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+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.
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