph Learning?
Generally speaking, graph learning refers to machine learn-
ing on graphs. Graph learning methods map the features
of a graph to feature vectors with the same dimensions in
the embedding space. A graph learning model or algorithm
directly converts the graph data into the output of the graph
learning architecture without projecting the graph into a low
dimensional space. Most graph learning methods are based
on or generalized from deep learning techniques, because
deep learning techniques can encode and represent graph data
into vectors. The output vectors of graph learning are in
continuous space. The target of graph learning is to extract
the desired features of a graph. Thus the representation of
a graph can be easily used by downstream tasks such as
node classification and link prediction without an explicit
embedding process. Consequently, graph learning is a more
powerful and meaningful technique for graph analysis.
In this survey paper, we try to examine machine learning
methods on graphs in a comprehensive manner. As shown
in Fig. 1, we focus on existing methods that fall into the
following four categories: graph signal processing (GSP) based
methods, matrix factorization based methods, random walk
based methods, and deep learning based methods. Roughly
speaking, GSP deals with sampling and recovery of graph, and
learning topology structure from data. Matrix factorization can
be divided into graph Laplacian matrix factorization and vertex
proximity matrix factorization. Random walk based methodsarXiv:2105.00696v1  [cs.LG]  3 May 2021IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 2
Fig. 1: The categorization of graph learning.
include structure-based random walk, structure and node in-
formation based random walk, random walk in heterogeneous
networks, and random walk in time-varying networks. Deep
learning based methods include graph convolutional networks,
graph attention networks, graph auto-encoder, graph generative
networks, and graph spatial-temporal networks. Basically, the
model architectures of these methods/techniques differ from
each other. This paper presents an extensive review of the
state-of-the-art graph learning techniques.
Traditionally, researchers adopt an adjacency matrix to
represent a graph, which can only capture the relationship
between two adjacent vertices. However, many complex and
irregular structures cannot be captured by this simple rep-
resentation. When we analyze large-scale networks, tradi-
tional methods are computationally expensive and hard to be
implemented in real-world applications. Therefore, effective
representation of these networks is a paramount problem to
solve [4]. Network Representation Learning (NRL) proposed
in recent years can learn latent features of network vertices
with low dimensional representation [7]–[9]. When the new
representation has been learned, previous machine learning
methods can be employed for analyzing the graph data as well
as discovering relationships hidden in the data.
When complex networks are embedded into a latent, low
dimensional space, the structural information and vertex at-
tributes can be preserved [4]. Thus the vertices of networks can
be represented by low dimensional vectors. These vectors can
be regarded as the features of input in previous machine learn-
ing methods. Graph learning methods pave the way for graph
analysis in the new representation space, and many graph
analytical tasks, such as link prediction, recommendation and
classification, can be solved efficiently [10], [11]. Graphical
network representation sheds light on various aspects of social
life, such as communication patterns, community structure,
and information diffusion [12], [13]. According to the at-
tributes of vertices, edges and subgraph, graph learning tasks
can be divided into three categories, which are vertices based,
edges based, and subgraph based, respectively. The relation-
ships among vertices in a graph can be exploited for, e.g.,classification, risk identification, clustering, and community
detection [14]. By judging the presence of edges between
two vertices in graphs, we can perform recommendation and
knowledge reasoning, for instance. Based on the classification
of subgraphs [15], the graph can be used for, e.g., polymer
classification, 3D visual classification, etc. For GSP, it is sig-
nificant to design suitable graph sampling methods to preserve
the features of the original graph, which aims at recovering the
original graph efficiently [16]. Graph recovery methods can
be used for constructing the original graph in the presence
of incomplete data [17]. Afterwards, graph learning can be
exploited to learn the topology structure from graph data. In
summary, graph learning can be used to tackle the following
challenges, which are difficult to solve by using traditional
graph analysis methods [18].
1)Irregular domains: Data collected by traditional sen-
sors have a clear grid structure. However, graphs lie in an
irregular domain (i.e., non-Euclidean space). In contrast
to regular domain (i.e., Euclidean space), data in non-
Euclidean space are not ordered regularly. Distance is
hence difficult to be defined. As a result, basic methods
based on traditional machine learning and signal pro-
cessing cannot be directly generalized to graphs.
2)Heterogeneous networks: In many cases, networks
involved in the traditional graph analysis algorithms
are homogeneous. The appropriate modeling methods
only consider the direct connection of the network and
strip other irrelevant information, which significantly
simplifies the processing. However, it is prone to cause
information loss. In the real world, the edges among
vertices and the types of vertices are usually diverse,
such as in the academic network shown in Fig. 2. Thus it
isn’t easy to discover potential value from heterogeneous
information networks with abundant vertices and edges.
3)Distributed algorithms: In big social networks, there
are often millions of vertices and edges [19]. Centralized
algorithms cannot handle this since the computational
complexity of these algorithms would significantly in-
crease with the growth of vertex number. The design ofIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 3
distributed algorithms for dealing with big networks is a
critical problem yet to be solved [20]. One major benefit
of distributed algorithms is that the algorithms can be
executed in multiple CPUs or GPUs simultaneously, and
hence the running time can be reduced significantly.
B. Related Surveys
There are several surveys that are partially related to the
scope of this paper. Unlike these surveys, we aim to provide
a comprehensive overview of graph learning methods, with a
focus on four specific categories. In particular, graph signal
processing is introduced as one approach for graph learning,
which is not covered by other surveys.
Goyal and Ferrara [21] summarized graph embedding meth-
ods, such as matrix factorization, random walk and their
applications in graph analysis. Cai et al. [22] reviewed graph
embedding methods based on problem settings and embedding
techniques. Zhang et al. [4] summarized NRL methods based
on two categories, i.e., unsupervised NRL and semi-supervised
NRL, and discussed their applications. Nickel et al. [23]
introduced knowledge extraction methods from two aspects:
latent feature models and graph based models. Akoglu et
al. [24] reviewed state-of-the-art techniques for event detec-
tion in data represented as graphs, and their applications in
the real world. Zhang et al. [18] summarized deep learning
based methods for graphs, such as graph neural networks
(GNNs), graph convolutional networks (GCNs) and graph
auto-encoders (GAEs). Wu et al. [25] reviewed state-of-the-
art GNN methods and discussed their applications in dif-
ferent fields. Ortega et al. [26] introduced GSP techniques
for representation, sampling and learning, and discussed their
applications. Huang et al. [27] examined the applications of
GSP in functional brain imaging and addressed the problem of
how to perform brain network analysis from signal processing
perspective.
In summary, none of the existing surveys provides a com-
prehensive overview of graph learning. They only cover some
parts of graph learning, such as network embedding and deep
learning based network representation. The NRL and/or GNN
based surveys do not cover the GSP techniques. In contrast, we
review GSP techniques in the context of graph learning, as it
is an important approach for GNNs. Specifically, this survey
paper integrates state-of-the-art machine learning techniques
for graph data, gives a general description of graph learning,
and discusses its applications in various domains.
C. Contributions and Organization
The contributions of this paper can be summarized as
follows.
A comprehensive overview of state-of-the-art graph
learning methods: we present an integral introduction
to graph learning methods, including, e.g., technical
sketches, application scenarios, and potential research
directions.
Taxonomy of graph learning: we give a technical clas-
sification of mainstream graph learning methods from the
perspective of theoretical models. Technical descriptions
Fig. 2: Heterogeneous academic network [28].
are provided wherever appropriate to improve understand-
ing of the taxonomy.
Insights into future directions in graph learning:
Besides qualitative analysis of existing methods, we shed
light on potential research directions in the field of graph
learning through summarizing several open issues and
relevant challenges.
The rest of this paper is organized as follows. An overview
of graph learning approaches containing graph signal pro-
cessing based methods, matrix factorization based methods,
random walk based methods, and deep learning based methods
is provided in Section II. The applications of graph learning
are examined in Section III. Some future directions as well
as challenges are discussed in Section IV. We conclude the
survey in Section V.
II. G RAPH LEARNING MODELS AND ALGORITHMS
The feature vectors that represent various categorical at-
tributes are viewed as the input in previous machine learning
methods. However, the mapping from the input feature vectors
to the output prediction results need to be handled by graph
learning [21]. Deep learning has been regarded as one of the
most successful techniques in artificial intelligence [29], [30].
Extracting complex patterns by exploiting deep learning from
a massive amount of irregular data has been found very useful
in various fields, such as pattern recognition and image pro-
cessing. Consequently, how to utilize deep learning techniques
to extract patterns from complex graphs has attracted lots of
attention. Deep learning on graphs, such as GNNs, GCNs,
and GAEs, has been recognized as a powerful technique for
graph analysis [18]. Besides, GSP has also been proposed
to deal with graph analysis [26]. One of the most typical
scenarios is that a set of values reside on a set of vertices,
and these vertices are connected by edges [31]. Graph signals
can be adopted to model various phenomena in real world. For
example, in social networks, users in Facebook can be viewed
as vertices, and their friendships can be modeled as edges. The
number of followers of each vertex is marked in this social
network. Based on this assumption, many techniques (e.g.,
convolution, filter, wavelet, etc.) in classical signal processing
can be employed for GSP with suitable modifications [26].
In this section, we review graph learning models and algo-
rithms under four categories as mentioned before, namely GSPIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 4
based methods, matrix factorization based methods, random
walk based methods, and deep learning based methods. In
Table I, we list the abbreviations used in this paper.
TABLE I: Definitions of abbreviations
Abbreviation Definition
PCA Principal component analysis
NRL Network representation learning
LSTM Long short-term memory (networks)
GSP Graph signal processing
GNN Graph neural network
GMRF Gauss markov random field
GCN Graph convolutional network
GAT Graph attention network
GAN Generative adversarial network
GAE Graph auto-encoder
ASP Algebraic signal processing
RNN Recurrent neural network
CNN Convolutional neural network
A. Graph Signal Processing
Signal processing is a traditional subject that processes
signals defined in regular data domain. In recent years, re-
searchers extend concepts of traditional signal processing into
graphs. Classical signal processing techniques and tools such
as Fourier transform and filtering can be used to analyze
graphs. In general, graphs are a kind of irregular data, which
are hard to handle directly. As a complement to learning
methods based on structures and models, GSP provides a new
perspective of spectral analysis of graphs. Derived from signal
processing, GSP can give an explanation of graph property
consisting of connectivity, similarity, etc. Fig. 3 gives a simple
example of graph signals at a certain time point, which is
defined as observed values. In a graph, the above mentioned
observed values can be regarded as graph signals. Each node is
then mapped to the real number field in GSP. The main task
of GSP is to expand signal processing approaches to mine
implicit information in graphs.
Fig. 3: The measurements of PM2.5 from different sensors on
July 5, 2014 (data source: https://www.epa.gov/).1) Representation on Graphs: A meaningful representation
of graphs has contributed a lot to the rapid growth of graph
learning. There are two main models of GSP, i.e., adjacency
matrix based GSP [31] and Laplacian based GSP [32]. An
adjacency matrix based GSP comes from algebraic signal
processing (ASP) [33], which interprets linear signal process-
ing from algebraic theory. Linear signal processing contains
signals, filters, signal transformation, etc. It can be applied
in both continuous and discrete time domains. The basic
assumption of linear algebra is extended to the algebra space in
ASP. By selecting signal model appropriately, ASP can obtain
different instances in linear signal processing. In adjacency
matrix based GSP, the signal model is generated from a shift.
Similar to traditional signal processing, a shift in GSP is a filter
in graph domain [31], [34], [35]. GSP usually defines graph
signal models using adjacency matrices as shifts. Signals of a
graph are normally defined at vertices.
Laplacian based GSP originates from spectral graph theory.
High dimensional data are transferred into a low dimensional
space generated by a part of the Laplacian basis [36]. Some
researchers exploited sensor networks [37] to achieve dis-
tributed processing of graph signals. Other researchers solved
the problem globally under the assumption that the graph is
smooth. Unlike adjacency matrix based GSP, Laplacian matrix
is symmetric with real and non-negative edge weights, which
is used to index undirected graphs.
Although the models use different matrices as basic shifts,
most of the notions in GSP are derived from signal processing.
Notions with different definitions in these models may have
similar meanings. All of them correspond to concepts in signal
processing. Signals in GSP are values defined on graphs, and
they are usually written as a vector, s= [s0;s1;:::;sN1]2
CN: N is the number of vertices, and each element in the
vector represents the value on a vertex. Some studies [26]
allow complex-value signals, even though most applications
are based on real-value signals.
In the context of adjacency matrix based GSP, a graph can
be represented as a triple G(V;E;W), whereVis the vertex
set,Eis the edge set and Wis the adjacency matrix. With
the definition of graphs, we can also define degree matrix
Dii=di, whereDis a diagonal matrix, and diis the degree
of vertexi. Graph Laplacian is defined as L=DW, and
normalized Laplacian is defined as Lnorm =D1=2LD1=2.
Filters in signal processing can be seen as a function that
amplifies or reduces relevant frequencies, eliminating irrele-
vant ones. Matrix multiplication in linear space equals to scale
changing, which is identical with filter operation in frequency
domain. It is obvious that we can use matrix multiplication as
a filter in GSP, which is written as sout=Hsin, whereH
stands for a filter.
Shift is an important concept to describe variation in sig-
nal, and time-invariant signals are used frequently [31]. In
fact, there are different choices of shifts in GSP. Adjacency
matrix based GSP uses Aas shift. Laplacian based GSP uses
L[32], and some researchers also use other matrices [38].
By following time invariance in traditional signal processing,
shift invariance is defined in GSP. If filters are commutative
with shift, they are shift-invariant, which can be written asIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 5
AH =HA . It is proved that shift-invariant filter can be
represented by the shift. The properties of shift are vital, and
they determine the fashion of other definitions such as Fourier
transform and frequency.
In adjacency matrix based GSP, eigenvalue decomposition
of shiftAisA=VV1.Vis the matrix of eigenvectors
[v0;v1;:::;vN1]and
=2
640
...
N13
75
is a diagonal matrix of eigenvalues. The Fourier transform
matrix is the inverse of V, i.e.,F=V1. Frequency of shift
is defined as total variation, which states the difference after
shift
TVG=jjvk1
maxAvkjj1;
where1
maxis a normalized factor of matrix. It means that the
frequencies of eigenvalue far away from the largest eigenval-
ues on complex plane are large. A large frequency means that
signals are changed with a large scale after shift filtering. The
differences between minimum and maximum can be seen in
Fig. 4. Generally, the total variation tends to be relatively low
with larger frequency, and vice versa. Eigenvectors of larger
eigenvalues can be used to construct low-frequency filters,
which capture fundamental characteristics, and smaller ones
can be employed to capture the variation among neighbor
nodes.
For topology learning problems, we can distinguish the
corresponding solutions depending on known information.
When topology information is partly known, we can use the
known information to infer the whole graph. In case the
topology information is unknown while we still can observe
the signals on the graph, the topology structure has to be
inferred from the signals. The former one is often solved as a
sampling and recovery problem, and blind topology inference
is also known as graph topology (or structure) learning.
2) Sampling and Recovery: Sampling is not a new concept
defined in GSP. In conventional signal processing, we normally
need to reconstruct original signals with the least samples
and retain all information of original signals for a sampling
problem. Few samples lead to the lack of information and more
samples need more space to store. The well-known Nyquist-
Shannon sampling theorem gives the sufficient condition of
perfect recovery of signals in time domain.
Researchers have migrated the sampling theories into GSP
to study the sampling problem on graphs. As the volume of
data is large in some real-world applications such as sensor
networks and social networks, sampling less and recover-
ing better are vital for GSP. In fact, most algorithms and
frameworks solving sampling problems require that the graph
models correlations within signals observed on it [39]. The
sampling problem can be defined as reconstructing signals
from samples on a subset of vertices, and signals in it are
usually band-limited. Nyquist-Shannon sampling theorem was
extended to graph signals in [40]. Based on the normalized
Laplacian matrix, sampling theorem and cut-off frequency are
(a) The maximum frequency
(b) The minimum frequency
Fig. 4: Illustration of difference between minimum and max-
imum frequencies.
defined for GSP. Moreover, the authors provided a method for
computing cut-off frequency from a given sampling set and a
method for choosing sampling set for a given bandwidth. It
should be noted that the sampling theorem proposed therein
is merely applied to undirected graph. As Laplacian matrix
represents undirected graphs only, sampling theory for directed
graph adopts adjacent matrix. An optimal operator with a
guarantee for perfect recovery was proposed in [35], and it
is robust to noise for general graphs.
One of the explicit distinctions between classical signal
processing and GSP is that signals of the former fall in regular
domain while the latter falls in irregular domain. For sampling
and recovery problems, classical signal processing samples
successive signals and can recover successive signals from
samplings. GSP samples a discrete sequence, and recovers the
original sequences from samplings. By following this order,
the solution is generally separated into two parts, i.e., findingIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 6
sampling vertex sets and reconstructing original signals based
on various models.
When the size of the dataset is small, we can handle the
signal and shift directly. However, for a large-scale dataset,
some algorithms require matrix decomposition to obtain fre-
quencies and save eigenvalues in the procedure, which are
almost impossible to realize. As a simple technique applicable
to large-scale datasets, a random method can also be used in
sampling. Puy et al. [41] proposed two sample strategies: a
non-adaptive one depending on a parameter and an adaptive
random sampling strategy. By relaxing the optimized con-
straint, they extended random sampling to large scale graphs.
Another common strategy is greedy sampling. For example,
Shomorony and Avestimehr [42] proposed an efficient method
based on linear algebraic conditions that can exactly compute
cut-off frequency. Chamon and Ribeiro [43] provided near-
optimal guarantee for greedy sampling, which guarantees the
performance of greedy sampling in the worst cases.
All of the sampling strategies mentioned above can be
categorized as selecting sampling, where signals are observed
on a subset of vertices [43]. Besides selecting sampling, there
exists a type of sampling called aggregation sampling [44],
which uses observations taken at a single vertex as input,
containing a sequential applications of graph shift operator.
Similar to classical signal processing, the reconstruction
task on graphs can also be interpreted as data interpolation
problem [45]. By projecting the samples on a proper signal
space, researchers obtain interpolated signals. Least squares
reconstruction is an available method in practice. Gadde and
Ortega [46] defined a generative model for signal recovery
derived from a pairwise Gaussian random field (GRF) and
a covariance matrix on graphs. Under sampling theorem, the
reconstruction of graph signals can be viewed as the maximum
posterior inference of GRF with low-rank approximation.
Wang et al. [47] aimed at achieving the distributed reconstruc-
tion of time-varying band limited signal, where the distributed
least squares reconstruction (DLSR) was proposed to recover
the signals iteratively. DLSR can track time-varying signals
and achieve perfect reconstruction. Di Lorenzo et al. [48]
proposed a linear mean squares (LMS) strategy for adaptive
estimation. LMS enables online reconstruction and tracking
from the observation on a subset of vertices. It also allows the
subset to vary over time. Moreover, a sparse online estimation
was proposed to solve the problems with unknown bandwidth.
Another common technique for recovering original signals
is smoothness. Smoothness is used for inferring missing values
in graph signals with low frequencies. Wang et al. [17]
defined the concept of local set. Based on the definition
of graph signals, two iterative methods were proposed to
recover band limited signals on graphs. Besides, Romero
et al. [49] advocated kernel regression as a framework for
GSP modeling and reconstruction. For parameter selection
in estimators, two multi-kernel methods were proposed to
solve a single optimization problem as well. In addition,
some researchers investigated different recovery problems with
compressed sensing [50].
In addition, there exists some researches on sampling of
different kinds of signals such as smooth graph signals, piece-wise constant signals and piece-wise smooth signals [51].
Chen et al. [51] gave a uniform framework to analyze graph
signals. The reconstruction of a known graph signal was stud-
ied in [52], where the signal is sparse, which means only a few
vertices are non-zeros. Three kinds of reconstruction schemes
corresponding to various seeding patterns were examined.
By analyzing single simultaneous injection, single successive
value injection, and multiple successive simultaneous injec-
tions, the conditions for perfect reconstruction on any vertices
were derived.
3) Learning Topology Structure from Data: In most appli-
cation scenes, graphs are constructed according to connections
of entity correlations. For example, in sensor networks, the cor-
relations between sensors are often consistent with geographic
distance. Edges in social networks are defined as relations such
as friends or colleagues [53]. In biochemical networks, edges
are generated by interactions. Although GSP is an efficient
framework for solving problems on graphs such as sampling,
reconstruction, and detection, there lacks a step to extract
relations from datasets. Connections exist in many datasets
without explicit records. Fortunately, they can be inferred in
many ways.
As a result, researchers want to learn complete graphs from
datasets. The problem of learning graph from a dataset is
stated as estimating graph Laplacian, or graph topology [54].
Generally, they require the graph to satisfy some properties,
such as sparsity and smoothness. Smoothness is a widespread
assumption in networks generated from datasets. Therefore, it
is usually used to constrain observed signals and provide a
rational guarantee for graph signals. Researchers have applied
it to graph topology learning. The intuition behind smoothness
based algorithms is that most signals on graph are stationary,
and the result filtered by shift tends to be the lowest frequency.
Dong et al. [55] adopted a factor analysis model for graph
signals, and also imposed a Gaussian prior on latent variables
to obtain a Principal Component Analysis (PCA) like represen-
tation. Kalofolias [56] formulated the objective as a weighted
l1problem and designed a general framework to solve it.
Gauss Markov Random Field (GMRF) is also a widely
used theory for graph topology learning in GSP [54], [57],
[58]. The models of GRMF based graph topology learning
select graphs that are more likely to generate signals which are
similar to the ones generated by GMRF. Egilmez et al. [54]
formulated the problem as a maximum posterior parameter
estimation of GMRF, and the graph Laplacian is a precision
matrix. Pavez and Ortega [57] also formulated the problem as
a precision matrix estimation, and the rows and columns are
updated iteratively by optimizing a quadratic problem. Both
of them restrict the result matrix, which should be Laplacian.
In [58], Pavez et al. chose a two steps framework to find
the structure of the underlying graph. First, a graph topology
inference step is employed to select a proper topology. Then,
a generalized graph Laplacian is estimated. An error bound of
Laplacian estimation is computed. In the next step, the error
bound can be utilized to obtain a matrix in a specific form as
the precision matrix estimation. It is one of the first work that
suggests adjusting the model to obtain a graph satisfying the
requirement of various problems.IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 7
Diffusion is also a relevant model that can be exploited
to solve the topology interfering problem [39], [59]–[61].
Diffusion refers to that the node continuously influences its
neighborhoods. In graphs, nodes with larger values will have
higher influence on their neighborhood nodes. Using a few
components to represent signals will help to find the main
factors of signal formation. The models of diffusion are often
under the assumption of independent identically-distributed
signals. Pasdeloup et al. [59] gave the concept of valid graphs
to explain signals and assumed that the signals are observed
after diffusion. Segarra et al. [60] agreed that there exists a
diffusion process in the shift, and the signals can be observed.
The signals in [61] were explained as a linear combination of
a few components.
For time series recorded in data, researchers tried to
construct time-sequential networks. For instance, Mei and
Moura [62] proposed a methodology to estimate graphs, which
considers both time and space dependencies and models them
by auto-regressive process. Segarra et al. [63] proposed a
method that can be seen as an extension of graph learning.
The aim of the paper was to solve the problem of joint
identification of a graph filter and its input signal.
For recovery methods, a well-known partial inference prob-
lem is recommendation [45], [64], [65]. The typical algorithm
used in recommendation is collaborative filtering (CF) [66].
Given the observed ratings in a matrix, the objective of CF is to
estimate the full rating matrix. Huang et al. [65] demonstrated
that collaborative filtering could be viewed as a specific band-
stop graph filter on networks representing correlations between
users and items. Furthermore, linear latent factor methods can
also be modeled as band limited interpretation problem.
4) Discussion: GSP algorithms have strict limitations on
experimental data, thus leading to less real-world applications.
Moreover, GSP algorithms require the input data to be exactly
the whole graph, which means that part of graph data cannot
be the input. Therefore, the computational complexity of this
kind of methods could be significantly high. In comparison
with other kinds of graph learning methods, the scalability of
GSP algorithms is relatively poor.
B. Matrix Factorization Based Methods
Matrix factorization is a method of simplifying a matrix into
its components. These components have a lower dimension
and could be used to represent the original information of
a network, such as relationships among nodes. Matrix fac-
torization based graph learning methods adopt a matrix to
represent graph characteristics like vertex pairwise similarity,
and the vertex embedding can be achieved by factorizing this
matrix [67]. Early graph learning approaches usually utilized
matrix factorization based methods to solve the graph embed-
ding problem. The input of matrix factorization is the non-
relational high dimensional data feature represented as a graph.
The output of matrix factorization is a set of vertex embedding.
If the input data lies in a low dimensional manifold, the graph
learning for embedding can be treated as a dimension-reduced
problem that preserves the structure information. There are
mainly two types of matrix factorization based graph learning.One is graph Laplacian matrix factorization, and the other is
vertex proximity matrix factorization.
1) Graph Laplacian Matrix Factorization: The preserved
graph characteristics can be expressed as pairwise vertex
similarities. Generally, there are two kinds of graph Laplacian
matrix factorization, i.e., transductive and inductive matrix
factorization. The former only embeds the vertices contained
in the training set, and the latter can embed the vertices that are
not contained in the training set. The general framework has
been designed in [68], and the graph Laplacian matrix factor-
ization based graph learning methods have been summarized
in [69]. The Euclidean distance between two feature vectors is
directly adopted in the initial Metric Multidimensional Scaling
(MDS) [70] to find the optimal embedding. The neighborhoods
of vertices are not considered in the MDS, i.e., any pair
of training instances are considered as connected. The data
feature is extracted by constructing a knearest neighbor graph,
and the subsequent studies [67], [71]–[73] tackle this issue.
The topksimilar neighbors of each vertex are connected with
itself. A similar matrix is calculated by exploiting different
methods, and thus the graph characteristics can be preserved
as much as possible.
Recently, researchers have designed more sophisticated
models. The performance of earlier matrix factorization model
Locality Preserving Projection (LPP) can be improved by
introducing an anchor taking advantage of Anchorgraph-based
Locality Preserving Projection (AgLPP) [74], [75]. The graph
structure can be captured by using a local regression model
and a global regression process based on Local and Global
Regressive Mapping (LGRM) [76]. The global geometry can
be preserved by using local spline regression [77].
More information can be preserved by exploiting the auxil-
iary information. An adjacency graph and a labelled graph
were constructed in [78]. The objective function of LPP
preserves the local geometric structure of the datasets [67].
An adjacency graph and a relational feedback graph were con-
structed in [79] as well. The graph Laplacian regularization,
k-means and PCA were considered in RF-Semi-NMF-PCA si-
multaneously [80]. Other works, e.g., [81], adopt semi-definite
programming to learn the adjacency graph that maximizes the
pairwise distances.
2) Vertex Proximity Matrix Factorization: Apart from solv-
ing the above generalized eigenvalue problem, another ap-
proach of matrix factorization is to factorize vertex proximity
matrix directly. In general, matrix factorization can be used
to learn the graph structure from non-relational data, and it is
applicable to learn homogeneous graphs.
Based on matrix factorization, vertex proximity can be
approximated in a low dimensional space. The objective of
preserving vertex proximity is to minimize the error. The
Singular Value Decomposition (SVD) of vertex proximity
matrix was adopted in [82]. There are some other approaches
such as regularized Gaussian matrix factorization [83], low-
rank matrix factorization [84], for solving SVD.
3) Discussion: Matrix factorization algorithms operate on
an interaction matrix to decompose several lower dimension
matrices. The process brings some drawbacks. For example,
the algorithms require a large memory when the decomposedIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 8
matrices become large. In addition, matrix factorization al-
gorithms are not applicable to supervised or semi-supervised
tasks with the training process.
C. Random Walk Based Methods
Random walk is a convenient and effective way to sample
networks [85], [86]. This method can generate sequences
of nodes meanwhile preserving original relations between
nodes. Based on network structure, NRL can generate feature
vectors of vertices so that downstream tasks can mine network
information in a low dimensional space. An example of NRL
is shown in Fig. 5. The image in Euclidean space is shown
in Fig. 5(a), and the corresponding graph in non-Euclidean
space is shown in Fig. 5(b). As one of the most successful
NRL algorithms, random walks play an important role in
dimensionality reduction.
(a) Image in Euclidean space
(b) Graph in non-Euclidean space
Fig. 5: An example of NRL mapping an image from Euclidean
space into non-Euclidean space.
1) Structure Based Random Walks: Graph-structured data
have various data types and structures. The information en-
coded in a graph is related to graph structure and vertex
attributes, which are the two key factors affecting the reason-
ing of networks. In real-world applications, many networks
only have structural information, but lack vertex attribute
information. How to identify network structure information
effectively, such as important vertices and invisible links,attracts the interest of network scientists [87]. Graph data
have high dimensional characteristics. Traditional network
analysis methods cannot be used for analyzing graph data in
a continuous space.
In recent years, various NRL methods have been proposed,
which preserve rich structural information of networks. Deep-
Walk [88] and Node2vec [7] are two representative methods
for generating network representation of basic network topol-
ogy information. These methods use random walk models
to generate random sequences on networks. By treating the
vertices as words and the generated random sequences of
vertices as word sequences (sentences), the models can learn
the embedding representation of the vertices by inputting these
sequences into the Word2vec model [89]–[91]. The principle
of the learning model is to maximize the co-occurrence prob-
ability of vertices such as Word2vec. In addition, Node2vec
shows that network has complex structural characteristics,
and different network structure samplings can obtain different
results. The sampling mode of DeepWalk is not enough
to capture the diversity of connection patterns in networks.
Node2vec designs a random walk sampling strategy, which
can sample the networks with the preference of breadth-first
sampling and depth-first sampling by adjusting the parameters.
The NRL algorithms mentioned above focused on the first-
order proximity information of vertices. Tang et al. [92]
proposed a method called LINE for large-scale network
embedding. LINE can maintain the first and second order
approximations. The first-order neighbor refers to the one-
hop neighbor between two vertices, and the second-order
neighbor is the neighbor with two hops. LINE is not a deep
learning based model, but it is often compared with these edge
modeling based methods.
It has been proved that the network structure information
plays an important role in various network analysis tasks. In
addition to this structural information, network attributes in
the original network space are also critical in modeling the
formation and evolution of the network [93].
2) Structure and Vertex Information Based Random Walks:
In addition to network topology, many types of networks also
have rich vertex information, such as vertex content or label
in networks. Yang et al. [84] proposed an algorithm called
TADW. The model is based on DeepWalk and considers the
text information of vertices. The MMDW [94] is another
model based on DeepWalk, which is a kind of semi-supervised
network embedding algorithm, by leveraging labelling infor-
mation of vertices to enhance the performance.
Focusing on the structural identity of nodes, Ribeiro et
al. [95] formulated a framework named Struc2vec. The frame-
work considers nodes with similar local structure rather than
neighborhood and labels of nodes. With hierarchy to evaluate
structural similarity, the framework constrains structural sim-
ilarity more stringently. The experiments indicate that Deep-
Walk and Node2vec are worse than Struc2vec which considers
structural identity. There are some other NRL models, such
as Planetoid [96], which learn network representation using
the feature of network structure and vertex attribute informa-
tion. It is well known that vertex attributes provide effective
information for improving network representation and helpIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 9
to learn embedded vector space. In the case of relatively
sparse network topology, vertex attribute information can be
used as supplementary information to improve the accuracy
of representation. In practice, how to use vertex information
effectively and how to apply this information to network vertex
embedding are the main challenges in NRL.
Researchers not only investigate random walk based NRL
on vertices but also on graphs. Adhikari et al. [97] proposed an
unsupervised scalable algorithm, Sub2Vec, to learn arbitrary
subgraph. To be more specific, they proposed a method to
measure the similarities between subgraphs without disturbing
local proximity. Narayanan et al. [98] proposed graph2vec,
which is a neural embedding framework. Modeling on neural
document embedding models, graph2vec takes a graph as a
document and the subgraph around words as vertices. By
migrating the model to graphs, the performance of graph2vec
significantly exceeds other substructure representation learning
algorithms.
Generally, random walk can be regarded as a Markov
process. The next state of the process is only related to last
state, which is known as Markov chain. Inspired by vertex-
reinforced random walks, Benson et al. [99] presented spacey
random walk, a non-Markovian stochastic process. As a spe-
cific type of a more general class of vertex-reinforced random
walks, it takes the view that the probability of time remained
on each vertex relates to the long term behavior of dynamical
systems. They proved that dynamical systems can converge to
a stationary distribution under sufficient conditions.
Recently, with the development of Generative Adversarial
Network (GAN), researchers combined random walks with
the GAN method [100], [101]. Existing research on NRL can
be divided into generative models and discriminative models.
GraphGAN [100] integrated these two kinds of models and
played a game-theoretical minimax game. With the process
of the game, the performance of the two models can be
strengthened. Random walk is used as a generator in the
game. NetGAN [101] is a generative model that can model
network in real applications. The method takes the distribution
of biased random walk as input, and can produce graphs with
known patterns. It preserves important topology properties and
does not need to define them in model definition.
3) Random Walks in Heterogeneous Networks: In reality,
most networks contain more than one type of vertex, and
hence networks are heterogeneous. Different from homoge-
neous NRL, heterogenous NRL should well reserve various
relationships among different vertices [102]. Considering the
ubiquitous existence of heterogeneous networks, many ef-
forts have been made to learn network representations of
heterogeneous networks. Compared to homogeneous NRL, the
proximity among entities in heterogeneous NRL is more than
a simple measure of distance or closeness. The semantics
among vertices and links should be considered. Some typical
scenarios include knowledge graphs and social networks.
Knowledge graph is a popular research domain in recent
years. A vital part in knowledge base population is relational
inference. The central problem of relational inference is infer-
ring unknown knowledge from the existing facts in knowledge
bases [103]. There are three types of common relationalinference method in general: statistical relational learning
(SRL), latent factor models (LFM) and random walk models
(RWM). Relational learning methods based on statistics lack
generality and scalability. As a result, latent factor model based
graph embedding and relational paths based random walk have
been adopted more widely.
In a knowledge graph, there exist various vertices and
various types of relationships among different vertices. For
example, in a scholar related knowledge graph [2], [28], the
types of vertices include scholar, paper, publication venue,
institution, etc. The types of relationships include coauthor,
citation, publication, etc. The key idea of knowledge graph
embedding is to embed vertices and their relationships into a
low dimensional vector space, while the inherent structure of
the knowledge graph can be reserved [104].
For relational paths based random walk, the path ranking
algorithm (PRA) is a path finding method using random walks
to generate relational features on graph data [105]. Random
walks in PRA are with restart, and combine features with a
logistic regression. However, PRA cannot predict connection
between two vertices if there does not exist a path between
them. Gardner et al. [106], [107] introduced two ways to
improve the performance of PRA. One method enables more
efficient processing to incorporate new corpus into knowledge
base, while the other method uses vector space to reduce
the sparsity of surface forms. To resolve cascade errors in
knowledge construction, Wang and Cohen [108] proposed a
joint information extraction and knowledge base based model
with a recursive random walk. Using latent context of the text,
the model obtains additional improvement. Liu et al. [109]
developed a new random walk based learning algorithm named
Hierarchical Random-walk inference (HiRi). It is a two-tier
scheme: the upper tier recognizes relational sequence pattern,
and the lower tier captures information from subgraphs of
knowledge bases.
Another widely-investigated type of heterogeneous net-
works is social networks, such as online social networks and
location based social networks. Social networks are heteroge-
neous in nature because of the different types of vertices and
relations. There are two main ways to embed heterogeneous
social networks, including meta path-based approaches and
random walk-based approaches.
A meta path in heterogeneous networks is defined as a
sequence of vertex types encoding significant composite re-
lations among various types of vertices. Aiming to employ
the rich information in social networks by exploiting various
types of relationships among vertices, Fu et al. [110] proposed
HIN2Vec, which is a representation learning framework based
on meta-paths. HIN2Vec is a neural network model and the
meta-paths are well embedded based on two independent
phases, i.e., training data preparation and representation learn-
ing. Experimental results on various social network datasets
show that HIN2Vec model is able to automatically learn vertex
vector in heterogeneous networks to support a variety of
applications. Metapath2vec [111] was designed by formalizing
meta-path based random walks to construct the neighborhoods
of a vertex in heterogeneous networks. It takes the advantage
of a heterogeneous skip-gram model to perform vertex em-IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 10
bedding.
Meta path based methods require either prior knowledge for
optimal meta-path selection or extended computations for path
length selection. To overcome these challenges, random walk
based approaches have been proposed. Hussein et al. [112]
proposed the JUST model, which is a heterogeneous graph
embedding approach using random walks with jump and stay
strategies so that the aforementioned bias can be overcomed ef-
fectively. Another method which does not require prior knowl-
edge for meta-path definition is MPDRL [113], meta-path
discovery with reinforcement earning. This method employs
the reinforcement learning algorithm to perform multi-hop rea-
soning to generate path instances and then further summarizes
the important meta-paths using the Lowest Common Ancestor
principle. Shi et al. [114] proposed the HERec model, which
utilizes the heterogeneous information network embedding
for providing accurate recommendations in social networks.
HERec is designed based on a random walk based approach
for generating meaningful vertex sequences for heterogeneous
network embedding. HERec can effectively adopt the auxiliary
information in heterogeneous information networks. Other
typical heterogeneous social network embedding approaches
include, e.g., PTE [115] and SHNE [116].
4) Random Walks in Time-varying Networks: Network is
evolving over time, which means that new vertices may emerge
and new relations may appear. Therefore, it is significant
to capture the temporal behaviour of networks in network
analysis. Many efforts have been made to learn time-varying
network embedding (e.g., dynamic networks or temporal net-
works) [117]. In contrast to static network embedding, time-
varying NRL should consider the network dynamics, which
means that old relationships may become invalid and new links
may appear.
The key of time-varying NRL is to find a suitable way to
incorporate the time characteristic into embedding via reason-
able updating approaches. Nguyen et al. [118] proposed the
CTDNE model for continuous dynamic network embedding
based on random walk with ”chronological” paths which can
only move forward as time goes on. Their model is more
suitable for time-dependent network representation that can
capture the important temporal characteristics of continuous-
time dynamic networks. Results on various datasets show
that CTDNE outperforms static NRL approaches. Zuo et
al. [119] proposed the HTNE model which is a temporal
NRL approach based on the Hawkes process. HTNE can well
integrate the Hawkes process into network embedding so that
the influence of historical neighbors on the current neighbors
can be accurately captured.
For unseen vertices in a dynamical network, Graph-
SAGE [120] was presented to efficiently generate embed-
dings for new vertices in network. In contrast to methods
that training embedding for every vertex in the network,
GraphSAGE designs a function to generate embedding for
a vertex with features of the neighborhoods locally. After
sampling neighbors of a vertex, GraphSAGE uses different
aggregators to update the embedding of the vertex. However,
current graph neural methods are proficient of only learning
local neighborhood information and cannot directly explorethe higher-order proximity and the community structure of
graphs.
5) Discussion: As mentioned before, random walk is a
fundamental way to sample networks. The sequences of nodes
could preserve the information of network structure. However,
there are some disadvantages of this method. For example,
random walk relies on random strategies, which creates some
uncertain relations of nodes. To reduce this uncertainty, it
needs to increase the number of samples, which will signifi-
cantly increase the complexity of algorithms. Some random
walk variants could preserve local and global information
of networks, but they might not be effective in adjusting
parameters to adapt to different types of networks.
D. Deep Learning on Graphs
Deep learning is one of the hottest areas over the past few
years. Nevertheless, it is an attractive and challenging task to
extend the existing neural network models, such as Recurrent
Neural Networks (RNNs) or Convolutional Neural Networks
(CNNs), to graph data. Gori et al. [121] proposed a GNN
model based on recursive neural network. In this model, a
transfer function is implemented, which maps the graph or its
vertices to an m-dimensional Euclidean space. In recent years,
lots of GNN models have been proposed.
1) Graph Convolutional Networks: GCN works on the ba-
sis of grid structure domain and graph structure domain [122].
Time Domain and Spectral Methods . Convolution is one
of a common operation in deep learning. However, since graph
lacks a grid structure, standard convolution over images or
text cannot be directly applied to graphs. Bruna et al. [122]
extended the CNN algorithm from image processing to the
graph using the graph Laplacian matrix, dubbed as spectral
graph CNN. The main idea is similar to Fourier basis for
signal processing. Based on [122], Henaff et al. [123] defined
kernels to reduced the learning parameters by analogizing
the local connection of CNNs on the image. Defferrard et
al. [124] provided two ways for generalizing CNNs to graph
structure data based on graph theory. One method is to reduce
the parameters by using polynomial kernel, and this method
can be accelerated by using Chebyshev polynomial approx-
imation. The other method is the special pooling method,
which is pooling on the binary tree constructed from vertices.
An improved version of [124] was introduced by Kipf and
Welling [125]. The proposed method is a semi-supervised
learning method for graphs. The algorithm employs an excel-
lent and straightforward neural network followed by a layer-
by-layer propagation rule, which is based on the first-order
approximation of spectral convolution on the graph and can
be directly acted on the graph.
There are some other time domain based methods. Based
on the mixture model of CNNs, for instance, Monti et
al. [126] generalized the CNN to non-Euclidean space. Zhou
and Li [127] proposed a new CNN graph modeling framework,
which designs two modules for graph structure data: K-
order convolution operator and adaptive filtering module. In
addition, the high-order adaptive graph convolution network
(HA-GCN) framework proposed in [127] is a general ar-
chitecture that is suitable for many applications of verticesIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 11
and graph centers. Manessi et al. [128] proposed a dynamic
graph convolution network algorithm for dynamic graphs.
The core idea of the algorithm is to combine the expansion
of graph convolution with the improved Long Short Term-
Memory networks (LSTM) algorithm, and then train and learn
the downstream recursive unit by using graph structure data
and vertex features. The spectral based NRL methods have
many applications, such as vertex classification [125], traffic
forecasting [129], [130], and action recognition [131].
Space Domain and Spatial Methods . Spectral graph
theory provides a convolution method on graphs, but many
NRL methods directly use convolution operation on graphs
in space domain. Niepert et al. [132] applied graph labeling
procedures such as Weisfeiler-Lehman kernel on graphs to
generate unique order of vertices. The generated sub-graphs
can be fed to the traditional CNN operation in space domain.
Duvenaud et al. [133] designed Neural fingerprints (FP), which
is a spatial method using the first-order neighbors similar to
the GCN algorithm. Atwood and Towsley [134] proposed an-
other convolution method, called diffusion-convolutional neu-
ral network, which incorporates transfer probability matrix and
replaces the characteristic basis of convolution with diffusion
basis. Gilmer et al. [135] reformulated existing models into
a single common framework, and exploited this framework to
discover new variations. Allamanis et al. [136] represented the
structure of code from syntactic and semantic, and utilized the
GNN method to recognize program structures.
Zhuang and Ma [137] designed dual graph convolution
networks (DGCN), which use diffusion basis and adjacency
basis. DGCN uses two convolutions: one is the characteristic
form of polynomial filter, and the other is to replace the
adjacency matrix with the PPMI (Positive Pointwise Mutual
Information) of the transition probability [89]. Dai et al. [138]
proposed the SSE algorithm, which uses asynchronous ran-
dom to learn vertex representation so as to improve learning
efficiency. In this model, a recursive method is adopted to
learn vertex latent representation and the sampled batch data
are utilized to update parameters. The recursive function of
SSE is calculated from the weighted average of historical state
and new state. Zhu et al. [139] proposed a graph smoothing
splines neural network which exploits non-smoothing node
features and global topological knowledge such as centrality
for graph classification. Gao et al. [140] proposed a large scale
graph convolution network (LGCN) based on vertex feature
information. In order to adapt to the scene of large scale
graphs, they proposed a sub-graph training strategy, which first
trained the sampled sub-graph in a small batch. Based on a
deep generative graph model, a novel method called DeepNC
for inferring the missing parts of a network was proposed
in [141].
A brief history of deep learning on graphs is shown in Fig. 6.
GNN has attracted lots of attention since 2015, and it is widely
studied and used in various fields.
2) Graph Attention Networks: In sequence-based tasks,
attention mechanism has been regarded as a standard [142].
GNNs achieve lots of benefits from the expanded model
capacity of attention mechanisms. GATs are a kind of spatial-
based GCNs [143]. It takes the attention mechanism into con-sideration when determining the weights of vertex’s neighbors.
Likewise, Gated Attention Networks (GAANs) also introduced
the multi-head attention mechanism for updating the hidden
state of some vertices [144]. Unlike GATs, GAANs employ a
self-attention mechanism which can compute different weights
for different heads. Some other models such as graph at-
tention model (GAM) were proposed for solving different
problems [145]. Take GAM as an example, the purpose of
GAM is to handle graph classification. Therefore, GAM is set
to process informative parts by visiting a sequence of signifi-
cant vertices adaptively. The model of GAM contains LSTM
network, and some parameters contain historical information,
policies, and other information generated from exploration of
the graph. Attention Walks (AWs) are another kind of learning
model based on GNN and random walks [146]. In contrast
to DeepWalk, AWs use differentiable attention weights when
factorizing the co-occurrence matrix [88].
3) Graph Auto-Encoders: GAE uses GNN structure to
embed network vertices into low dimensional vectors. One
of the most general solutions is to employ a multi-layer
perception as the encoder for inputs [147]. Therein the decoder
reconstructs neighborhood statistics of the vertex. PPMI or
the first and the second nearest neighborhood can be taken
into statistics [148], [149]. Deep neural networks for graph
representations (DNGR) employ PPMI. Structural deep net-
work embedding (SDNE) employs stacked auto-encoder to
maintain both the first-order and the second-order proximity.
Auto-encoder [150] is a traditional deep learning model,
which can be classified as a self-supervised model [151].
Deep recursive network embedding (DRNE) reconstructs some
vertices’ hidden state rather than the entire graph [152]. It has
been found that if we regard GCN as an encoder, and combine
GCN with GAN or LSTM with GAN, then we can design
the auto-encoder for graphs. Generally speaking, DNGR and
SDNE embed vertices by the given structure features, while
other methods such as DRNE learn both topology structure
and content features [148], [149]. Variational graph auto-
encoder [153] is another successful approach that employs
GCN as an encoder and a link prediction layer as a decoder.
Its successor, adversarially regularized variational graph auto-
encoder [154], adds a regularization process with an adversar-
ial training approach to learn a more robust embedding.
4) Graph Generative Networks: The purpose of graph
generative networks is to generate graphs according to the
given observed set of graphs. Many previous methods of graph
generative networks have their own application domains. For
example, in natural language processing, the semantic graph
or the knowledge graph is generated based on the given
sentences. Some general methods have been proposed recently.
One kind of them considers the generation process as the for-
mation of vertices and edges. Another kind is to employ gener-
ative adversarial training. Some GCNs based graph generative
networks such as molecular generative adversarial networks
(MolGAN) integrate GNN with reinforcement learning [155].
Deep generative models of graphs (DGMG) achieves a hidden
representation of existing graphs by utilizing spatial-based
GCNs [156]. There are some knowledge graph embedding
algorithms based on GAN and Zero-Shot Learning [157]. VyasIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 12
Fig. 6: A brief history of algorithms of deep learning on graphs.
et al. [158] proposed a Generalized Zero-Shot learning model,
which can find unseen semantic in knowledge graphs.
5) Graph Spatial-Temporal Networks: Graph spatial-
temporal networks simultaneously capture the spatial and tem-
poral dependence of graphs. The global structure is included in
the spatial-temporal graphs, and the input of each vertex varies
with the change of time. For example, in traffic networks, each
sensor records the traffic speed of a road continuously as a
vertex, in which the edge of the traffic networks is determined
by the distance between the sensor pairs [129]. The goal of a
spatial-temporal network can be to predict future vertex values
or labels, or to predict spatial-temporal graph labels. Recent
studies in this direction have discussed the use of GCNs,
the combination of GCNs with RNN or CNN, and recursive
structures for graph structures [130], [131], [159].
6) Discussion: In this context, the task of graph learning
can be seen as optimizing the objective function by using
gradient descent algorithms. Therefore the performance of
deep learning based NRL models is influenced by gradient
descent algorithms. They may encounter challenges like local
optimal solutions and the vanishing gradient problem.
III. A PPLICATIONS
Many problems can be solved by graph learning methods,
including supervised, semi-supervised, unsupervised, and re-
inforcement learning. Some researchers classify the applica-
tions of graph learning into three categories, i.e., structural
scenarios, non-structural scenarios, and other application sce-
narios [18]. Structural scenarios refer to the situation where
data are performed in explicit relational structures, such as
physical systems, molecular structures, and knowledge graphs.
Non-structural scenarios refer to the situation where data are
with unclear relational structures, such as images and texts.
Other application scenarios include, e.g., integrating models
and combinatorial optimization problems. Table II lists the
neural components and applications of various graph learning
methods.
A. Datasets and Open-source Libraries
There are several datasets and benchmarks used to evaluate
the performance of graph learning approaches for various tasks
such as link prediction, node classification, and graph visual-
ization. For instance, datasets like Cora1(citation network),
1https://relational.fit.cvut.cz/dataset/CORAPubmed2(citation network), BlogCatalog3(social network),
Wikipedia4(language network) and PPI5(biological network)
include nodes, edges, labels or attributes of nodes. Some
research institutions developed graph learning libraries, which
include common and classical graph learning algorithms. For
example, OpenKE6is a Python library for knowledge graph
embedding based on PyTorch. The open-source framework has
the implementations of RESCAL, HolE, DistMult, ComplEx,
etc. CogDL7is a graph representation learning framework,
which can be used for node classification, link prediction,
graph classification, etc.
B. Text
Many data are in textual form coming from various re-
sources like web pages, emails, documents (technical and
corporate), books, digital libraries and customer complains,
letters, patents, etc. Textual data are not well structured for
obtaining any meaningful information as text often contains
rich context information. There exist abundant applications
around text, including text classification, sequence labeling,
sentiment classification, etc. Text classification is one of
the most classical problems in natural language processing.
Popular algorithms proposed to handle this problem include
GCNs [120], [125], GATs [143], Text GCNs [160], and
Sentence LSTM [161]. Sentence LSTM has also been applied
to sequence labeling, text generation, multi-hop reading com-
prehension, etc [161]. Syntactic GCN was proposed to solve
semantic role labeling and neural machine translation [162].
Gated Graph Neural Networks (GGNNs) can also be used to
address neural machine translation and text generation [163].
For relational extraction, Tree LSTM, graph LSTM, and GCN
are better solutions [164].
C. Images
Graph learning applications pertaining to images include
social relationship understanding, image classification, visual
question answering, object detection, region classification, and
semantic segmentation, etc. For social relationship understand-
ing, for instance, graph reasoning model (GRM) is widely
2https://catalog.data.gov/dataset/pubmed
3http://networkrepository.com/soc-BlogCatalog.php
4https://en.wikipedia.org/wiki/Wikipedia:Database download
5https://openwetware.org/wiki/Protein-protein interaction databases
6https://github.com/thunlp/OpenKE
7https://github.com/THUDM/cogdl/IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 13
TABLE II: Summary of graph learning methods and their applications
Categories Algorithms Neural Component Applications
Time Domain and Spectral MethodsSNLCN [122] Graph Neural Network Classification
DCN [123] Spectral Network Classification
ChebNet [124] Convolution Network Classification
GCN [125] Spectral Network Classification
HA-GCN [127] GCN Classification
Dynamic GCN [128] GCN, LSTM Classification
DCRNN [129] Diffusion Convolution Network Traffic Forecasting
ST-GCN [131] GCN Action Recognition
Space Domain and Spatial MethodsPATCHY-SAN [132] Convolutional Network Runtime Analysis,
Feature Visualization,
Graph Classification
Neural FP [133] Sub-graph Classification
DCNN [134] DCNN Classification
DGCN [137] Graph-Structure-Based
Convolution, PPMI-Based
Convolution.Classification
SSE [138] Vertex Classification
LGCN [140] Convolutional Neural Network Vertex Classification
STGCN [130] Gated Sequential Convolution Traffic Forecasting
Deep Learning Model Based MethodsGATs [143]
Attention Neural NetworkClassification
GAAN [144] Vertex Classification
GAM [145] Graph Classification
Aws [146]
Auto-encoder Neural NetworkLink Prediction,
Sensitivity Analysis,
Vertex Classification
SDNE [149] Classification,
Link Prediction,
Visualization
DNGR [148] Clustering, Visualization
DRNE [152] Regular Equivalence Predic-
tion,
Structural Role Classifica-
tion,
Network Visualization
MolGAN [155]
Generative Neural NetworkGenerative Model
DGMG [156] Molecule Generation
DCRNN [129] Diffusion Convolution Network Traffic Forecasting
STGCN [130] Gated Sequential Convolution
ST-GCN [131] GCNs Action Recognition
used [165]. Since social relationships such as friendships
are the basis of social networks in real world, automatically
interpreting these relationships is important for understanding
human behaviors. GRM introduces GGNNs to learn a propa-
gation mechanism. Image classification is a classical problem,
in which GNNs have demonstrated promising performance.
Visual question answering (VQA) is a learning task that in-
volves both computer vision and natural language processing.
A VQA system takes the form of a certain pictures and its
open natural language question as input, in order to generate
a natural language answer as output. Generally speaking, VQA
is question-and-answer for a given picture. GGNNs have been
exploited to help with VQA [166].D. Science
Graph learning has been widely adopted in science. Model-
ing real-world physical systems is one of the most fundamental
perspectives in understanding human intelligence. Represent-
ing objects as vertices and relations as edges between them
is a simple but effective way to perform physics. Battaglia et
al. [167] proposed interaction networks (IN) to predict and
infer abundant physical systems, in which IN takes objects
and relationships as input. Based on IN, the interactions can
be reasoned and the effects can be applied. Therefore, physical
dynamics can be predicted. Visual interaction networks (VIN)
can make predictions from pixels by firstly learning a stateIEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 14
code from two continuous input frames per object [168].
Other graph networks based models have been developed to
address chemistry and biology problems. Calculating molecu-
lar fingerprints, i.e., using feature vectors to represent molec-
ular, is a central step. Researchers [169] proposed neural
graph fingerprints using GCNs to calculate substructure feature
vectors. Some studies focused on protein interface prediction.
This is a challenging issue with significant applications in
biology. Besides, GNNs can be used in biomedical engineering
as well. Based on protein-protein interaction networks, Rhee et
al. [170] used graph convolution and protein relation networks
to classify breast cancer subtypes.
E. Knowledge Graphs
Various heterogeneous objects and relationships are re-
garded as the basis for a knowledge graph [171]. GNNs can
be applied in knowledge base completion (KBC) for solving
the out-of-knowledge-base (OOKB) entity problem [172]. The
OOKB entities are connected to existing entities. Therefore,
the embedding of OOKB entities can be aggregated from
existing entities. Such kind of algorithms achieve reasonable
performance in both settings of KBC and OOKB. Likewise,
GCNs can also be used to solve the problem of cross-lingual
knowledge graph alignment. The main idea of the model is to
embed entities from different languages into an integrated em-
bedding space. Then the model aligns these entities according
to their embedding similarities.
Generally speaking, knowledge graph embedding can be
categorized into two types: translational distance models and
semantic matching models. Translational distance models aim
to learn the low dimensional vector of entities in a knowledge
graph by employing distance-based scoring functions. These
methods calculate the plausibility as the distance between
two entities after a translation measured by the relationships
between them. Among current translational distance models,
TransE [173] is the most influential one. TransE can model the
relationship of entities by interpreting them as translations op-
erating on the low dimensional embedding. Inspired by TranE,
TranH [174] was proposed to overcome the disadvantages
of TransE in dealing with 1-to-N, N-to-1, and N-to-N rela-
tions by introducing relation-specific hyperplanes. Instead of
hyperplanes, TransR [175] introduces relation-specific spaces
to solve the flows in TransE. Meanwhile, various extensions
of TransE have been proposed to enhance knowledge graph
embeddings, such as TransD [176] and TransF [177]. On the
basis of TransE, DeepPath [178] incorporates reinforcement
learning methods for learning relational paths in knowledge
graphs. By designing a complex reward function involving
accuracy, efficiency and path diversity, the path finding process
is better controlled and more flexible.
Semantic matching models utilize the similarity-based scor-
ing functions. They measure the plausibility among entities
by matching latent semantics of entities and relations in low
dimensional vector space. Typical models of this type include
RESCAL [179], DistMult [180], ANALOGY [181], etc.F . Combinatorial Optimization
Classical problems such as traveling salesman problem
(TSP) and minimum spanning tree (MST) have been solved
by using different heuristic solutions. Recently, deep neural
networks have been applied to these problems. Some solutions
make further use of GNNs thanks to their structures. Bello et
al. [182] first proposed such kind of methods to solve TSP.
Their method mainly contains two steps, i.e., a parameterized
reward pointer network and a strategy gradient module for
training. Khalil et al. [183] improved this work with GNN
and achieved better performance by two main procedures.
First, they used structure2vec to achieve vertex embedding and
then input them into Q-learning module for decision-making.
This work also proves the embedding ability of GNN. Nowak
et al. [184] focused on the secondary assignment problem,
i.e., measuring the similarity of two graphs. The GNN model
learns each graph’s vertex embedding and uses the attention
mechanism to match the two graphs. Other studies use GNNs
directly as the classifiers, which can perform the intensive
prediction on graphs with two sides. The rest of the model
facilitates diverse choices and effective training.
IV. O PEN ISSUES
In this section, we briefly summarize several future research
directions and open issues for graph learning.
Dynamic Graph Learning : For the purpose of graph learn-
ing, most existing algorithms are suitable for static networks
without specific constraints. However, dynamic networks such
as traffic networks vary over time. Therefore, they are hard to
deal with. Dynamic graph learning algorithms have rarely been
studied in the literature. It is of significant importance that
dynamic graph learning algorithms are designed to maintain
good performance, especially in the case of dynamic graphs.
Generative Graph Learning : Inspired by the generative
adversarial networks, generative graph learning algorithms can
unify the generative and discriminative models by playing a
game-theoretical min-max game. This generative graph learn-
ing method can be used for link prediction, network evolution,
and recommendation by boosting the performance of genera-
tive and discriminative models alternately and iteratively.
Fair Graph Learning : Most graph learning algorithms rely
on deep neural networks, and the resulting vectors may have
captured undesired sensitive information. The bias existing
in the network is reinforced, and hence it is of significant
importance to integrate the fair metrics into the graph learning
algorithms to address the inherent bias issue.
Interpretability of Graph Learning : The models of graph
learning are generally complex by incorporating both graph
structure and feature information. The interpretability of graph
learning (based) algorithms remains unsolved since the struc-
tures of graph learning algorithms are still a black box. For
example, drug discovery can be achieved by graph learning al-
gorithms. However, it is unknown how this drug is discovered
as well as the reason behind this discovery. The interpretability
behind graph learning needs to be further studied.IEEE TRANSACTIONS ON ARTIFICIAL INTELLIGENCE, VOL. 00, NO. 0, 2021 15
V. C ONCLUSION
This survey gives a general description of graph learning,
and provides a comprehensive review of the state-of-the-art
graph learning methods. We examined existing graph learning
methods under four categories: graph signal processing based
methods, matrix factorization based methods, random walk
based methods, and deep learning based methods. The ap-
plications of graph learning methods mainly under these four
categories in areas such as text, images, science, knowledge
graphs, and combinatorial optimization are outlined. We also
discuss some future research directions in the field of graph
learning. Graph learning is currently a hot area which is grow-
ing at an unprecedented speed. We do hope that this survey
will help researchers and practitioners with their research and
development in graph learning and related areas.
ACKNOWLEDGMENTS
The authors would like to thank Prof. Hussein Abbass at
University of New South Wales, Yuchen Sun, Jiaying Liu,
Hao Ren at Dalian University of Technology, and anonymous
reviewers for their valuable comments and suggestions.
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Feng Xia (M’07SM’12) received the B.Sc. and
Ph.D. degrees from Zhejiang University, Hangzhou,
China. He is currently an Associate Professor and
Discipline Leader in School of Engineering, IT and
Physical Sciences, Federation University Australia.
Dr. Xia has published 2 books and over 300 scientific
papers in international journals and conferences. His
research interests include data science, computa-
tional intelligence, social computing, and systems
engineering. He is a Senior Member of IEEE and
ACM.
Ke Sun received the B.Sc. and M.Sc. degrees from
Shandong Normal University, Jinan, China. He is
currently Ph.D. Candidate in Software Engineering
at Dalian University of Technology, Dalian, China.
His research interests include deep learning, network
representation learning, and knowledge graph.
Shuo Yu (M’20) received the B.Sc. and M.Sc.
degrees from Shenyang University of Technology,
China, and the Ph.D. degree from Dalian University
of Technology, Dalian, China. She is currently a
Post-Doctoral Research Fellow with the School of
Software, Dalian University of Technology. She has
published over 30 papers in ACM/IEEE conferences,
journals, and magazines. Her research interests in-
clude network science, data science, and computa-
tional social science.
Abdul Aziz received the Bachelor’s degree in com-
puter science from COMSATS Institute of Informa-
tion Technology, Lahore Pakistan in 2013 and Mas-
ter degree in Computer science from National Uni-
versity of Computer & Emerging Sciences Karachi
in 2018. He is currently a PhD student at the
Alpha Lab, Dalian University of Technology, China.
His research interests include big data, information
retrieval, graph learning, and social computing.
Liangtian Wan (M’15) received the B.S. degree
and the Ph.D. degree from Harbin Engineering
University, Harbin, China, in 2011 and 2015, re-
spectively. From Oct. 2015 to Apr. 2017, he has
been a Research Fellow at Nanyang Technological
University, Singapore. He is currently an Associate
Professor of School of Software, Dalian University
of Technology, China. He is the author of over 70
papers. His current research interests include data
science, big data and graph learning.
Shirui Pan received a Ph.D. in computer science
from the University of Technology Sydney (UTS),
Australia. He is currently a lecturer with the Fac-
ulty of Information Technology, Monash University,
Australia. His research interests include data mining
and machine learning. Dr Pan has published over 60
research papers in top-tier journals and conferences.
Huan Liu (F’12) received the B.Eng. degree in
computer science and electrical engineering from
Shanghai Jiaotong University and the Ph.D. degree
in computer science from the University of Southern
California. He is currently a Professor of computer
science and engineering at Arizona State Univer-
sity. His research interests include data mining,
machine learning, social computing, and artificial
intelligence, investigating problems that arise in
many real-world applications with high-dimensional
data of disparate forms. His well-cited publications
include books, book chapters, and encyclopedia entries and conference, and
journal papers. He is a Fellow of IEEE, ACM, AAAI, and AAAS.