July 2020 xx xx xx.xxxx/xxxxxxxxx April, 2020 May, 2020 11affiliationtext:
Department of Statistics, North Carolina State University, Raleigh, United
States 22affiliationtext: School of Computer Science, Fudan University,
Shanghai, China

# A Review on Graph Neural Network Methods in Financial Applications

Jianian Wang  Sheng Zhang  Yanghua Xiao Corresponding author. Email:
<EMAIL_ADDRESS>Rui Song 111Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract

With multiple components and relations, financial data are often presented as
graph data, since it could represent both the individual features and the
complicated relations. Due to the complexity and volatility of the financial
market, the graph constructed on the financial data is often heterogeneous or
time-varying, which imposes challenges on modeling technology. Among the graph
modeling technologies, graph neural network (GNN) models are able to handle
the complex graph structure and achieve great performance and thus could be
used to solve financial tasks. In this work, we provide a comprehensive review
of GNN models in recent financial context. We first categorize the commonly-
used financial graphs and summarize the feature processing step for each node.
Then we summarize the GNN methodology for each graph type, application in each
area, and propose some potential research areas.

###### keywords:

Deep learning; Finance; Graph convolutional network; Graph representation
learning.

##  1 Introduction

As the data collection techniques grow, graph data are commonly collected in
many areas including social sciences, transportation systems, chemistry, and
physics (Wu et al., 2020a). Representing complex relational data, graph data
contain both the individual node information and the structural information.
Recently, there are growing interests in developing machine-learning methods
to model the graph data of various domains. Among them, graph neural network
(GNN) methods could achieve great performance on various tasks, including node
classification, edge prediction, and graph classification (Kipf and Welling,
2017; Zhang and Chen, 2018; Xu et al., 2018). Performing node aggregation and
updates, graph neural network models extend the deep learning methodology to
graphs and are gaining popularity.

A financial system is a complex system with many components and sophisticated
relations, which may be frequently updated. To represent the relational data
in the financial domain, graphs are commonly constructed, including the
transaction network (Weber et al., 2019), user-item review graph (Dou et al.,
2020), and stock relation graph (Feng et al., 2019). By converting the
financial task into a node classification task, GNN methods are commonly
utilized since it performs well among graph modeling methods (Liu et al.,
2019). For instance, GNN could be utilized in a stock prediction task, by
formulating it as a node classification task, where each node represents a
stock and edges represent relations between companies. Figure 1 demonstrates
the workflow of a stock prediction task using GNN methods. However, the
complex nature of financial systems may result in multiple data sources and
complicated graph structures, which imposes challenges on feature processing,
graph construction, and graph neural network modeling. Represented as
numerical sequences or textual information, financial data need to be
processed with caution to keep the temporal pattern or semantic meanings.
Also, the multi-facet nature of financial relations make it hard to construct
a graph to capture the relations. Moreover, the financial-related graph is
often heterogeneous or time-varying, which impose challenges on existing graph
neural network models. What’s more, to reflect some financial patterns (e.g.
device aggregation pattern, see Section 5.4 for details), GNN methods may need
to be modified such as changing losses and adding additional layers. Since
financial systems process unique characteristics and receive great attention,
it is of significant importance to discuss and summary the GNN methodology
developed for financial tasks.

Figure 1: Workflow for stock movement prediction task using GNN methodology.
The graph construction and feature processing steps present stock information
in a graph and a feature matrix, which is then used as the input for the GNN
model.  In the graph, nodes are connected if there exist some relationships
between stocks, such as supplier, competitor, shake-holder, etc. A multi-layer
perception layer (MLP) is used to output the price prediction result.

There are several recent reviews on graph neural networks. Among them, Wu et
al. (2020a) present a comprehensive review on graph neural networks and
categorize the GNNs into four categories: recurrent graph neural networks,
convolutional graph neural networks, graph auto-encoders, and spatial-temporal
graph neural networks. Zhou et al. (2020) provide a taxonomy on GNN models
based on graph type, training methods, and propagation steps. There is also
literature focusing on limited types of GNNs. Zhang et al. (2019) focus on
graph convolutional networks (GCN) and introduce two taxonomies to group the
existing GCNs. Lee et al. (2019) survey the literature on graph attention
models and provided detailed examples on each type of method. However, the
aforementioned reviews focus on the general methodology and provide little
details for applications, seldom mentioning the financial application. Without
covering GNN models developed based on financial contexts, the reviewed models
may not be applicable to financial tasks due to the complexity of financial
data. On the other side, review papers focusing on the financial domain
haven’t covered GNN methodologies in detail yet. Ozbayoglu et al. (2020)
summarize the machine learning and deep learning models in the financial
field, without mentioning the GNN methodologies. Huang et al. (2020) survey
the financial deep learning models in the finance and bank industry, and the
GNN models are not covered. Jiang (2021) review stock prediction-related
machine-learning mythologies and mention GNN models very briefly. In summary,
existing GNN surveys focus on modeling methodology and do not emphasize the
financial application of GNN methods, while surveys on financial applications
don’t cover the GNN models in detail. To fill the gap, in this survey, we
provide a systematic and comprehensive review of graph neural network methods
in the financial application.

In this paper, we present a thorough survey on graph neural network models
with financial application. We provide a comprehensive review of graph neural
networks and summarize the corresponding methods. This survey has
contributions as follows.

  * •

We systemically categorize the commonly-used financial graphs based on graph
characteristics and provide a thorough list of graphs. Graphs are categorized
into five groups: homogeneous graph, directed graph, bipartite graph, multi-
relation graph, and dynamic graph. We also present the GNN models according to
their graph types, so that this review could serve as a guide for implementing
GNNs on real-life datasets.

  * •

We provide a comprehensive list of financial applications that GNN methods are
applied. We categorized the applications into five categories: stock movement
prediction, loan default risk prediction, recommender system of e-commerce,
fraud detection, and event prediction.

  * •

We summarize various aspects of information for each application, including
features, graphs, GNN models, and available codes. A GitHub222Github link:
https://github.com/jackieD14/Graph-models-in-finance-application page is built
to document the collection of information. This work could be considered as a
resource to understand, implement and develop GNN models on multiple financial
tasks.

  * •

We identify five challenges and discuss the recent progress. We also suggest
future directions for these problems.

The rest of the paper is organized as follows. Section 2 classifies financial
graphs into different categories based on its characteristics. Section 3
summarizes the commonly-used feature processing techniques for each node in
the graph. Section 4 presents the GNN methodology used for each graph type.
Section 5 provides a collection of application areas. Section 6 proposes some
challenges that could be future directions of research.

##  2 Graph categorization

When preparing the data, how to construct the graph to represent the
structural information is essential and the type for the constructed graph
could determine the follow-up modeling methodology. In this section, we
present the categorization of the graph based on its construction methods and
graph types. Table LABEL:table_summary presents a comprehensive list of graphs
for financial tasks.

###  2.1 Graph-related definition

In this section, we provide some graph-related definitions for better
understanding of this article.

######  Definition 2.1 (Graph).

A graph $\mathcal{G}$ is defined by a pair:
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, where
$\mathcal{V}=\\{v_{1},...,v_{n}\\}$ is a set of $n$ nodes and $\mathcal{E}$ is
a set of edges, where $e_{ij}=(v_{i},v_{j})\in\mathcal{E}$ denotes an edge
joining node $v_{i}$ and node $v_{j}$.

######  Definition 2.2 (Adjacency matrix).

An adjacency matrix A is an $n\times n$ matrix, where $\textbf{A}_{ij}$
represents the connection status between node $v_{i}$ and node $v_{j}$. For an
unweighted graph, the adjacency matrix could be an binary matrix where
$\textbf{A}_{ij}=1$ if $e_{ij}\in\mathcal{E}$ and 0 otherwise.

######  Definition 2.3 (Undirected graph and Directed graph).

A undirected graph is a graph where the edges are undirected. A directed graph
is a graph where the edges have orientations.
$e_{ij}=(v_{i},v_{j})\in\mathcal{E}$ denotes an edge pointing from node
$v_{i}$ to node $v_{j}$.

######  Remark.

Undirected graph has a symmetric adjacency matrix, i.e.,
$\textbf{A}_{ij}=\textbf{A}_{ji}$.

######  Definition 2.4 (Bipartite graph).

A Bipartite graph is a graph whose nodes could be divided into two non-empty
and disjoint sets $\mathcal{U},\mathcal{W}$, such that every edge connects a
node in $\mathcal{U}$ and a node in $\mathcal{W}$.

######  Definition 2.5 (Homogeneous graph and Heterogeneous graph).

In a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, we can assign a type to
each node and edge; in this case, the graph is denoted as
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A},\mathcal{R})$, where each
node $v_{i}\in\mathcal{V}$ is associated with its type $a_{i}\in\mathcal{A}$,
and each edge $e_{ij}\in\mathcal{E}$ is associated with its type
$r_{ij}\in\mathcal{R}$. A homogeneous graph is a graph whose nodes are of the
same type and edges are of the same type. Otherwise, the graph is
heterogeneous.

######  Definition 2.6 (Multi-relation graph).

A Multi-relation graph is a graph where edges have different types.

######  Definition 2.7 (Dynamic graph).

A dynamic graph is defined as a sequence of graphs
$\mathcal{G}^{seq}=\\{\mathcal{G}_{1},...,\mathcal{G}_{T}\\}$, where
$\mathcal{G}_{i}=(\mathcal{V}_{i},\mathcal{E}_{i}),\text{for }i=1,...,T$,
where $\mathcal{V}_{i},\mathcal{E}_{i}$ are the set of nodes and edges for
$i^{th}$ graph in the sequence respectively.

###  2.2 Graph categorization by construction methods

In this section, we elaborate on frequently-used graph construction methods so
that researchers could better understand, choose and construct graph data.

####  2.2.1 Data-based construction

Some types of data could be naturally represented as a graph since they
contain relations among data objects. For instance, Wang et al. (2019)
construct a user-relationship graph, where users are linked by an edge if they
are labeled classmates, friends, or workmates in the data. Liu et al. (2018)
construct an account-device network, where nodes are either accounts or
devices. Edges connect an account node to a device node if the account has
activities on the device. This type of construction method is based on the
nature of data and could be used on data where relations are clearly defined.

####  2.2.2 Knowledge-based construction

Sometimes, data may not contain relational information, but relations could be
found in knowledge bases. A knowledge base is a collection of descriptive data
and contains numerous entities and their relations. For example, Wikidata
(Vrandečić and Krötzsch, 2014) is one of the largest open-domain knowledge
bases which provides support for Wikipedia, an online encyclopedia. A graph
could be built utilizing the relations extracted from knowledge bases. In
order to predict stock movement, Feng et al. (2019) extract company relations
from Wikidata, such as supplier, provider, partner, etc, and constructed a
company-based relation network. This type of graph-construction method brings
new information into the graph by utilizing the knowledge bases, which may
improve modeling performance. However, it may take some effort to process the
complicated data structure and the massive amount of information of knowledge
bases.

####  2.2.3 Similarity-based construction

There also exist cases that neither the data nor knowledge bases contain
relations, but there may be some hidden relationships in the data. To mine the
underlying relationship, a commonly-used approach is to calculate a similarity
measure of the features for different observations and construct relations if
the similarity value is greater than a threshold. For instance, Li et al.
(2020a) construct a stock correlation graph based on the cosine similarity of
stocks’ historic market price. Two stocks are then connected if the absolute
value of their correlation is larger than a threshold. This type of
construction method could be easily implemented and understood. However, in
this type of construction, feature information is represented both in the
graph adjacency matrix and the feature matrix. The overlapping information may
lead to doubts that whether the graph representation is still necessary. Also,
how to set the threshold is an issue, and justification may be needed for the
selected similarity threshold value.

###  2.3 Graph categorization by graph types

In this section, we categorize graphs into five categories based on their
characteristics and provide examples in the financial context. Since different
types of graphs may impose various challenges on modeling technology, we
discuss GNN methods for each graph type in section 4 to provide solutions
respectively. Figure 2 provides visualization for each graph type: homogeneous
graph, directed graph, bipartite graph, multi-relation graph, and dynamic
graph. It is also worth mentioning that a graph may be categorized into
multiple types.

Figure 2: Graph categorization based on graph characteristics. Each color of
the circle represents a node type and each color of the line represents an
edge type. Arrows represent directed edges. A homogeneous graph is a graph
with one type of node and one type of edge. A directed graph is a graph with
directed edges. A bipartite graph is a graph with two types of nodes and edges
only exist between nodes of different types. A multi-relation graph has edges
with different types. A dynamic graph is a sequence of graphs.

####  2.3.1 Homogeneous graph

A homogeneous graph is a graph with one type of node and one type of edge. For
instance, Liou et al. (2021) construct a financial news co-occurrence graph,
where two companies are connected if they are tagged in the same news
articles. Li et al. (2020b) build a transaction network, where nodes are
accounts and are connected when there exist transactions between them. This
type of graph has a relatively simple structure and the majority of the GNN
methods could be applied to model this type of graph.

####  2.3.2 Directed graph

A directed graph is a graph where edges have orientations. For instance, Cheng
et al. (2019) construct a guarantee network where nodes are the companies and
edges represent the guarantee relationship. Since the guarantor has the
obligation to pay the debt for the borrower, but not the other way round, this
type of guarantee relationship is one-sided and could be represented in a
directed edge. In general, a directed graph may have an asymmetric adjacency
matrix and thus cannot be semi-definite. Since some GNN mythologies are
developed for semi-definite adjacency matrices, they may not be suitable for
the directed graph.

####  2.3.3 Bipartite graph

A bipartite graph is a graph with two types of nodes and edges only exist
between nodes of different types. For instance, Liu et al. (2018) construct an
account-device network in a fraud detection task. Nodes could be either
accounts or devices, with edges connecting them if the account has activities
on the device. Li et al. (2019a) extract a user-item network using the rating
data, where nodes could be either a user or an item. An edge exists if a user
has rated the item. A bipartite graph is commonly used, when data can be
divided into two groups and the interaction between two groups matters. A
bipartite graph could be seen as a specific case of a multi-relation graph
discussed in the next session and GNN methods developed for the multi-relation
graph are applicable on a bipartite graph as well.

####  2.3.4 Multi-relation graph

Sometimes, edges may have multiple types to represent different relations
between nodes. For instance, Wang et al. (2019) construct a user-relation
graph, where nodes are the users of an e-commerce platform. There are multiple
edge types representing various relationships including friendship, workmates
and classmates. Dou build a review graph where users are represented as nodes
in the graph. Three types of relations between users are defined to capture
their behavioral patterns: reviewing the same product, having the same star
rating, and having similar texts. With multiple edge types, this type of graph
contains more information about the relationship between nodes, and thus how
to capture this information is critical when developing the GNN methodology.

####  2.3.5 Dynamic graph

A dynamic graph is a sequence of graphs , where each graph could have an
adjacency matrix and feature matrix. For example, Cheng_IJCAI construct a
temporal guarantee network, representing the guarantee relationship in each
time step. This type of graph is commonly used to represent the changes in
both relations and features, as time goes. Since both nodes and edges could
appear and disappear, it is hard to perform some graph operations that require
fixed dimensions of matrices. Thus, capturing the dynamically of the graph is
challenging and requires a more sophisticated methodology.

Table 1: Summary of financial-related graphs Graph |  Construction method 333Knowledge, Similarity, Data denotes knowledge-based, similarity-based and data-based construction respectively. |  Graph type 444Multi denotes multi-relation graph and Homo denotes homogeneous graph. |  Application 555Stock denotes stock movement prediction. Loan denotes loan default risk prediction. E-comm denotes recommender system of e-commerce. Fraud denotes fraud detection.666 Details for financial applications could be found in Section 5. |  Reference
---|---|---|---|---
Sector-industry stock relation network |  Knowledge |  Multi |  Stock |  Feng
Wiki company-based relation network |  Knowledge |  Multi |  Stock |  Feng; Sawhney; Ying
Supplier, customer, partner and shareholder relation graph |  Knowledge |  Multi |  Stock |  Matsunaga
Corporation shareholder network |  Knowledge |  Homo |  Stock |  Chen
Stock correlation graph |  Similarity |  Multi |  Stock |  Li
Stock earning call graph |  Knowledge |  Bipartite |  Stock |  Sawhney_audio
News co-occurrence graph |  Similarity |  Homo |  Stock |  Liou
User-relation graph |  Data |  Homo |  Loan |  Wang
User-app graph |  Data |  Bipartite |  Loan |  Wang
User-nickname graph |  Data |  Bipartite |  Loan |  Wang
User-address graph |  Data |  Bipartite |  Loan |  Wang
Guarantee network |  Data |  Directed |  Loan |  Cheng_network_loan; Cheng_KDD
Temporal guarantee network |  Data |  Dynamic |  Loan |  Cheng_IJCAI
Temporal small business entrepreneur network |  Data |  Dynamic |  Loan |  Yang
Alipay user and applet graph |  Data |  Dynamic |  Loan |  Hu_Loan
User relationship graph |  Data |  Multi |  Loan |  Liang_WSDM
Auto relation network |  Data |  Multi |  Loan |  Xu
Loan application event graph |  Data |  Directed |  Loan |  Harl
Borrower relations’ network |  Similarity |  Directed |  Loan |  Lee
Xianyu comment graph |  Similarity |  Homo |  E-comm |  Li_spam
Yelp review network |  Data |  Bipartite |  E-comm |  Zhang_Fraud; Dou
Amazon review network |  Data |  Bipartite |  E-comm |  Zhang_Fraud; Dou; Kudo
E-commerce user-item network |  Data |  Bipartite |  E-comm |  Li_hierarchical
Taobao user-item network |  Data |  Bipartite |  E-comm |  Li_bipartite
Device sharing graph |  Data |  Bipartite |  Fraud |  Liang
JD Finance anti-fraud graph |  Data |  Multi |  Fraud |  Lv
Transaction records graph |  Data |  Multi |  Fraud |  Rao1
Iqiyi user network |  Data |  Multi |  Fraud |  Zhu
CMU simulated user activity network |  Similarity |  Homo |  Fraud |  Jiang_anomaly
Alipay one-month account-device network |  Data |  Bipartite |  Fraud |  Liu_18
Alipay one-week account-device network |  Data |  Bipartite |  Fraud |  Liu_19
Account-registration graph |  Data |  Dynamic |  Fraud |  Rao
Bitcoin-alpha graph |  Data |  Directed |  Fraud |  Zhao_loss

##  3 Feature processing

With diverse data sources in the financial field, node features are commonly
formatted as sequential numerical features or textual information. These data
formats impose challenges on the feature processing step since GNN methods
could not be directly applied to these data formats. In this section, we
summarize the commonly-used feature processing techniques and how they solve
these challenges.

(a) Sequential numerical features

(b) Textual information

Figure 3: Feature processing for sequential features and textual information.
For sequential numerical features, recurrent neural network (RNN) based
approaches are commonly used to capture the temporal dependencies. For text
features, it is often processed utilizing natural language processing (NLP)
methods including word embedding, sentence embedding, and language models, to
convert the unstructured data to structured ones.

###  3.1 Sequential numerical data

Updating information as time goes, the financial industry has a rich source of
time-series data. Indexed using timestamps, features could be seen as
sequential which requires appropriate modeling. Consider the feature matrix at
time $s$, $X^{s}\in\mathbb{R}^{n\times p\times l}$, where $n$ is the number of
nodes, $p$ is the dimension of features at each time point, $l$ is the length
of the sequence. For example, in a stock prediction task, we have a stock
relation graph, with $n$ stocks as nodes. The feature matrix $X^{s}$
represents the $p$ features in the past $l$ days from time $s$ for these $n$
stocks. To encode the numerical sequence for each node, a recurrent neural
network (RNN) is frequently used due to its superior performance in predicting
time-series data. The literature could be summarized into two lines of work,
long short-term memory (LSTM) based approach and gated recurrent unit (GRU)
based approach.

####  3.1.1 LSTM based approach

As a special form of recurrent neural network, long short-term memory(LSTM)
(LSTM) is capable to capture the long-term dependencies and avoid the
vanishing gradient problem. Using memory cells and gate units, it has the
following expression:

$\displaystyle f_{t}$ $\displaystyle=\sigma(W_{f}x_{t}+U_{f}h_{t-1}+b_{f}),$
$\displaystyle i_{t}$ $\displaystyle=\sigma(W_{i}x_{t}+U_{i}h_{t-1}+b_{i}),$
$\displaystyle o_{t}$ $\displaystyle=\sigma(W_{o}x_{t}+U_{o}h_{t-1}+b_{o}),$
$\displaystyle\tilde{c}_{t}$
$\displaystyle=\text{tanh}(W_{c}x_{t}+U_{c}h_{t-1}+b_{c}),$ $\displaystyle
c_{t}$ $\displaystyle=f_{c}\circ c_{t-1}+i_{t}\circ\tilde{c}_{t-1},$
$\displaystyle h_{t}$ $\displaystyle=o_{t}\circ\text{tanh}(c_{t}),$

where $x_{t}\in\mathbb{R}^{D}$ is the input vector at time $t$ and $D$ is the
number of features, $f_{t},i_{t},o_{t},\tilde{c}_{t},c_{t},h_{t}$ denotes the
forget gate, output gate, cell input, cell state and hidden state vectors
respectively, $W_{f},W_{i}$, $W_{o},W_{c}$ $,U_{f},U_{i},U_{o},U_{c}$ are
trainable weight matrices and $b_{f},b_{i},b_{o},b_{c}$ are trainable bias
vectors, $\sigma(\cdot)$represents the sigmoid activation function, and
$\circ$ denotes the element-wise product. The hidden state of the LSTM on day
$t$ is denoted by: $h_{t}=LSTM(x_{t},h_{t-1}),s-l\leq t\leq s$.

Since the LSTM updates the hidden state to capture the structural information,
a common approach to encode the historical data is generating sequential
embedding $E^{s}$ using the last hidden state of LSTM,
$E^{s}=\text{LSTM}(X^{s})\in\mathbb{R}^{n\times u}$, where $u$ is the
dimension of the output feature. Then the encoded pricing information is used
as input to the graph neural network. For instance, Chen used the generated
sequential embedding $E^{s}$ as the input feature matrix for graph
convolutional network and Feng utilized $E^{s}$ as the input feature matrix
for their proposed temporal graph convolutional layer. Using the last hidden
state as an input feature, this type of method could capture the information
in the past days while having an appropriate format to feed into the GNN
model.

####  3.1.2 GRU based approach

Gated recurrent unit (GRU) (GRU) is another variant of RNN models. It also
applies gating mechanism and has fewer parameters. It has the following
structure:

$\displaystyle r_{t}$ $\displaystyle=\sigma(W_{r}x_{t}+U_{r}h_{t-1}+b_{r}),$
$\displaystyle z_{t}$ $\displaystyle=\sigma(W_{z}x_{t}+U_{z}h_{t-1}+b_{z}),$
$\displaystyle\tilde{h}_{t}$
$\displaystyle=\text{tanh}(W_{h}x_{t}+U_{h}(r_{t}\circ h_{t-1})+b_{h}),$
$\displaystyle h_{t}$ $\displaystyle=(1-z_{t})\circ
h_{t-1}+z_{t}\circ\tilde{h}_{t},$

where $x_{t}\in\mathbb{R}^{D}$ is the input vector at time $t$ for stock $i$
and $D$ is the number of features, $r_{t},z_{t},\tilde{h}_{t},h_{t}$ denotes
the reset gate, update gate, candidate activation and hidden state vectors
respectively, $W_{r},W_{z},W_{h}$ $,U_{r},U_{z},U_{h}$ are trainable weight
matrices and $b_{r},b_{z},b_{h}$ are trainable bias vectors, $\sigma(\cdot)$
represents the sigmoid activation function, and $\circ$ denotes the element-
wise product. The hidden states of the GRU on day $t$ is denoted by:
$h_{t}=\text{GRU}(x_{t},h_{t-1})$.

Utilizing GRU to encode the past numerical information, we could obtain the
hidden state for each day. Since past days’ impact on the current-day
representation may differ, an attention mechanism is frequently used to assign
weights differently. For instance, Sawhney use an additive attention mechanism
to aggregate the hidden states across time. Cheng_KDD utilize the concatenated
attention method to incorporate different importance of time. The attention
mechanism aggregates the hidden states of past days and assigns different
weights across time. To get the feature representation at time $s$, it rewards
the influential days when aggregating the hidden states from time $s-l$ to
time $s$, and thus take temporal dependencies into account. The obtained node
representation is then used as a feature in the graph neural network model.

###  3.2 Textual information

In the financial industry, a large amount of information is of textual form,
including financial news, financial statements, and customer reviews. How to
translate the texts into vector representation while preserving semantic
information, is essential. The following section summarizes the commonly-used
natural language processing (NLP) methodology to convert the unstructured
texts into a vector form.

####  3.2.1 Word embedding and sentence embedding

Word embedding methods are widely used to represent a word as a fixed-length
vector. Then, to learn the sentence representation, a recurrent neural network
model is often utilized to capture local semantic information. For instance,
to embed the news headlines, Li encode the word as word embedding using GloVe
(Glove). Then LSTM method is applied with an attention mechanism to create the
sentence representation. However, since a word may have multiple meanings,
word embedding methods may cause problems by assigning the same vector to
words with different meanings. Thus, there are also approaches to embed at a
sentence level in order to alleviate the problem. For instance, Sawhney
generate sentence-level embedding for each Tweet using Universal sentence
encoders (USE).

####  3.2.2 Language model

Instead of focusing on generating vectors for words, language models focus on
capturing the pattern of languages to predict the word based on its
surrounding words. Taking the contexts into account, language models achieve
the state of art performance on many NLP tasks and are widely used in the
literature. For instance, Liou use bidirectional encoder representations from
transformers (BERT) model (BERT) to encode the entire news article and
generated a news embedding. Without modifying the model architecture, the pre-
trained BERT model is able to be fine-tuned and produce state-of-art
performance. For example, FinBERT (Finbert), a BERT model pre-trained specific
to the financial domain, is utilized to encode the text scripts in company
earning calls (Sawhney_audio).

##  4 Graph neural network models

Proposed by Gori, a graph neural network is a neural network model capable of
processing graphs. Unlike network embedding methods whose major aim is to
generate a vector to represent each node, graph neural network models are
designed for a variety of tasks, including node classification, edge
prediction, and graph classification. Due to its wide application and superior
performance, graph neural network models have drawn great attention. In this
section, we present the commonly-used GNN models for each type of graphs,
since the methodology may vary with different graph characteristics. In the
supplementary materials, we present a figure demonstrating the major GNN
methodology used for each graph type.

###  4.1 Homogeneous graph

Proposed by GCN, graph convolutional network (GCN) is a widely-used graph
neural network model and could encode both local graph structure and node
features. Extending the convolution concepts to graphs, graph convolution
could be seen as message passing and information propagation. Aggregating
neighbors’ feature information, graph convolutional networks could be
represented with the following layer-wise propagation rule:

$\displaystyle
H^{l+1}=\sigma(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{l}W^{l}),$

where $\tilde{A}$ is the adjacency matrix with added self-connections,
$\tilde{D}=\text{diag}(\sum_{j}\tilde{A}_{ij})$ $W^{l}$ is trainable weight
matrix of $l^{th}$ layer, $H^{l}$ is the node hidden feature matrix in the
$l^{th}$ layer and $\sigma(\cdot)$ is the activation function. With its
relatively simple model structure and great performance, the GCN model is
often used as a benchmark method to compare with. Among the reviewed
literature, over half of them have applied the GCN method as a benchmark
method.

While GCN equally treats the neighbors of the target node, it often occurs
that some neighbors may be more influential than others. Considering various
impacts of the neighbor nodes, GAT propose graph attention networks (GAT) and
it is able to assign different weights to nodes in the same neighborhood as
follows:

$\displaystyle h_{i}^{l+1}$
$\displaystyle=\sigma(\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}^{l}W^{l}h_{j}^{l}),$
$\displaystyle\alpha_{ij}^{l}$
$\displaystyle=\frac{\exp(\text{LReLU}(a^{T}[W^{l}h_{i}^{l}\|W^{l}h_{j}^{l}]]))}{\sum_{k\in\mathcal{N}_{i}}\exp(\text{LReLU}(a^{T}[W^{l}h_{i}^{l}\|W^{l}h_{k}^{l}]},$

where $h^{l}$ is the hidden feature vector for node $i$ in the $l^{th}$ layer,
$W^{l}$ is trainable weight matrix, $a$ is a learnable vector,
$\mathcal{N}_{i}$ is the neighborhoods of node $i$, $\alpha_{ij}^{l}$
represents attention coefficient of node $j$ to $i$ at $l^{th}$ layer,
$\sigma(\cdot)$ is the activation function, $\|$ denotes vector concatenation,
and LReLU denotes the leaky ReLU activation function. It is also worth
mentioning that, since GAT is able to learn the weights of the neighboring
node, we could interpret the learned attention weights as a relative
importance measure, to better understand the model. Similar to GCN, GAT is
also often used as a benchmark method in the reviewed papers with about 40%
coverage.

###  4.2 Directed graph

A undirected graph has a symmetric adjacency matrix that guarantees a semi-
definite Laplacian matrix, which lays the foundation for applying GCN. The
directed graph, on the other hand, may has asymmetric adjacency matrix and
could be handled by spatial-based GNN methods (Wu_review), such as GAT. In
practice, there is not much work developing methodologies for directed graph,
since it could be processed by spatial-based GNN methods or making the
adjacency matrix symmetric. However, there is also a line of work developing
GNN methodology to predict sequences of events represented as a directed
graph.

A sequence of events could be naturally formed as a directed graph, where each
node is one type of event and an edge points from one event to the following
one. Given the graph, which could be seen as a partial sequence of events, how
to encode the features and predict the rest of the sequence is a challenge.
The aforementioned GCN and GAT models aim at representation learning and are
used to produce a single output instead of outputting a sequence. To fill the
gap, GGNN propose a gated graph neural network (GGNN) that could produce
sequential outputs. It applies gated recurrent unit (GRU) as a recurrent
function and is constructed as follows:

$\displaystyle a_{t}$ $\displaystyle=A^{T}h_{t-1}+b,$ $\displaystyle r_{t}$
$\displaystyle=\sigma(W_{r}a_{t}+U_{r}h_{t-1}),$ $\displaystyle z_{t}$
$\displaystyle=\sigma(W_{z}a_{t}+U_{z}h_{t-1}),$ $\displaystyle\tilde{h}_{t}$
$\displaystyle=\text{tanh}(W_{h}a_{t}+U_{h}(r_{t}\circ h_{t-1})),$
$\displaystyle h_{t}$ $\displaystyle=(1-z_{t})\circ
h_{t-1}+z_{t}\circ\tilde{h}_{t},$

where $h_{t}$ is the updated event representation at $t^{th}$ step, $a_{t}$
contains information transferred from both directions’ edges,
$r_{t},z_{t},\tilde{h}_{t}$ denotes the reset gate, update gate, and candidate
activation vectors at $t^{th}$ respectively, $W_{r},W_{z},W_{h}$
$,U_{r},U_{z},U_{h}$ are trainable weight matrices, $b$ is a trainable bias
vector, $\sigma(\cdot)$ is the sigmoid function, $\circ$ denotes the element-
wise product, and $tanh$ denotes the hyperbolic tangent function.
Incorporating the adjacency matrix $A$, GGNN aggregates the structural
information in every propagation step. Unrolling the recurrence function to a
fixed number, the GGNN ensures convergence without constraining the
parameters.

###  4.3 Bipartite graph

The aforementioned methods focus on homogeneous graphs whose nodes and edges
are all of one type. In real-life applications, graphs could be heterogeneous.
As a running example, in a spam detection task, we could have a user-item
network, where nodes are either users or items. An edge $e_{ij}$ denotes that
user $i$ has rated on item $j$. This type of graph is well known as a
bipartite graph $\mathcal{G}$ with the following notation:
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\\{U,W\\}$ is a
set of nodes and could be divided into two non-empty and disjoint sets
$\mathcal{U,W}$. $\mathcal{E}$ is a set of edges, where every edge joins a
node in $\mathcal{U}$ and a node in $\mathcal{W}$.

Based on the characteristics of a bipartite graph, the commonly-used
methodologies could be categorized into a framework as follows. In each
iteration, the edge information $h_{uw}^{l}$ is updated aggregating its
previous hidden state $h_{uw}^{l-1}$ and hidden states of the two nodes it
links ($h_{u}^{l-1}$, $h_{w}^{l-1}$), as shown in equation (1). At each
iteration, a user node $u$ aggregates information from its rated items
$h_{\mathcal{N}(u)}^{l}$ and its past hidden state $h_{u}^{l-1}$, as in
equation (3). While the representation of the rated items
$h_{\mathcal{N}(u)}^{l}$ is updated aggregating the information from the
linking edges $h_{uw}^{l}$ and the item nodes $h_{w}^{l-1}$, as in equation
(2). Then, the item node $w$ is updated respectively as shown in the following
equations:

$\displaystyle h_{uw}^{l}$
$\displaystyle=\sigma(W_{E}^{l}\cdot\text{AGG}_{E}(h_{uw}^{l-1},h_{u}^{l-1},h_{w}^{l-1})),$
(1) $\displaystyle h_{\mathcal{N}(u)}^{l}$
$\displaystyle=\sigma(W_{\mathcal{N}(U)}^{l}\cdot\text{AGG}_{U}(\text{AGG}_{UW}(h_{w}^{l-1},h_{uw}^{l}))),\qquad$
$\displaystyle\forall w\in\\{\mathcal{N}(u)\\},$ (2) $\displaystyle h_{u}^{l}$
$\displaystyle=\text{concat}(W_{U}^{l}\cdot
h_{u}^{l-1},h_{\mathcal{N}(u)}^{l}),$ (3) $\displaystyle h_{N(w)}^{l}$
$\displaystyle=\sigma(W_{\mathcal{N}(W)}^{l}\cdot\text{AGG}_{W}(\text{AGG}_{WU}(h_{u}^{l-1},h_{uw}^{l}))),\qquad$
$\displaystyle\forall u\in\\{\mathcal{N}(w)\\},$ (4) $\displaystyle h_{w}^{l}$
$\displaystyle=\text{concat}(W_{W}^{l-1}\cdot h_{w}^{l},h_{N(w)}^{l}),$ (5)

where $e_{uw}$ represents the edge representation for edge linking node $u$
and $w$,
$h_{uw}^{l},h_{u}^{l},h_{w}^{l},h_{\mathcal{N}(u)}^{l},h_{\mathcal{N}(w)}^{l}$
are hidden states at $l^{th}$ layer,
$W_{E}^{l},W_{U}^{l},W_{\mathcal{N}(U)}^{l},W_{N(W)}^{l},W_{W}^{l}$ are
trainable weight matrices at $l^{th}$ layer, $\text{AGG}_{E}(\cdot)$,
$\text{AGG}_{U}(\cdot),\text{AGG}_{UW}(\cdot)$,
$\text{AGG}_{W}(\cdot),\text{AGG}_{WU}(\cdot)$ are user-chosen aggregation
functions, $\sigma(\cdot)$ is the activation function, and concat denotes the
vector concatenation.

There exist many modeling methodologies for bipartite graphs that could fit
into the above framework. For instance, Zhang_Fraud propose a similar model
structure as the framework and apply the attention mechanism as the
aggregation function, since different items may have different impacts when
learning uses’ representations. Instead of using all neighbors, Li_spam
utilize a sampling technique when aggregating neighbors’ information in each
iteration. There are also literature using the above framework as a building
block and combining clustering methodology to learn a hierarchical
representation of the graph, since hierarchical representation with various
GNN models could achieve satisfactory performance. For instance,
Li_hierarchical utilize the node embedding generated from the framework to
cluster users into different communities and make a recommendation based on
both community information and user information. Specifically, the user
information is decomposed into two orthogonal spaces representing community-
level information and individualized user preferences. Li_bipartite treat the
framework as a GNN module and stack it in a hierarchical fashion. With the
embedding generated from the framework, clustering algorithms are performed to
generate a coarsened graph which is used as an input for the next GNN layer.

###  4.4 Multi-relation graph

Instead of a simple homogeneous graph where all nodes and edges have the same
type, in real life, there may exist multiple relations between nodes. For
example, in a malicious account detection task, we could construct an Amazon
review network. Users are the nodes in the graph and there are three relations
between users: reviewing the same product, having the same star rating, and
having similar texts. The multi-relation graph is denoted as $\mathcal{G}$ :
$\mathcal{G}=(\mathcal{V},\mathcal{E}_{1:R})$, where $\mathcal{V}$ is the set
of nodes, $\mathcal{E}_{1:R}$ is the set of edges, $R$ is the number of node
types, $e_{i,j}^{r}\in\mathcal{E}_{r}$ is an edge between node $i,j$ with a
relation $r\in\\{1,...,R\\}$.

A frequently used approach is to transform the heterogeneous graph into
multiple homogeneous graphs by extracting $R$ subgraphs
{$\mathcal{G}^{r}=(\mathcal{V},\mathcal{E}_{r}),r=1,...,R$} from it. Each
subgraph $\mathcal{G}^{r}$only preserves one type of edge and thus is
homogeneous. This line of work could be unified in a two-step framework. The
first step implements the sub-graph aggregation to aggregate neighbor
information in each sub-graph as shown in equation (6). The second step is to
conduct the inter-relation aggregation to aggregate relation-specific
embeddings as equation (7),

$\displaystyle h^{l}_{i,r}$
$\displaystyle=f(\text{AGG}_{r}\\{h^{l-1}_{j,r}\\}),\qquad$
$\displaystyle\forall j\text{ s.t.}(i,j)\in\mathcal{E}_{r},$ (6)
$\displaystyle h^{l}_{i}$
$\displaystyle=g(\text{AGG}\\{h^{l}_{i,(1:R)},h^{l-1}_{i}\\}),$ (7)

where $h^{l}_{i,r}$ is the subgraph-specific embedding of node $i$ in subgraph
$r$ in $l^{th}$ layer, $h^{l}_{i}$ is the general embedding of node $i$ in
$l^{th}$ layer, $\text{AGG}_{r}(\cdot)$ is the aggregation function in
subgraph $r$, $\text{AGG}(\cdot)$ is the inter-relation aggregation function,
and $f(\cdot),g(\cdot)$ are user-defined functions.

There are multiple methods that could be categorized into the above two-step
framework. For instance, in a fraud classification task, Liu_18 observe that
fraudsters tend to congregate in topology and thus use weighted sum for
within-relation aggregation to capture this congregation pattern. They then
apply an attention mechanism for inter-relation aggregation to learn the
significance for each sub-graph as follows:

$\displaystyle h^{l}_{i,r}$ $\displaystyle=\sigma(\text{Weighted
mean}\\{h^{l-1}_{j,r}\\}),\qquad$ $\displaystyle\forall j\text{
s.t.}(i,j)\in\mathcal{E}_{r},$ $\displaystyle h^{l}_{i}$
$\displaystyle=\sigma(X_{i}W+\text{Attention}\\{h^{l}_{i,(1:R)}\\}),$

where $X_{i}$ is the feature vector for node $i$, $W$ is a trainable matrix,
$\sigma(\cdot)$is the activation function, and Attention denotes the attention
aggregator.

To incorporate neighbors’ information, Dou use the mean aggregator for within-
relation aggregation. To reduce the computational cost and keep the relational
importance information, they apply a pre-calculated parameter $p_{r}^{l}$ as
the weight in the intra-relation aggregation step. They also test several
aggregating functions when aggregating relation-specific embeddings with the
following structure:

$\displaystyle h^{l}_{i,r}$
$\displaystyle=\sigma(\text{Mean}\\{h^{l-1}_{j,r}\\}),\qquad$
$\displaystyle\forall j\text{ s.t.}(i,j)\in\mathcal{E}_{r},$ $\displaystyle
h^{l}_{i}$ $\displaystyle=\sigma(h^{l-1}_{i}+\text{AGG}\\{h^{l}_{i,r}\cdot
p_{r}^{l}\\}),\qquad$ $\displaystyle\forall r\in(1,...,R),$

where $p_{r}^{l}$ is a pre-trained weight.

Since different relations provide various facets of user characteristics,
relationship-specific embedding may have different statistical properties,
which may cause trouble when aggregating them in a lower-level space. To deal
with that, Wang project the relation-specific node embedding to higher spaces
using multi-layer perception (MLP) and concatenate them with relation-level
attention:

$\displaystyle h_{i,r}^{l}$
$\displaystyle=\text{MLP}\\{h_{i,r}^{l-1}\\},\text{where
}h_{i,r}^{1}=\text{Attention}\\{x_{j,r}:\forall j\text{
s.t.}(i,j)\in\mathcal{E}_{r}\\},$ $\displaystyle h^{l}_{i}$
$\displaystyle=\text{Concatenation with attention}\\{h^{l}_{i,(1:R)}\\},$

where Attention denotes the attention aggregator.

###  4.5 Dynamic graph

The previously mentioned neural network models generally focus on a static
graph. However, in real-life settings, a graph may be dynamically evolving
since relations may be updated with time. For example, in order to predict the
loan default risk, a guarantee network needs to be updated, adding newly-
constructed guarantee relationships and removing companies that have fully
paid the loan. With the rapid development of graph neural network
methodologies on static graphs, there emerges a trend to extend GNN models to
a dynamic setting. Consider a sequence of $T$ graphs
$\mathcal{G}^{seq}=\\{\mathcal{G}_{1},...,\mathcal{G}_{T}\\}$, where
$\mathcal{G}_{i}=(\mathcal{V}_{i},\mathcal{E}_{i})$ represent the graph at
$i^{th}$ time point. The feature matrices are represented as
$X=\\{\textbf{X}_{1},...,\textbf{X}_{T}\\}$ and the adjacency matrices are
$A=\\{\textbf{A}_{1},...,\textbf{A}_{T}\\}$.

To capture the sequential pattern in the dynamic graph, a common approach is
to train a GNN to generate the node embedding at each time stamp and then
utilize a recurrent neural network to aggregate the information. For example,
Cheng_IJCAI obtain the node embeddings at each time step by training a GCN
with multi-head attention. Then, they utilize the GRU to capture the
sequential pattern with a temporal attention layer to capture the temporal
variation over timestamps. Similarly, Yang first aggregate node and edge
information in each snapshot and then employ a LSTM operator to capture the
temporal variations in the node enbeddings.

In the aforementioned methods, a graph neural network is learned for feature
aggregation and an RNN model is trained to capture the sequential pattern of
the node embeddings. However, in reality, a node may appear and disappear,
which may worsen the performance of the RNN model when updating the node
representation. In a guarantee network, for example, a company that has
borrowed loans could disappear from the graph after it pays all the debts and
could appear again backing up other companies’ loans. To overcome the
limitation, EvolveGCN proposed EvolveGCN which utilizes a recurrent neural
network to evolve the GCN parameters instead of updating the node embeddings.
For each time point $t$, a GCN model is constructed as follows to fit the
graph $\mathcal{G}_{t}$:

$\displaystyle
H^{l+1}_{t}=\sigma(\tilde{D}_{t}^{-\frac{1}{2}}\tilde{A_{t}}\tilde{D}_{t}^{-\frac{1}{2}}H^{l}_{t}W^{l}_{t}),$

where $\tilde{A}_{t}$ is the adjacancy matrix with added self-connections,
$\tilde{D}_{t}=\text{diag}(\sum_{j}\tilde{A}_{ij})$ is the degree matrix,
$W^{l}_{t}$ is trainable weight matrix of $l^{th}$ layer, $H^{l}_{t}$ is the
matrix of activation in the $l^{th}$ layer, and $\sigma(\cdot)$ is the
activation function.

To update the weight matrix $W^{l}_{t}$, EvolveGCN propose two methods. The
first method considers $W^{l}_{t}$ as a hidden state of the dynamics and
update it using a GRU model, as shown in equation (8). The second method
treats $W^{l}_{t}$ as an output state which is updated using a LSTM method, as
shown in equation (9). The structure for both methods is as follows:

$\displaystyle W^{l}_{t}$ $\displaystyle=\text{GRU}(H_{t}^{l},W^{l}_{t-1}),$
(8) $\displaystyle W^{l}_{t}$ $\displaystyle=\text{LSTM}(W^{l}_{t-1}).$ (9)

Compared to the second method, the first method incorporates the updated node
embedding in the recurrent neural network and it may lead to better
performance when node features are informative.

##  5 Application

In this section, we have detailed some financial applications that the GNN
methods have been commonly applied on. We have also summarized features,
graphs, methods, evaluation metrics, and baselines used in each financial
application in the supplementary materials.

###  5.1 Stock movement prediction

Though there are still debates on whether stocks are predictable, stock
prediction receives great attention and there are rich literature on
predicting stock movements using machine learning methods. However, the task
of stock prediction is challenging due to the volatile and non-linear nature
of the stock market. Traditionally, there are two major approaches to handle
the task: technical analysis and fundamental analysis (Sawhney). Technical
analysis utilizes numerical features such as closing prices and trading
volumes, while the fundamental analysis approach includes non-numerical
information, such as news and earning calls. The limitation of these non-graph
approaches is that they often have a hidden assumption that the stocks are
independent. To take the dependence into account, there is an increasing trend
to represent the stock relations in a graph where each stock is represented as
a node and an edge would exist if there are relations between two stocks.
Predicting multiple stocks’ movements could then be formed as a node
classification task and GNN models could be utilized to make the prediction.

In this section, we summarize the literature on stock prediction applying GNN
methods, where table LABEL:stock presents the key features and graphs used in
this application. There also exist challenges to apply the GNN methods in the
stock prediction task. Unlike other fields where the benchmark graphs are
available, to the best of our knowledge, there is no off-the-shelf graph
representing inter-stock relations. With abundant relations existing in the
financial system, it becomes challenging to obtain and select the relation for
graph construction. Moreover, owing to the volatility of the stock market, how
to model the sequential features and capture the temporal patterns are also
critical. Also, the financial industry has rich data sources including
financial statements, news and pricing information, which impose difficulty on
modeling the data.

There are multiple ways to construct the stock relational graph. For instance,
believing that correlation on historical prices reflects the inter-stock
relation, Li construct the graph using the correlation matrix of historic data
to predict the movement of Tokyo stock price index. On the other hand,
Matsunaga borrow information from knowledge bases and construct supplier,
customer, partner, and shareholder relational graphs. With multiple ways of
graph construction, there doesn’t exist a "best" graph due to the lack of
graph evaluation methods. Future work could be done to design a graph
evaluation method to help researchers better construct a relational graph.

To effectively process the sequential data and incorporate related
corporations’ information Chen propose a joint model using LSTM and GCN to
predict the stock movement. However, Chen’s approach assumes that the
relations between stocks are static, which may not reflect the reality.
Instead, Feng propose a temporal graph convolution layer to capture the stock
relations in a time-sensitive manner, so that the strength of relation could
be evolving over time. The relations are then updated based on historical
pricing sequences and the proposed method obtained better performance compared
to GCN. Believing that stock description documents also contain information
reflecting the changes in companies’ effect, Ying capture the temporal
relation by both sequential features and stock document attributes with a
time-aware relational attention network.

The aforementioned methods focus on capturing the temporal dependencies, while
Sawhney focus on fusing data from different sources. Sawhney propose a
multipronged attention network to jointly learn from historical price, social
media, and inter stock relations. Encoded pricing and textual information are
used as node feature inputs to GAT, where the graph information comes from the
Wiki company-based relations. The attention mechanisms is applied to allocate
different weights on various data sources and latent correlations may learned
via the attention layers.

###  5.2 Loan default risk prediction

For commercial banks and financial regularity institutions, monitoring and
assessing the default risk is at the heart of risk controlling process. As one
of the credit risks, default risk is the probability that the borrower fails
to pay the interest and principal on time. With a binary outcome, loan default
prediction could be seen as a classification problem and is commonly addressed
utilizing user-related features with classifiers including neural network (nn)
and gradient boosted trees (gbm). Since the probability that a borrower
defaults may be influenced by other related individuals, there is plenty of
literature forming a graph to reflect the interactions between borrowers. With
the rapid growth of GNN methods, GNN methods are widely applied on the graph
structure for loan default predicting problems. There are currently three
lines of work focusing on various types of loans: guarantee loans, e-commerce
loans, and other loans.

The guarantee loan allows small entrepreneurs to back each other in order to
increase their credibility. It has a debt obligation contract that specifies
that if one corporation fails to pay the debt, its guarantor needs to pay for
it. A guarantee network naturally arises where each node is a company and
directed edges represent the guarantee relationship. To learn a better
representation of the network, Cheng_network_loan utilize the graph attention
layer and design a objective function, so that vertices with similar
structures will be closer in the learned feature space. Since the guarantee
relations changes with time, Cheng_IJCAI forms a dynamic guarantee network to
represent the dynamics. A recurrent graph neural network layer is developed to
learn the temporal pattern and attentional weights are learned for each time
point via an attention architecture.

Unlike guarantee loans that the loan information could be naturally
represented in a directed graph, other loan types may not have a clear graph
structure and researchers need to construct the graph based on interactive
information. For instance, Xu construct a user relation graph where users are
connected by various relationships, such as social connections, transactions,
and device usage. However, the interactive graph may also contain noisy data,
which may be irrelevant. Since the massive interactive information may be
noisy and the impacting supply-chain information is deficient, Yang extract
supply-chain relations while predicting loan defaults. Forming the interaction
data as a graph, Yang formulate the supply chain mining task as a link
prediction task and thus construct a supply chain network, which is then used
to predict the default probability with GNN methodology.

With the rise of e-commerce, e-commerce consumer lending service is gaining
popularity to enhance consumers’ purchasing power. Able to obtain information
from multiple facets, the e-commerce platform could have multi-view data and
multi-relation networks, which may require sophisticated modeling methodology.
For instance, in order to predict the default probability for each consumer
with multi-view data, Liang_WSDM utilize a hierarchical attention mechanism to
encode the features on each view. Exploring multiplex relations, Hu propose an
attributed multiplex graph-based model with relation-specific layer and
attention mechanism to jointly model multiple relations. To simultaneously
model the labeled and unlabeled data, Wang proposed a semi-supervised graph
neural network approach and obtain interpretable results.

###  5.3 Recommender system of e-commerce

With the rapid growth of e-commerce, customers gradually get used to shopping
online and are exposed to a numerous range of products. To alleviate the
burden of users choosing the appropriate item, the recommender system is
developed to suggest products to users based on predicted item ratings.
Presenting the user and item information in a graph, GNN methodologies are
widely used in recommender system related tasks including click rate
prediction and fake review detection.

To accurately predict users’ preferences and recommend the appropriate item,
it is vital to exploit information for users, items, and their interactions. A
widely-used representation of that information is a bipartite graph, where
nodes are of two types, user and item, and edges represent that there are
relationships between user and item nodes. Noticing that the community the
user belongs to may affect the shopping decision, for example, a user belongs
to the traveler group may purchase travel-related items, Li_hierarchical
combine the bipartite graph modeling algorithm with clustering techniques to
reflect the community impact and the individual preference. However,
Li_hierarchical’s approach only considers the hierarchy in the user side,
while items may also have hierarchical information. To fill the gap,
Li_bipartite stack several GNN modules hierarchically to capture the
hierarchical structure in both user and item perspectives. Embeddings learned
from the GNN layer are clustered and used as an input for the next GNN layer,
which could preserve the high-order hierarchical connections.

The recommender system is mainly based on the past history of the user,
including its rating and reviews on the item. However, fake ratings and
feedback may be posted by the fraudsters to seek financial benefits. To detect
fraudulent reviews on the e-commerce platform, Kudo construct a directed and
signed comment graph with a signed graph convolutional network approach.
Compared to Kudo’s approach which only considers the comment graph, Li_spam
integrate both the bipartite user-item graph and a comment graph to capture
the local and global context of the comments. Noticing that camouflage
behaviors of the fraudsters may deteriorate the performance of fraud detection
mechanism and has seldom been considered by prior works, Dou propose a model
against both feature and relation camouflage. For each node, only informative
neighbors are selected for the next aggregation step, utilizing a similarity
measure and a reinforcement learning mechanism. While the above literature
focus on the fraud review detection side, there are also works that accomplish
both fraud review detection and item recommendation tasks. For example,
Zhang_Fraud propose a GCN-based framework that performs both item
recommendation and fraud detection in an end-to-end manner, while each of the
tasks is beneficial for the other one.

###  5.4 Fraud detection

Including payment fraud, identity theft, financial scam and insurance fraud,
financial fraud has a variety of types and has an increasing trend (Kurshan).
Observing that fraudsters tend to have abnormal connectivity with other users,
there is a trend to present users’ relations in a graph and thus, the fraud
detection task could be formulated as a node classification task.

Aiming to detect the malicious accounts, who may attack the online services to
seek excessive profits, Liu_18 find out that fraudsters have two patterns:
device aggregation and activity aggregation. Due to economic constraints,
attackers tend to use limited number of devices and perform activities in a
limited time, which may be reflected in the local graph structure. With this
observation, Liu_18 propose a variant of GCN and use sum operators to capture
the aggregation pattern. While the malicious accounts may be aggregated
together, Liu_19 argue that the normal accounts could also be connected with
the malicious account and may be mislabeled as malicious, which adds noisy
signals to the graph. Taking into account that the graph could be noisy and
nodes may have different impact, Liu_19 propose an adaptive path layer to
adaptively select the important neighbor nodes that contribute most to the
target node. Applying the GNN methodologies proposed by Liu_18, Liang stack
multiple adaptive path layers to aggregate neighbors’ features and have great
performance in a insurance fraud detection task.

The above literature focus on a bipartite graph, where nodes are either
account or devices, and there are also literature utilizing other types of
graphs. For instance, Jiang_anomaly construct a homogeneous user network
connecting user nodes based on their similarity of behaviors and apply the GCN
model. Rao focus on a dynamic graph of registration records and implement GCN
layers on structural and temporal subgraphs.

Besides the literature developing GCN-based methodologies, there are also
fraud-detection related literature applying GNN models with other structures.
Since GCN may suffer from over-smoothing problem and have shallow model
structure, Lv propose to replace the graph convolutional matrix with auto-
encoder to increase the depth of the neural network. Zhao_loss argues that GNN
models with random-walk based losses have poor performance in anomaly
detection task, since nodes of the same label may not be closer. They then
propose a loss function which leads to better performance and has bounded
prediction error.

###  5.5 Event prediction

Financial events, including revenue growth, acquisition and bankruptcy, could
provide valuable information on market trends and could be used to predict
future stock movement. Therefore, it draws great attention on how to predict
next financial event based on past events and currently GGNN model is often
used to accomplish the task. For example, given a sequence of financial events
of Chinese listed companies, Yang_event aim to predict the next event type and
construct an event graph where each node is a financial event and edges are
weighted using the frequency of the event pairs. Harl transform a binary
classification problem into an event prediction task by dividing the loan
application process into several events and predicting whether the next event
would be accepting or rejecting the application. Utilizing the GGNN model,
they obtain the predictions with high accuracy.

##  6 Challenges

###  6.1 Graph evaluation methods

To justify the inclusion of a graph, a commonly-used evaluation method is to
compare the outcome for a graph-based machine learning method with a graph-
free machine learning method (Feng; Li). However, there is little discussion
on comparing different graphs’ effects and quality, while existing literature
discussing graph comparisons are often inadequate. For example, Liang
visualize the structural patterns in different graphs, concluding that the
device sharing graph is more appropriate based on the observed patterns.
Without presenting the evaluation metric for each graph, the graph comparison
based on the visualization may not be adequate. The problem may be more severe
in similarity-based graph construction since a threshold needs to be set to
determine whether an edge exists. Different threshold values may lead to
completely different graphs and thus affect the model performance. Thus,
justification on threshold setup during graph construction is of great
importance. Some efforts have been made to tackle this problem, such as
utilizing a reinforcement learning approach to automatically select the
optimal threshold (Dou). More attention may need to be drawn to develop a
framework to assess the graph quality systematically.

###  6.2 Explainability

Combining both graph structural information and feature information, GNN
models are often complicated and it is challenging to make an interpretation.
Recently, there are some literature focusing on the explainability of GNN
models. GNNExplainer (GNNExplainer), for example, is proposed to provide an
interpretable explanation on trained GNN models such as GCN and GAT. Model
explainability in financial tasks is of great importance, since understanding
the model could benefit decision-making and reduce economic losses. However,
there is little literature studying the explainability of GNN models in a
financial application, which often accompanies by heterogeneous and dynamic
graphs. Current literature focuses on relatively simpler graphs. For example,
Li_explain extend the GNNExplainer to a weighted directed graph and apply it
on a Bitcoin transaction graph. Rao1 propose an explainable fraud prediction
system that could operate on heterogeneous graphs consisting of different node
and edge types. More work could be done on the explainability of GNN models
with edge-attributed graphs and dynamic graphs, which are not yet considered.

###  6.3 Task type

Tasks of GNN models are commonly classified into three categories: node-level
task, edge-level task, and graph-level task (Wu). However, the GNN methods
applied on financial applications are mostly focused on the node-level task.
For example, the stock movement prediction task is often formulated as a node
classification task, where stocks are represented as nodes. There exist some
literature focus on other types of task, for instance, Yang aim to mine the
underlying supply chain network and formulate it as a link prediction task,
but this line of work is rare. Since there are rich literature developing GNN
methodologies on edge prediction or graph classification tasks, there are rich
opportunities on applying recently-developed GNN methods on financial fields
if financial tasks could be formulated into an edge or graph level task.

###  6.4 Data availability

To pursue reproducibility, it is a common practice to release both data
sources and codes when publishing the paper. However, in the reviewed papers,
only about 24% of them release the code. Since the financial tasks are
commonly based on real-world problems, data may have some restrictions due to
the privacy obligation of the related corporations. Lack of open-source codes
and datasets, it is hard to reproduce previous works and compare their
methodologies in the latter literature. Thus, it is of great value to
construct benchmark datasets, so that methods could be compared on the same
data.

###  6.5 Scalability

In the real-world financial scenario, commercial data are often of large
scales. For instance, AliGraph utilize data from a popular e-commerce platform
and it contains about 483 million nodes with 231 million edges. How to improve
the scalability of GNNs is vital but challenging. Computing the Laplacian
matrix becomes hard with millions of nodes and for a graph of irregular
Euclidean space, optimizing the algorithm is also difficult. Sampling
techniques may partially solve the problem with the cost of losing structural
information. Thus, how to maintain the graph structure and improve the
efficiency of GNN algorithms are worth further exploration.

## Supplementary material

In the supplementary materials, we present materials that are not covered in
the main text. The supplementary materials contain the summary table for each
financial application, figures categorizing major GNN methodologies for each
graph type and acronyms used in the text.

### Summary tables for financial applications

Table 2: Summary of stock movement prediction literature Reference |  Feature 777The red color format represents the acronym format and the full form could be found in the supplementary material section by clicking the acronyms. |  Graph |  Method |  Evaluation metric |  Baseline method
---|---|---|---|---|---
Chen |  7-day open, high, low, close prices and volume for CSI listed companies |  Corporation shareholder network |  LSTM + GCN |  Accuracy |  LSTM, GCN
Feng |  5/10/20/30 days moving average of close price for SP500/NYSE listed companies |  Wiki company-based relation network & sector-industry stock relation network |  RSR |  MSE, IRR, MRR |  SFM, LSTM, GBR, GCN
Matsunaga |  5/10/20/30 days moving average of close prices for Nikkei 225 listed companies |  Supplier, customer, partner and shareholder relation graph |  RSR |  Return ratio, Sharpe ratio |  Model with subsets of relations
Ying |  5/10/20/30 days moving average of close prices and stock description documents for SP500 or NYSE listed companies |  Wiki company-based relation network |  TRAN |  MSE, MRR, IRR |  RSR, GCN, LSTM
Li |  News of TPX 500/100 listed companies |  Stock correlation graph |  LSTM-RGCN |  Accuracy |  Naïve Bayes, LR, RF, HAN, transformer, S-LSTM
Sawhney |  Price and social media information for companies listed in the SP500 index or NYSE or NASDAQ markets |  Wiki company-based relation network |  MAN-SF |  Accuracy, F1, MCC |  ARIMA, RF, TSLDA, HAN, StockNet, LSTM+GCN
Sawhney_audio |  Text and audio features of earning calls for companies in the SP500 index |  Stock earning call graph |  VolTAGE |  MSE, R-squared |  LSTM, HAN, MDRM, HTML
Liou |  News and attributes for stock tags |  News co-occurance graph |  HAN |  Accuracy, MCC |  RF
Table 3: Summary of loan default risk prediction literature Reference |  Feature |  Graph |  Method |  Evaluation metric |  Baseline method
---|---|---|---|---|---
Wang |  / |  Multiple user networks |  SemiGNN |  AUC, KS |  Xgboost, LINE, GCN, GAT
Cheng_network_loan |  Active loan behavior, historical behavior, and user profile |  Guarantee network |  HGAR |  AUC, Precision@k |  GF, DW, node2vec, AANE, SNE, GAT
Cheng_KDD |  Customer profile, loan information, guarantee profile, and loan contract |  Guarantee network |  TRACER |  F1, Precision@k |  LR, GBDT, DNN
Cheng_IJCAI |  Loan behavior and company profile |  Temporal guarantee network |  DGANN |  AUC |  GF, GCN, node2vec, GAT, SEAL, RNN, GRNN
Yang |  Credit-related features, spatial features, and temporal features |  Temporal smallbuiness en-trepreneur network |  ST-GNN |  AUC, KS |  GBDT, GAT, STAR
Hu_Loan |  User credit exposure features |  Alipay user and applet graph |  AMG-DP |  AUC, KS |  MLP, Xgboost, node2vec, GraphSAGE, GAT, HAN, SemiGNN
Liang_WSDM |  Device information and trading features |  User relationship graph |  MvMoE |  AUC |  GBDT
Xu |  User profile and transaction summary |  Auto loan network |  GRC |  Precision, recall, F1 |  SVM, MLP, GCN
Lee |  Loan information, credit history, and soft information |  Borrower’s relation network |  / |  Accuracy, precision, recall, F1, AUC |  SVM, RF, Xgboost, MLP, GCN
Table 4: Summary of literature on recommendation system of e-commerce Reference |  Feature |  Graph |  Method |  Evaluation metric |  Baseline method
---|---|---|---|---|---
Li_spam |  Features for item, user and comment |  Xianyu comment graph |  GAS |  AUC, F1, recall |  GBDT
Zhang_Fraud |  Behavioral features |  Yelp review network, Amazon review network |  GraphRfi |  MAE, RMSE, precision, recall, F1 |  RCF, GCMC, PMF, ICF, MF
Dou |  Behavioral features |  Yelp review network, Amazon review network |  CARE-GNN |  AUC, recall |  GCN, GAT, RGCN, GraphSAGE, SemiGNN
Kudo |  Behavioral features |  Amazon review network |  GCNEXT |  AUC |  RGCN, SIDE, SGCN
Li_hierarchical |  Purchasing power, number of transactions, item category, and shipping costs |  E-commerce user-item network |  Bi-HGNN |  Accuracy, AUC, F1 |  GraphSAGE, Diffpool
Li_bipartite |  Click and transaction logs |  Taobao user-item network |  HiGNN |  AUC |  DIN, GE
Table 5: Summary of fraud detection prediction literature Reference |  Feature |  Graph |  Method |  Evaluation metric |  Baseline method
---|---|---|---|---|---
Liang |  Insurance claim history, shipping history and shopping history |  Device sharing graph |  / |  F1, DE |  GBDT, node embeddings
Lv |  Purchase history of 1000 commodity categories |  JD Finance anti-fraud graph |  AutoGCN |  AUC |  GCN, GAT, GCN with attention
Rao1 |  Individual risk features |  Transaction records graph |  xFraud |  AUC |  LR, DNN, GAT, GCN, HGT
Zhu |  User features |  Iqiyi user network |  HMGNN |  Precision, recall, F1, AUC |  LR, Xgboost, MLP, GCN, GAT, ASGCN, mGCN
Jiang_anomaly |  User behavioral features and content based features |  Simulated user activity network |  / |  Accuracy, precision, recall |  RF, SVM, LR, CNN
Liu_18 |  User activities |  Alipay one-month account-device network |  GEM |  F1, AUC, precision-recall |  Connected subgraph, GBDT, GCN
Liu_19 |  User activities |  Alipay one-week account-device network |  GeniePath |  Accuracy |  MLP, node2vec, GCN, GraphSAGE, GAT
Rao |  Registration profile and transaction features |  Account-registration graph |  DHGReg |  Precision |  MLP, GCN, GAT
Zhao_loss |  / |  Bitcoin-alpha graph |  GAL |  Precision, recall, F1, AUC |  GCN, GAT, GraphSAGE, DOMINANT
Table 6: Summary of event prediction literature Reference |  Feature |  Graph |  Method |  Evaluation metric |  Baseline method
---|---|---|---|---|---
Harl |  Event features |  Loan application event graph |  GGNN |  Accuracy |  /
Yang_event |  Financial news |  Financial event graph |  GGNN |  Accuracy, precision, recall, F1 |  PMI, DW, LSTM

### Figures on GNN methods for each graph type

(a) Graph neural network models for homogeneous graphs. With a graph adjacency
matrix and a feature matrix as inputs, the graph convolutional process
generates a representation for each target node by aggregating the information
of its neighbors. A multi-layer perception layer is then applied to generate
the predicted labels. (a) Graph neural network models for bipartite graphs.
For nodes having the same type, the graph convolutional process updates their
node representations using their own information and their neighbors’
information. Aggregating functions are utilized to generate the node
representations from the representations for both node types. A multi-layer
perception layer is then applied to generate the predicted labels. (a) Graph
neural network models for multi-relation graphs. A multi-relation graph is
split into several sub-graphs in which edges are of the same type. Within each
sub-graph, the graph convolutional process takes place to update the node
representations. Then between-graph aggregation is utilized to obtain the
final node representations. A multi-layer perception layer is then applied to
generate the predicted labels.  (a) Graph neural network models for dynamic
graphs. A dynamic graph is often a sequence of graphs ordered by time. The
graph convectional process generates node representations for graph at each
time stamp. To get the most recent representation, node representations are
aggregated using functions including the recurrent neural network. A multi-
layer perception layer is then applied to generate the predicted labels.
Figure 7: Graph neural network models for different graph types. The term Conv
denotes graph convolution process. The term MLP denotes the multi-layer
perception. The term Agg denotes the aggregation process. The term Split
denotes data splitting according to its characteristics.

## Acronyms

AANE

    Accelerated Attributed Network Embedding
AMG-DP

    Attributed Multiplex Graph based loan Default Prediction approach
ARIMA

    AutoRegressive Integrated Moving Average
ASGCN

    Adaptive Sampling Graph Convolutional Network
AUC

    Area Under the Curve
AutoGCN

    Auto-encoder based Graph Convolutional Networks
Bi-HGNN

    Bipartite Hierarchical bipartite Graph Neural Network
CARE-GNN

    CAmouflage-REsistant GNN
CNN

    Convolutional Neural Network
CSI

    China Securities Index
DE

    Detection Expansion
DGANN

    Dynamic Graph-based Attention Neural Network
DHGReg

    Dynamic Heterogeneous Graph Neural Network
Diffpool

    Differentiable graph pooling
DIN

    Deep Interest Network
DNN

    Deep Neural Network-based model
DOMINANT

    Deep anOMaly detectIoN on Attributed NeTworks
DW

    DeepWalk
GAL

    Graph Anomaly Loss
GAS

    GCN-based Anti-Spam model
GAT

    Graph Attention Network
GBDT

    Gradient Boosting Decision Tree
GBR

    Graph-Based Ranking
GCMC

    Graph Convolutional Matrix Completion
GCN

    Graph Convolutional Network
GCNEXT

    Graph Convolutional Network with Expended Balance Theory
GE

    Graph Embedding-based method
GEM

    Graph Embeddings for Malicious accounts
GF

    Graph Factorization
GGNN

    Gated Graph Neural Network
GraphRfi

    GCN-based user Representation learning framework
GraphSAGE

    SAmple and aggreGatE
GRC

    Graph neural network with a Role-constrained Conditional random field
GRNN

    Graph Recurrent Neural Network
HAN

    Hierarchical Attention Networks
HAN

    Hybrid Attention Network
HGAR

    High-order Hraph Attention Representation
HGT

    Heterogeneous Graph Transformer
HiGNN

    Hierarchical bipartite Graph Neural Network
HMGNN

    Heterogeneous Mini-Graphs Neural Network
HTML

    Hierarchical Transformer-based Multi-task Learning
ICF

    Item-based Collaborative Filtering
IRR

    Investment Return Ratio
KS

    Kolmogorov-Smirnov distance
LINE

    Large-scale Information Network Embedding
LR

    Linear Regression
LSTM-RGCN

    LSTM Relational Graph Convolutional Network
LSTM

    Long Short Term Memory
MAE

    Mean Absolute Error
MAN-SF

    Multipronged Attention Network for Stock Forecasting
MCC

    Matthew’s Correlation Coefficient
MDRM

    Multimodal Deep Regression Model
MF

    Matrix Factorization
mGCN

    modified version of Graph Convolutional Network
MLP

    Multi-Layer Perceptron
MRR

    Mean Reciprocal Rank
MSE

    Mean Squared Error
MvMoE

    Multi-view-aware Mixture-of-Experts network
NASDAQ

    NASDAQ stock exchange
Nikkei 225

    Nikkei Stock Average
NYSE

    New York Stock Exchange
PMF

    Probabilistic Matrix Factorization
PMI

    model based on Pairwise Mutual Information
Precision@k

    Precision of the top k nodes
RCF

    Robust Collaborative Filtering model
RF

    Random Forest
RGCN

    Relational Graph Convolutional Network
RMSE

    Root Mean Squared Error
RNN

    Recurrent Neural Network
RSR

    Relational Stock Ranking
S-LSTM

    Sentence-state LSTM
SEAL

    learning from Subgraphs, Embeddings and Attributes for Link prediction
SemiGNN

    Semisupervised attentive Graph Neural Network
SFM

    State Frequency Memory
SGCN

    Signed Graph Convolutional Network
SIDE

    SIgned Directed network Embedding
SNE

    attributed Social Network Embedding
SP500

    Standard and Poor’s 500
ST-GNN

    Spatial-Temporal aware Graph Neural Network
STAR

    Spatio-Temporal Attentive Recurrent neural network
SVM

    Support Vector Machine
TPX

    Tokyo Stock Price Index
TRACER

    TempoRal Attention Contagion chain Enhanced Rating model
TRAN

    Time-aware graph Relational Attention Network
TSLDA

    Topic Sentiment Latent Dirichlet Allocation
VolTAGE

    Volatility forecasting via Text-Audio fusion with Graph convolution networks for Earnings calls
Xgboost

    eXtreme gradient boosting

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