# Signed Directed Graph Contrastive Learning with Structure and Laplacian
Augmentation

Taewook Ko<EMAIL_ADDRESS>Seoul National UniversityRepublic of Korea ,
Yoonhyuk Choi<EMAIL_ADDRESS>Seoul National UniversityRepublic of Korea
and  Chong-Kwon Kim<EMAIL_ADDRESS>Korea Institute of Energy
TechnologyRepublic of Korea

###### Abstract.

Graph contrastive learning has become a powerful technique for several graph
mining tasks. It learns discriminative representation from different
perspectives of augmented graphs. Ubiquitous in our daily life, singed-
directed graphs are the most complex and tricky to analyze among various graph
types. That is why singed-directed graph contrastive learning has not been
studied much yet, while there are many contrastive studies for unsigned and
undirected. Thus, this paper proposes a novel signed-directed graph
contrastive learning, SDGCL. It makes two different structurally perturbed
graph views and gets node representations via magnetic Laplacian perturbation.
We use a node-level contrastive loss to maximize the mutual information
between the two graph views. The model is jointly learned with contrastive and
supervised objectives. The graph encoder of SDGCL does not depend on social
theories or predefined assumptions. Therefore it does not require finding
triads or selecting neighbors to aggregate. It leverages only the edge signs
and directions via magnetic Laplacian. To the best of our knowledge, it is the
first to introduce magnetic Laplacian perturbation and signed spectral graph
contrastive learning. The superiority of the proposed model is demonstrated
through exhaustive experiments on four real-world datasets. SDGCL shows better
performance than other state-of-the-art on four evaluation metrics.

graph contrastive learning, magnetic Laplacian, signed directed graph, link
sign prediction

##  1\. Introduction

Various types of online social graphs are developing as the use of the
internet increases. Social graph research attracts attention from researchers
with its usefulness for various downstream tasks such as prediction,
classification, and recommendation(Fout et al., 2017; Zhang et al., 2018; Fan
et al., 2019; He et al., 2020a). Signed-directed graphs, which express the
relationship between users as like/dislike or trust/distrust, have the most
information and are the most valuable among several graph types. However, it
has difficulties to analyze them caused of severe noise and complexity.
Many network embedding and graph convolution studies for signed-directed
graphs have been proposed based on the two well-known social theories, balance
and status.(Heider, 1946; Holland and Leinhardt, 1971) Balance theory is the
concept that ”A friend of my friend is my friend, and an enemy of my friend is
my enemy.” Furthermore, status theory sets the relative user ranks with the
relation of positive and negative edges. The ideas of most existing studies,
including SiNE(Wang et al., 2017), SGCN(Derr et al., 2018), SNEA(Li et al.,
2020), and SDGNN(Huang et al., 2021), are derived from the two social
theories. Recently, some studies that do not use them have been proposed.
ROSE(Tong et al., 2020b) proposed a user-role-based methodology by
transforming a graph into an unsigned bipartite. SDGCN(Ko, 2022) proposed a
spectral convolution via a signed magnetic Laplacian matrix.
Contrastive learning(Chen et al., 2020a; He et al., 2020b) is a successful
learning mechanism in computer vision research with fewer labels or a self-
supervised environment. It learns discriminative representations through
contrastive loss from the data itself. They maximize the mutual information of
representations of augmented images. Recently, many graph studies utilizing
contrastive learning have been suggested. Graph contrastive learning models
generate graph views in their own ways, such as edge or node deletion(You et
al., 2020), addition(Zeng and Xie, 2021), random-walk(Qiu et al., 2020), or
attribute masking(Zhu et al., 2021). JOAO(You et al., 2021) and AD-GCL(Suresh
et al., 2021) learn optimal perturbing distribution according to the dataset
and SimGRACE(Xia et al., 2022) adds gaussian noise on the trained parameters
for encoder perturbation. They define contrastive loss with the augmented
representations of nodes or graphs. Graph contrastive learning also aims to
maximize the mutual information of different graph views. However, not much
research has been conducted on signed-directed graphs.
SGCL(Shu et al., 2021) is the first study to apply contrastive learning on
signed-directed. They make two graph views by giving random perturbations on
edge signs and directions. Then the views are divided into positive graph
views and negative graph views. The positive graph view is constructed with
only the positive edges of a graph view, and the same for the negative graph
view. Then run graph convolution on the different graph views to get augmented
node representations. Though SGCL is innovative, there are some limitations.
First, it loses the context of the original graph if we separate the edges
into different graphs by the edge types. The mutual influence between edge
types cannot be trained. For example, valuable patterns like the balance
theory cannot be reflected. Second, most real-world signed graphs have a
nearly 90% of positive edge ratio(Kumar et al., 2016). Therefore, the negative
views are unsuitable for graph convolutions due to the sparsity issue. In this
study, we propose a Signed Directed Graph Contrastive Learning(SDGCL), which
improves these limitations.
SDGCL utilizes two levels of perturbation. We apply random edge perturbation
similar to SGCN. It changes edge signs and directions to generate augmented
graph views. Then we introduce magnetic Laplacian perturbation. It changes a
phase parameter $q$ of the magnetic Laplacian matrix. $q$ controls the phase
angle between the real- and imaginary axis(Ko, 2022). Magnetic Laplacian was
studied in the field of quantum physics(Shubin, 1994; Olgiati, 2017; Colin de
Verdière, 2013). Thanks to its Hermitian properties, it has been used in graph
convolutions for directed and signed-directed(Zhang et al., 2021; Singh and
Chen, 2022). Signed magnetic Laplacian encodes graph sturucture, including
edge signs and directions. It reflects the graph information and allows the
model to learn the patterns of nodes and edges. We define a spectral
convolution layer with the perturbed Laplacian by the idea of graph signal
processing(Defferrard et al., 2016; Hammond et al., 2011; Kipf and Welling,
2016). SDGCL gets the two augmented node representations with the graph
encoder and defines node-level contrast loss(Zhu et al., 2021; Shu et al.,
2021) for contrastive learning.
The performance of the proposed model was evaluated with the link sign
prediction task, which is widely selected to evaluate the signed-directed
graph embedding. It was tested with four real-world graphs and showed
excellent performance compared to various baselines for signed-directed graph
embedding and graph contrast learning.

The followings are the contributions of this paper

  * •

This paper proposes a novel signed-directed graph contrastive learning model,
SDGCL.

  * •

It augments node representations via two stages, graph structure perturbation
and magnetic Laplacian perturbation.

  * •

To the best of our knowledge, it is the first to introduce magnetic Laplacian
perturbation and is the first spectral signed graph contrastive model.

  * •

The proposed model has the best link sign prediction performance in real-world
graph experiments.

##  2\. Related Works

###  2.1. Signed-Directed Graph Embedding

Most signed-directed graph studies utilize sociological stories.(Heider, 1946;
Holland and Leinhardt, 1971) SiNE(Wang et al., 2017) is an undirected model
that uses balance theory, defining an objective function: friend nodes
increase their similarity. BESIDE(Chen et al., 2018) focused on the bridge
edges to overcome the limitation of triads and utilized both social theories.
SGCN(Derr et al., 2018) defined a novel balanced and unbalanced path for
neighbor aggregation. SNEA(Li et al., 2020) and SiGAT(Huang et al., 2019) used
attention mechanism. SDGNN defined four different weight matrices to
distinguish the edge signs and directions. And they proposed triad loss based
on the theories. Balance and status theories are developed with the patterns
of the user triads. About 60-70% of triads on real-world graphs satisfy the
theories(Leskovec et al., 2010; Javari et al., 2017). Though the theories are
crucial paradigms in signed-directed graph research, not all the triads
satisfy them. Moreover, it is not a rule that all users should follow. Models
that depend on the theories are not working well with users with little or no
triads. On top of that, such models require much computational cost to find
triads or paths. Recently, ROSE(Javari et al., 2020) proposed a methodology
using user-role by transforming graphs into unsigned bipartite. SDGCN(Ko,
2022) proposed a spectral convolution model by proving the positive
semidefinite of signed magnetic Laplacian.

Figure 1. Example of a signed-directed graph and its structure perturbation.
Dashed edges indicate perturbed edges.

###  2.2. Magnetic Laplacian

Magnetic Laplacian was first introduced in the field of quantum mechanics to
analyze the charged particle under magnetic flux.(Shubin, 1994; Olgiati, 2017;
Lieb and Loss, 1993; Colin de Verdière, 2013) But thanks to its Hermitian
properties, the magnetic Laplacian has been utilized in directed graph
studies.(Guo and Mohar, 2017; Liu and Li, 2015) Some application studies such
as graph clustering(Cucuringu et al., 2020; Cloninger, 2017) node(Zhang et
al., 2021) and graph representation learning(Furutani et al., 2019) are
delivered. MagNet(Zhang et al., 2021) proved the positive semidefinite
property of the magnetic Laplacian and proposed a spectral graph convolution.
Recently, some studies extended the magnetic Laplacian to the signed graph.
SDGCN(Ko, 2022) and SigMagNet(Fiorini et al., 2022) define signed magnetic
Laplacian. The signed magnetic Laplacian matrix uniquely encodes edge types of
signed-directed graphs with real- and imaginary numbers. They proposed
spectral graph convolution with the signed magnetic Laplacian.

###  2.3. Graph Contrastive Learning

SimCLR(Chen et al., 2020a) and MoCo(He et al., 2020b) proposed contrastive
learning and achieved great success in image classification. Many follow-up
studies were proposed thanks to the increasing learning efficiency via self-
supervised. DGI(Velickovic et al., 2019), GMI(Peng et al., 2020) adopted the
contrastive learning to graph studies. They measure mutual information of
input and node representations. InfoGraph(Sun et al., 2019) proposed patch-
level representation. GCC employed random-walk sampling to make positive and
negative samples. Since then, several graph augmentations(You et al., 2020;
Zeng and Xie, 2021; Zhu et al., 2021) and graph encoders(Shu et al., 2021; Xia
et al., 2022) have been proposed for contrastive learning. There are
contrastive studies for graph clustering(Pan and Kang, 2021; Zhong et al.,
2021), node embedding(Zhu et al., 2021), DDI prediction(Wang et al., 2021; Li
et al., 2022), and recommendation(Lin et al., 2022). However, signed-directed
graph contrastive studies are not discussed much. To the best of our
knowledge, SGCL(Shu et al., 2021) is the only one. SGCL makes graph views by
randomly perturbing the edges. Then they construct positive- and negative-
graph views by separating edges by the edge type. However, the edges of
signed-directed graphs have meaningful relationships with each other.
Therefore, the context of the data cannot be properly interpreted when we
arbitrarily separate graph views by edge types. Moreover, they are inefficient
in that they need to find positively or negatively linked nodes for every
aggregation stage. To overcomes these limitations, this study proposes a novel
graph contrastive learning model.

##  3\. Problem Formulation

Let $\text{G}=(V,\mathcal{E}^{+},\mathcal{E}^{-})$ be a signed-directed graph
where $V$ is a set of nodes like users in social graphs.
$\mathcal{E}^{+}\subseteq V\times V$ indicates positive edge matrix, and
$\mathcal{E}^{-}\subseteq V\times V$ is of negative edges. Positive and
negative edges represent user relationships, such as like/dislike or
trust/distrust. For example $\mathcal{E}^{+}_{u,v}$ equals 1 if there is a
positive edge from node $u$ to node $v$; otherwise, 0. Similarly, if there is
a negative edge from node $v$ to $u$, $\mathcal{E}^{-}_{v,u}$ equals 1. Note
that $\mathcal{E}^{+}\cap\mathcal{E}^{-}=\varnothing$. Because we do not
accept users having two different edges to the other user simultaneously, such
as ”John loves Jane and also hates her.” A user may be connected to other
users by one of the three relations(none, positive, negative). Thus, there are
nine-edge types between a user pair. The goal of this paper is to map the
nodes $u\in V$ into the low-dimensional embedding vectors
$z_{u}\in\mathbb{R}^{d}$ with a given graph G as:

$f(\text{G})=Z,$

$f$ is a transformation function and $Z\in\mathbb{R}^{|V|\times d}$ is an
embedding matrix. Each row of $Z$ represents the node embedding with dimension
size $d$.

##  4\. Model Framework

###  4.1. Graph Augmentation

####  4.1.1. Structure Perturbation

There are two methods for structure perturbation, edge sign perturbing and
edge direction perturbing. In edge sign perturbing, we change the edge signs
randomly from the given graph. For example, we sample $p\%$ of positive edges
and change their signs to negative. And the same for the negative edges.
Similarly, we sample $r\%$ of edges and change their directions to inverse. If
an edge is bidirectional, we change it to directional arbitrary. With this
random edge perturbation, we get structurally perturbed graph views,
$\widetilde{\text{G}}=(V,\tilde{\mathcal{E}^{+}},\tilde{\mathcal{E}^{+}})$.
Figure 1 shows an example of structure perturbation. As already known by
balance and status theories, there is important information in the edge sign
and directions. We may harm the vital triad patterns via random perturbation.
However, we expect some amount of perturbation would be helpful to learn
robust representations from noisy real-world data. Moreover, the perturbing
can discover and exploit some relationships that might exist. The model can
improve the generalization performance through random perturbations.

####  4.1.2. Signed-Directed Magnetic Laplacian

Before introducing Laplacian Perturbation, we define the signed-directed
magnetic Laplacian. Graph Laplacian is useful to encode graph sturucture with
degree and adjacency matrices, $\textbf{L}=\textbf{D}-\textbf{A}$. They are
not only positive semidefinite but also have non-negative eigenvalues and
associate orthonormal eigenvectors. With that properties, (Kipf and Welling,
2016) and (Hammond et al., 2011) proposed the idea of spectral graph
convolution. However, graph Laplacian is asymmetric when a graph is directed
or signed-directed. They have complex eigenvalues and do not satisfy the
properties for spectral convolution. Thus, (Ko, 2022; Singh and Chen, 2022;
Fiorini et al., 2022) proposed a novel magnetic Laplacian matrix representing
the structure of signed-directed graphs and satisfying positive semidefinite.
First of all, we define a complex Hermitian adjacency matrix as follow,

$\mathbf{H}^{q}=\mathbf{A}_{s}\odot\mathbf{P}^{q}.$

$\mathbf{A}_{s}:=\frac{1}{2}(\mathbf{A}+\mathbf{A^{\intercal}})\subseteq
V\times V$ is a symmetrized adjacency matrix, and $\mathbf{P}^{q}\subseteq
V\times V$ is a phase matrix with complex numbers. $\odot$ is an element-wise
multiplication operation. The definition of the phase matrix is,

$\mathbf{P}^{q}(u,v):=\frac{\text{exp}(i\Theta_{uv}^{q})\text{A}_{uv}+\text{exp}(i\overline{\Theta}^{q}_{uv})\text{A}_{vu}}{|\text{exp}(i\Theta_{uv}^{q})\text{A}_{uv}+\text{exp}(i\overline{\Theta}^{q}_{uv})\text{A}_{vu}|+\epsilon}.$

$\Theta_{uv}^{q}=q\mathcal{E}^{+}_{uv}+(\pi+q)\mathcal{E}^{-}_{uv}$ and
$\overline{\Theta_{uv}}^{q}=-q\mathcal{E}^{+}_{vu}+(\pi-q)\mathcal{E}^{-}_{vu}$
Be careful with the subscripts orders. $q$ is a hyperparameter lies in
[0,$\pi/2$]. The symmetrized adjacency matrix encodes the node connectivity,
and the phase matrix encodes link directions and signs with different phase
values. Figure 2(b) shows the edge encodings of the defined Hermitian
adjacency matrix $\textbf{H}^{q}$. They uniquely encode the nine-edge types of
signed-directed graphs. It tells node connections and edge types as well. We
can see that $\textbf{H}^{q}$ is a complex numbered skew-symmetric form, a
complex Hermitian matrix. Then, we define the signed-directed magnetic
Laplacians with this Hermitian adjacency by,

$\mathbf{L}^{q}_{U}:=\mathbf{D}_{s}-\mathbf{H}^{q}=\mathbf{D}_{s}-\mathbf{A}_{s}\odot\mathbf{P}^{q}$
$\mathbf{L}^{q}_{N}:=\mathbf{I}-(\mathbf{D}^{-\frac{1}{2}}_{s}\mathbf{A}_{s}\mathbf{D}^{-\frac{1}{2}}_{s})\odot\mathbf{P}^{q},$

$\mathbf{D}_{s}$ is a symmetric degree matrix, similar to $\mathbf{A}_{s}$.
$\mathbf{L}^{q}_{U}$ and $\mathbf{L}^{q}_{N}$ are unnormalized and normalized
signed-directed magnetic Laplacians. It is well known that the skew-symmetric,
complex Hermitian matrix is positive semidefinite(Ko, 2022; Fiorini et al.,
2022; Singh and Chen, 2022). Thus, the Laplacians are positive semidefinite
and diagonalizable by spectral decomposition. For example, the normalized
Laplacian is diagonalized by,

$\textbf{L}^{q}_{N}=\textbf{U}\Lambda\textbf{U}^{\dagger}.$

Each column of $\mathbf{U}$ is eigenvector $\mathbf{u}_{k}$ and
$\mathbf{U}^{\dagger}$ is a conjugate transpose of $\mathbf{U}$. $\Lambda$ is
a diagonal matrix where the elements are $k$-th eigenvalues
$\Lambda_{k,k}=\mathbf{\lambda}_{k}$. The eigenvalues and eigenvectors contain
the structural information of the signed-directed graph. We leverage this
matrix to define spectral graph convolution.

Figure 2. Edge encoding and meaning of $q$. It shows edge encoding values of a
complex Hermitian adjacency matrix. $q$ controls the phase angle. Figure 3.
Model overview

####  4.1.3. Laplacian Perturbation

Figure 2(a) describes the meaning of $q$ in magnetic Laplacian. $q$ controls
the phase angle of encoded values between the real- and imaginary axes. If $q$
is small, the Laplacian puts less attention on directional information. It
becomes an undirected model when the $q$ is 0. However, the high $q$ value
also harms the encoding validity. $q$ influences the encoding effectiveness of
the signed-directed magnetic Laplacian. In this subsection, we introduce
Laplacian perturbation via $q$ variation. We randomly select $q$ value from 0
to 0.4$\pi$. The Laplacian matrix varies according to the $q$ variations,
though the meaning of the graph structure that the Laplacian represents is
still the same. This kind of Laplacian perturbation is a useful technique that
makes graph augmentation without distortion of the original data(Tong et al.,
2020a).
In short, we make two structurally perturbed graph views
$\widetilde{\textbf{G}}_{1}$ and $\widetilde{\textbf{G}}_{2}$, through random
edge changes. Then we get perturbed signed-directed magnetic Laplacian
$\widetilde{\textbf{L}}^{q_{1}}$ and $\widetilde{\textbf{L}}^{q_{2}}$ from the
graph views with randomly chosen $q$ values. With these Laplacian matrices, we
define graph encoders.

###  4.2. Graph Encoder

####  4.2.1. Spectral Convolution via Magnetic Laplacian

The signed-directed magnetic Laplacian $\mathbf{L}$, is diagonalizable with
eigenvector matrix $\mathbf{U}$, and diagonal eigenvalues $\Lambda$. Several
graph convolution studies (Defferrard et al., 2016; Hammond et al., 2011)
utilize the eigenvectors as discrete Fourier modes of graph signal processing.
The graph signals are transformed into the Fourier domain through graph
Fourier transform $\mathbf{x}:V\to\mathbb{C}$ by
$\mathbf{\hat{\textbf{x}}}=\mathbf{U}^{\dagger}\mathbf{x}$. The inverse
Fourier transform formula is defined as follow thanks to the unitarity of
$\mathbf{U}$,

$\mathbf{x}=\mathbf{U}\mathbf{\hat{x}}=\sum_{k=1}^{N}\hat{\mathbf{x}}(k)\mathbf{u}_{k}.$

The spectral convolution operation of graph signal processing is described as

$\mathbf{g}_{\theta}\ast\mathbf{x}=\mathbf{U}\mathbf{g}_{\theta}\mathbf{U}^{\dagger}\mathbf{x},$

where $\mathbf{g}_{\theta}=diag(\theta)$ is a trainable filter. For efficient
calculation, (Hammond et al., 2011) proposed a truncated Chebyshev polynomial
expansion of the filter by,

$\mathbf{g}_{\theta^{\prime}}(\mathbf{\Lambda})\approx\sum_{k=0}^{K}\theta^{\prime}_{k}T_{k}(\mathbf{\tilde{\Lambda}}).$

$T_{0}(x)=1,T_{1}(x)=x$, and $T_{k}=2xT_{k-1}(x)+T_{k-2}(x)$ for $k\geq 2$.
$k$ is an expansion order.
$\mathbf{\tilde{\Lambda}}=\frac{2}{\lambda_{max}}\mathbf{\Lambda}-\mathbf{I}$
is a normalized eigenvalue matrix, and $\lambda_{max}$ is the largest
eigenvalue. $\theta^{\prime}_{k}$ are Chebyshev coefficients. We have a
simplified form of spectral graph convolution by,

$\mathbf{g}_{\theta^{\prime}}\ast\mathbf{x}=\sum_{k=0}^{K}\theta^{\prime}_{k}T_{k}(\mathbf{\tilde{L}})x,$

where $\mathbf{\tilde{L}}=\frac{2}{\lambda_{max}}\mathbf{L}-\mathbf{I}$
analogous to $\mathbf{\tilde{\Lambda}}$.

####  4.2.2. Spectral Convolution Layer

We define the spectral convolution layer with the approximated signed-directed
spectral convolution operation. We set $k$ as 1, the maximum polynomial order,
and $\lambda_{max}$ is assumed $2$ to make it practical. Similar to GCN(Kipf
and Welling, 2016), we set $\theta=\theta_{0}^{\prime}=-\theta_{1}^{\prime}$.
Then we have

$\mathbf{g}_{\theta^{\prime}}\ast\mathbf{x}\approx\theta(\mathbf{I}+(\mathbf{D}^{-\frac{1}{2}}_{s}\mathbf{A}_{s}\mathbf{D}^{-\frac{1}{2}}_{s})\odot\mathbf{P}^{q})\mathbf{x}.$

The spectral convolution layer is defined as

$\textbf{H}=(\mathbf{\tilde{D}}^{-\frac{1}{2}}_{s}\mathbf{\tilde{A}}_{s}\mathbf{\tilde{D}}^{-\frac{1}{2}}_{s}\odot\mathbf{P}^{q})\mathbf{X}\mathbf{W}.$

$\textbf{H}\in\mathbb{R}^{N\times F}$ is the convoluted graph signals or
representations. $\mathbf{X}\in\mathbb{R}^{N\times C}$ is the input signal.
$C$ and $F$ are the numbers of input and output channels.
$\textbf{W}\in\mathbb{R}^{C\times F}$ is a learnable matrix. It is a
renormalization trick that
$\mathbf{I}+(\mathbf{D}^{-\frac{1}{2}}_{s}\mathbf{A}_{s}\mathbf{D}^{-\frac{1}{2}}_{s})\odot\mathbf{P}^{q}\to\mathbf{\tilde{D}}^{-\frac{1}{2}}_{s}\mathbf{\tilde{A}}_{s}\mathbf{\tilde{D}}^{-\frac{1}{2}}_{s}\odot\mathbf{P}^{q}$
where $\mathbf{\tilde{A}_{s}}=\mathbf{A}_{s}+\mathbf{I}$ and
$\mathbf{\tilde{D}_{s}}(i,i)=\sum_{j}{\mathbf{\tilde{A}_{s}}(i,j)}$. It
prevents gradient vanishing and exploding problems.

####  4.2.3. Signed-Directed Graph Encoder

The graph encoder stacks $L$ layers of proposed spectral convolution layer.
The $l$-th layer feature vector $\mathbf{x}^{(l)}$ is defined as,

$\mathbf{x}_{j}^{(l)}=\sigma(\sum_{i=1}^{F_{l-1}}\mathbf{Y}_{ij}^{(l)}\mathbf{x}_{i}^{(l-1)}+\mathbf{b}_{j}^{(l)}).$

We use a novel activation function for the complex elements. The activation
function $\sigma$ is an complex version of ReLU, $\sigma(z)=z$, if
$-\pi/2\leq\text{arg}(z)\leq\pi/2$, otherwise, $\sigma(z)=0$. The feature
matrix $\mathbf{X}^{(L)}$ has both real and imaginary values. Here we
introduce unwinding operation. It converts real and imaginary features into
the same domain.

$\mathbf{X}_{\text{unwind}}^{(L)}=[\text{real}(\mathbf{X}^{(L)})||\text{imag}(\mathbf{X}^{(L)})\otimes(-i)].$

We add a fully connected layer after unwinding to get the node representation.

$\textbf{Z}=\sigma(\mathbf{X}_{\text{unwind}}^{(L)}\mathbf{W}^{L+1}+\mathbf{B}^{(L+1)})$

$\textbf{Z}\in\mathbb{R}^{N\times D}$ is the output node representation. The
augmented two graph views are fed into the spectral graph encoder and makes
the augmented node representations.

###  4.3. Contrastive Objective

There are two graph views after perturbations, and the graph encoder makes
node representations of them $\textbf{Z}^{1}$ and $\textbf{Z}^{2}$. The goal
of the contrastive objective is that the embeddings of the same nodes agree
with each other. At the same time, they distinguish from the other nodes. We
apply a MLP projection layer to improve the discriminative power.(Jacovi et
al., 2021; Chen et al., 2020b) We define the contrastive losses with the
projected representation M.

####  4.3.1. Inter-view Loss

It is considered that the two identical nodes from different views are inter-
positive pairs. On the other hand, the rest of the nodes are considered inter-
negative pairs. For example, a node $u$ from graph view 1 and another node $u$
from graph view 2 are inter-positive pairs. And others nodes $v\in\text{V}$
and $v\neq u$ from graph view 2 are inter-negative pairs with a node $u$ from
graph view 1. Even though the augmented representations from different graph
views are different, they represent the same nodes. We want to maximize the
agreement of the positive pair representations $\textbf{m}^{v_{1}}_{u}$ and
$\textbf{m}^{v_{2}}_{u}$. While minimize the agreements of the negative pairs
$\textbf{m}^{v_{1}}_{u}$ and $\textbf{m}^{v_{2}}_{v}$. The goal of the inter-
view objective is to maximize the similarity of positive pairs and minimize
the negative pairs.

$\mathcal{L}_{inter}=-\frac{1}{N}\sum^{N}_{i=i}\text{log}\frac{\text{exp}(sim(\textbf{m}^{v_{1}}_{i},\textbf{m}^{v_{2}}_{i})/\tau)}{\sum^{N}_{j=1,j\neq
i}\text{exp}(sim(\textbf{m}^{v_{1}}_{i},\textbf{m}^{v_{2}}_{j})/\tau)}$

$sim$ is a cosine similarity operation, and $\tau$ is a temperature parameter.

####  4.3.2. Intra-view Loss

We define intra-view loss as similar to inter-view loss. The inter-view loss
compares the projected node representations between the two different graph
views. In contrast, inter-view loss calculates the discriminative loss within
a graph view. Each node in a graph view is a unique user. All nodes have their
representations, and we need to distinguish them from others. Therefore, we
define inter-view loss as,

$\mathcal{L}_{intra}=-\frac{1}{N}\sum^{N}_{i=i}\text{log}\frac{1}{\sum^{N}_{j=1,j\neq
i}\text{exp}(sim(\textbf{m}^{v}_{i},\textbf{m}^{v}_{j})/\tau)}.$

####  4.3.3. Contrastive Loss

The whole contrastive loss is the sum of the inter- and intra-view loss
functions. The goal of the contrastive loss is to let the model learn
discriminative power from the projected node representations and maximize the
positive agreements/mutual information.

$\mathcal{L}_{contrastive}=\mathcal{L}_{inter}+\mathcal{L}_{intra}$

###  4.4. Prediction and Label Loss

For model training, we not only use contrastive loss but also utilize label
loss. The augmented two graph views are inputs of graph encoder and makes two
node representations $\textbf{Z}^{1}$ and $\textbf{Z}^{2}$. The
representations are concatenated and fed into output layer. The output layer
makes the final node representation.

$\textbf{R}=\sigma([\textbf{Z}^{1}\parallel\textbf{Z}^{2}]\textbf{W}^{out}+\textbf{B}^{out})$

$\textbf{R}\subseteq V\times d$ and $\textbf{W}^{out}\subseteq 2d\times d$
where $d$ is embedding dimension. We make prediction results with this output
node representation. The prediction layer is defined as,

$\hat{y}_{u,v}=\sigma([\textbf{r}^{1}_{u}||\textbf{r}^{2}_{v}]{\textbf{W}}^{pred}+\textbf{B}^{pred})$

Prediction value estimates the link sign when there is an edge from node $u$
to $v$. We define the label loss with the prediction error.

$\mathcal{L}_{label}=-\sum_{{u,v}\in\mathcal{E}^{+}}^{|\mathcal{E}^{+}|}y_{u,v}\text{log}\hat{y}_{u,v}-\sum_{{u,v}\in\mathcal{E}^{-}}^{|\mathcal{E}^{-}|}(1-y_{u,v})\text{log}(1-\hat{y}_{u,v})$

SDGCL is trained with the following objective function,

$\mathcal{L}=\alpha\times\mathcal{L}_{contrastive}+\mathcal{L}_{label}.$

$\alpha$ is weight of contrastive loss. Note that, unlike other graph
contrastive studies, it leverages supervised labels.

##  5\. Experiments

###  5.1. Datasets and Metrics

We evaluate the model with four real-world signed directed graph datasets
widely used in the link sign prediction task. Bitcoin-Alpha111 http://www.btc-
alpha.com and Bitcoin-OTC222http://www.bitcoin-otc.com(Kumar et al., 2016) are
extracted from Bitcoin trading platforms. Nodes are users, and edges are user
relationships. Users can score the others on a scale of -10 to +10. Edges
higher than 0 are treated as positive edges, otherwise negative edges.
Epinions333http://www.epinions.com(Guha et al., 2004) is a who-trust-whom
network crawled from a consumer review site. Users can notate trust or
distrust to reviews of other users. Slashdot444http://www.slashdot.com(Kunegis
et al., 2009) is a social network of user community site. Especially they
share new information. Users tag others as friends or foes, and we can
construct positive and negative edges with this information. Moreover, we
adopt four metrics, AUC, macro-F1, micro-F1, and binary-F1, for unbiased
evaluation.

Dataset | # nodes | # pos links | # neg links | positive ratio
---|---|---|---|---
Bitcoin-Alpha | 3,783 | 22,650 | 1,536 | 0.937
Bitcoin-OTC | 5,881 | 32,029 | 3,563 | 0.900
Epinions | 131,828 | 717,667 | 123,705 | 0.853
Slashdot | 82,144 | 425,072 | 124,130 | 0.774
|  |  |  |

Table 1. Dataset statistics.

|  | Signed Convolution | Contrastive Learning | Signed Contrastive Learning
---|---|---|---|---
Dataset | Metric | SGCN | SDGNN | SDGCN | GraphCL | GCA | SimGRACE | SGCL | SDGCL-s | SDGCL-l | SDGCL
Bitcoin-Alpha | AUC | 0.782 | 0.835 | 0.858 | 0.814 | 0.838 | 0.823 | 0.849 | 0.896 | 0.883 | 0.886
| Macro-F1 | 0.668 | 0.683 | 0.723 | 0.653 | 0.671 | 0.657 | 0.712 | 0.740 | 0.744 | 0.754
| Micro-F1 | 0.899 | 0.909 | 0.923 | 0.907 | 0.913 | 0.919 | 0.923 | 0.947 | 0.942 | 0.949
| Binary-F1 | 0.941 | 0.947 | 0.958 | 0.950 | 0.953 | 0.957 | 0.959 | 0.973 | 0.969 | 0.971
Bitcoin-OTC | AUC | 0.832 | 0.879 | 0.887 | 0.852 | 0.868 | 0.859 | 0.893 | 0.914 | 0.902 | 0.910
Macro-F1 | 0.710 | 0.751 | 0.773 | 0.725 | 0.743 | 0.725 | 0.781 | 0.803 | 0.796 | 0.802
Micro-F1 | 0.886 | 0.902 | 0.911 | 0.904 | 0.907 | 0.906 | 0.920 | 0.935 | 0.930 | 0.937
Binary-F1 | 0.924 | 0.938 | 0.950 | 0.948 | 0.948 | 0.948 | 0.956 | 0.964 | 0.962 | 0.965
Epinions | AUC | 0.848 | 0.914 | 0.939 | 0.839 | 0.911 | 0.913 | 0.876 | 0.941 | 0.943 | 0.942
Macro-F1 | 0.741 | 0.831 | 0.850 | 0.726 | 0.814 | 0.812 | 0.798 | 0.861 | 0.865 | 0.863
Micro-F1 | 0.893 | 0.912 | 0.925 | 0.887 | 0.913 | 0.915 | 0.909 | 0.934 | 0.936 | 0.936
Binary-F1 | 0.937 | 0.944 | 0.956 | 0.936 | 0.950 | 0.951 | 0.948 | 0.962 | 0.963 | 0.963
Slashdot | AUC | 0.740 | 0.849 | 0.886 | 0.813 | 0.870 | 0.865 | 0.783 | 0.900 | 0.891 | 0.902
Macro-F1 | 0.688 | 0.729 | 0.780 | 0.667 | 0.750 | 0.745 | 0.683 | 0.792 | 0.785 | 0.789
Micro-F1 | 0.786 | 0.823 | 0.855 | 0.813 | 0.842 | 0.833 | 0.811 | 0.864 | 0.859 | 0.863
Binary-F1 | 0.869 | 0.889 | 0.908 | 0.887 | 0.902 | 0.895 | 0.884 | 0.915 | 0.911 | 0.914
|  |  |  |  |  |  |  |  |  |  |

Table 2. Link sign prediction performance. Bold and underline indicate the
best and the second performance respectively. The performances are the average
score of 10 experiments with different seed sets.

Table 1 shows the statistics of the datasets. BitCoin datasets have relatively
small size of nodes and edges. The positive and negative ratios are highly
imbalanced. Three of the datasets have nearly 90% of positives among the
edges. Otherwise, Slashdot has a 23% share of negative edges. The four
datasets have various graph sizes and positive ratios. It is suitable for
unbiased evaluation.

###  5.2. Baselines

We implemented seven baselines to compare the model performance. There are
three signed graph convolutions, three constative learnings, and one signed
graph contrastive model.

  * •

SGCN(Derr et al., 2018) defines (un)balanced path based on the balanced theory
for neighbor aggregation. It is the first signed convolution model but does
not consider the edge directions.

  * •

SDGNN(Huang et al., 2021) proposed four weight matrices to aggregate the
neighbor features according to the edge types adaptively.

  * •

SDGCN(Ko, 2022) is the first spectral convolution for signed-directed graphs.
They proposed a signed magnetic Laplacian to overcome the disadvantage of
traditional graph Laplacian.

  * •

GraphCL(You et al., 2020) is a graph contrastive model with node and edge
augmentations. They randomly perturbed graph structures by dropping or adding
edges and nodes.

  * •

GCA(Zhu et al., 2021) proposed score-based graph augmentation methods and
introduced novel node-level contrastive objectives.

  * •

SimGRACE(Xia et al., 2022) tried to overcome the cumbersome augmentation
search. They introduced a novel graph encoder perturbation rather than graph
augmentation.

  * •

SGCL(Shu et al., 2021) is a graph contrastive model for signed-directed
graphs. It perturbs the edge sign and directions to get positive and negative
graph views. Each view makes node representations and defines a node-level
contrastive loss.

For a fair comparison, all the baselines were implemented under the same
environments, such as embedding dimension, learning epochs, and convolution
layers. Though the graph contrastive baselines are designed for self-
supervised learning, we train them with the same supervised loss. Moreover, we
removed the read-out process of GraphCL and SimGRACE, which are designed for
graph embedding.

Figure 4. Structure perturbing analysis. x-axis indicates perturbing ratio.
Figure 5. Laplacian perturbing analysis. x-axis indicates noise variance.

###  5.3. Implementation Details

We follow the hyperparameter settings of the original papers of each model.
The node embedding dimension is set to 64 for all the baselines to make the
same learning capacity. The edges are split into 60:20:20 for training,
validation, and test sets. However, we did not use all the positive edges as
training instances during the training stage. We sampled positive edges at the
ratio of 3:1. It is because the ratio of the positive edge is too high. If we
use all positive edges for training, the model would be easy to make positives
only. Nevertheless, we utilize all edge instances for the validation and test
phase. The structure perturbing ratio $p$ and $r$ are set to 0.1 for all
datasets. And the magnetic Laplacian phase $q$ is randomly selected from [0,
0.1$\pi$, 0.2$\pi$, 0.3$\pi$, 0.4$\pi$] for every iterations. The influence of
the perturbing ratio and $q$ variation is analyzed in Figure 5. The
contrastive loss weight $\alpha$ is set to 0.2. Graph encoder stacks two
signed-directed spectral convolution layers. We use Adam optimizer with
learning rate = 0.001, weight decay = 0.001. All experiments are run 10 times
with different seed sets to avoid randomness and get the average score. The
experiments are conducted on Xeon E5-2660 v4 and accelerated via Nvidia Titan
XP 12G GPU. The software is implemented via Ubuntu v16.4 with python v3.6 and
Pytorch v1.8.0.

##  6\. Results

###  6.1. Link Sign Prediction

We summarize the prediction results in Table 2. There are three proposed
models. SDGCL-s is a model only with structure augmentation, and SDGCL-l is a
model with Laplacian augmentation. SDGCL is the one that adopts both
augmentations. Consequently, the proposed SDGCL and its variants always show
the best performance in all datasets and all metrics. For Bitcoin-Alpha,
Bitcoin-OTC, and Slashdot datasets, SDGCL-composite and SDGCL-structure have
the highest score alternatively. SDGCL-composite and SDGCL-Laplacian get the
highest score in the Epinions dataset.
SGCL or SDGCN shows the best performance among the baselines. SDGCN could
achieve better performance than others thanks to the signed-directed spectral
convolution. They could fully enjoy the sign and direction information. SGCL
is the only model for signed graph contrastive. They could improve the
performance via a contrastive learning mechanism than other baselines.
However, SGCL shows poor performance on Epinions and Slashdot. We think there
are two reasons. First, the advantage of contrastive learning is maximized
with scarce labeled datasets. That is why the model can learn more from the
contrastive loss. However, if the label is rich enough, the utility of
contrastive learning is lessened. Second, we ruin the original dataset context
by random perturbation of SGCL. Epinions and Slashdot have larger sizes of
edges. The hidden patterns and joint distributions of edges are far more
complicated. We harm the original data information by perturbing many edges
and splitting them into different graph views. Unlike SGCL, our proposed model
minimizes the structure perturbation by proposing Laplacian augmentation.
Furthermore, we do not split positive and negative edges in different graph
views.

###  6.2. Perturbing Analysis

Here we analyze the effect of structure and Laplacian augmentation. Figure 4
shows that the performance varies according to the edge perturbing ratio.
Laplacian perturbation is not adopted in this experiment to get the effect of
edge perturbing. When the perturbing ratios are zero, AUC and Macro-F1 scores
are low. We can see the importance of structure perturbing. The performances
go up with a small perturbing ratio and go down with a large value. Figure 5
shows the performances of magnetic Laplacian phase $q$ variation. We set
default $q$ as 0.1$\pi$ and add some Gaussian noise with standard deviation.
At the same time, the edge perturbing ratios are set to zeros. The x-axis
shows the standard deviation values. We cap the maximum value of $q$ to
$0.5\pi$. Similar to structure perturbation, zero standard deviations show low
performances. Zero standard deviation means there is no Laplacian
perturbation. They have similar trends that go up and go down. However,
Laplacian perturbation is not as sensitive as structure perturbation. Though
graph augmentations are compulsory in contrastive learning, much perturbation
is detrimental to training. Structure perturbation gives direct contrastive
information while it may lose the data context. Laplacian perturbation gives
indirect perturbation of graph data but is still effective. SDGCL combines
these two augmentations for better and more stable graph augmentation.

###  6.3. Ablation Study

Table 3 shows the AUC scores of ablation studies. We check the effects of
SDGCL components. w/o Laplacian, w/o structure, and w/o augmentation are the
variation of graph augmentations. Especially, w/o augmentation shows the
lowest performance. As we expected, the model leverages the advantage of
contrastive learning. It shows that augmentations are important in our model.
w/o contrastive loss is a model with $\alpha=0$ and the model uses label loss
only. Note that it does not mean that the model does not utilize the benefits
of contrastive learning. Even though the contrastive loss weight is zero,
label loss is calculated with augmented representations of graph views. Thus,
the model is still leveraging the contrastive effects. w/o projection is a
model without a projection layer. It is well known that the projection layer
is useful for robust contrastive learning(Jacovi et al., 2021; Chen et al.,
2020b).

| Bitcoin-Alpha | Bitcoin-OTC | Epinions | Slashdot
---|---|---|---|---
SDGCL | 0.886 | 0.910 | 0.942 | 0.902
w/o structure aug | 0.883 | 0.902 | 0.943 | 0.891
w/o Laplacian aug | 0.896 | 0.914 | 0.941 | 0.900
w/o augmentation | 0.846 | 0.889 | 0.936 | 0.884
w/o contrastive loss | 0.878 | 0.901 | 0.941 | 0.891
w/o projection | 0.872 | 0.904 | 0.939 | 0.897
|  |  |  |

Table 3. Ablation Study

##  7\. Conclusion

This paper proposed SDGCL, a signed-directed graph contrastive learning model.
To make augmented node representations, we introduced two levels of the
perturbation process. Structure perturbation randomly changes the signs and
directions of edges. Even though it may lose some vital information from the
original graphs, we check that it makes the model noise-robust. Laplacian
perturbation changes the phase parameter $q$ for every training iteration to
give perturbation. It does not directly influence the graph information but
perturbs the Laplacian matrix. Then we define the inter- and intra-view
contrastive objectives with the augmented node representations. Unlike other
signed graph studies, our model does not depend on social theories or
assumptions. Also, there are no arbitrary graph-splitting processes. Our graph
encoder is based on the Laplacian matrix; it only enjoys the structural
information. Therefore it could reduce computational costs. We evaluate the
proposed model with four real-world datasets and seven baselines. The
experimental results show that our proposed model has superior performance to
others. SDGCL and their variants ranked at the top for every dataset and
metric.

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# Face-on Map of the Molecular Disc and 3-kpc Expanding Ring of the Galaxy
based on a High-Accuracy Rotation Curve

Yoshiaki Sofue
Institute of Astronomy  The University of Tokyo  Mitaka  Tokyo 186-0015  Japan

(Accepted; Received YYY; in original form; 2023)

###### Abstract

We analyze the longitude-velocity diagram (LVD) of 12CO line emission from
archival data and use the most accurate rotation curve (RC) of the Milky Way
to transform radial velocity to face-on position in the galactic plane. We
point out that the face-on transformation is highly sensitive to the adopted
RC, especially in the inner Milky Way, in the sense that deviations of the RC
from the true rotation velocity lead either to an artifact hole or overcrowded
concentration along the tangent circle for over- or under-estimated RC,
respectively. Even if the RC is sufficiently accurate, non-circular motion
such as with the 3 kpc expanding ring introduces significant artifacts in the
resulting face-on-map, as long as a circular rotation is assumed. On the other
hand, if we properly take into account the non-circular motion, it can be used
to solve the near-far degeneracy problem of determination of kinematic
distance. We thus propose a new method to solve the degeneracy by
incorporating the expanding motion of a ring or arms. We apply the method to
the LVD of the 3-kpc expanding ring and present its face-on map projected onto
the galactic plane for the first time.

###### keywords:

ISM: molecules — Galaxy: centre — Galaxy: disc — Galaxy: kinematics and
dynamics — radio lines: ISM

##  1 Introduction

The 3-kpc expanding ring of interstellar gas in the Milky Way is recognized as
a tilted elliptical feature in the longitude-velocity diagram (LVD) of the HI
and CO line emissions in the Galactic plane (Shane, 1972; Sanders &
Prendergast, 1974; Oort, 1977; Dame & Thaddeus, 2008). The oval LV feature has
been attributed to arm or ring structure formed either by an expanding motion
of a shock front produced by an explosive event in the Galactic Center
(Sanders & Prendergast, 1974; Sofue, 1976, 1977) or by non-circular motion of
gas in an oval potential of a stellar bar (Fux, 1999). Due to the highly non-
circular velocities, the kinematic method to determine the distance by
transformation of the radial velocity cannot be applied to this particular
structure.

Since the pioneering work by Oort et al. (1958), there have been extensive
studies for mapping the face-on structure of the Milky Way using the radial
velocities of the HI and molecular gases on the assumption of circular
rotation (hereafter, ”face-on transformation, or FOT”) (Burton & te Lintel
Hekkert, 1986; Nakanishi & Sofue, 2003, 2006; Kalberla et al., 2007; Nakanishi
& Sofue, 2016; Sofue & Nakanishi, 2016). Because these works have aimed at
mapping the global spiral structure in the entire Galaxy, the inner arm
structures have not necessarily been resolved well, although the molecular map
can trace the 4-kpc molecular ring (Nakanishi & Sofue, 2016; Sofue &
Nakanishi, 2016).

Recently, Fujita et al. (2023) have analyzed the molecular gas distribution in
the inner Galaxy between $l=10^{\circ}$ and $62^{\circ}$ using the high-
resolution 12CO -line data from FUGIN (Four-receiver system Unbiased Galactic
plane Imaging survey with the Nobeyama 45-m telescope)(Umemoto et al., 2017).
By applying a machine-learning near-far distinguishing method of molecular
clouds, they have obtained a high-resolution face-on map of the molecular gas
using FOT for a flat circular rotation. They have identified several inner
spiral arms, and noticed an artificial hole in the central region. Besides the
identified arms, an arm-like structure is evident in their face-on map, which
apparently extends from the Scutum arm, crosses the far-side solar circle at
$l\sim 12^{\circ}$, and extends beyond the circle, composing a massive
”leading” spiral structure.

In this paper, we seek for the reason to produce such irregular structures
often appearing in the current face-on maps of the Milky Way by considering
the non-circular motion associated with the 3-kpc expanding arm. We further
derive a face-on distribution of the molecular gas in the Galactic plane by
analyzing the archival 12CO line data.

Figure 1: [Top] LVD of 12CO -line $T_{\rm B}$ in K at $b=0^{\circ}$ from
Columbia survey (Dame et al., 2001). [Middle] Same, but from $l=-10^{\circ}$
to $+50^{\circ}$. [Bottom] LVD from FUGIN with 45-m telescope in the same
longitude range as above (Umemoto et al., 2017) and GC surveys (Tokuyama et
al., 2019). All diagrams are smoothed, not presenting the original
resolutions.

##  2 Longitude-velocity diagrams

We make use of the archival 12CO line data cubes from the Columbia and FUGIN
surveys. The radial velocity $V_{\rm lsr}$ used in this paper is the LSR
(local standard of rest) velocity. The LVD of the entire disc is taken from
the Columbia survey (Dame et al., 2001), which had angular resolution of
$9^{\prime}$ and velocity coverage of $\pm 300$ km s-1, and the used LVD at
$b=0^{\circ}$ had $2881\times 493$ pixel sizes in the longitude and velocity
directions, which was resized to a $1000\times 1000$ pixel sized LVD, having
$0^{\circ}.36$ and 0.6 km s-1pixel sizes.

LVDs from FUGIN surveys integrated between $b=-1^{\circ}$ and $+1^{\circ}$
have been published, in which various known galactic structures and molecular
clouds have been identified (Torii et al., 2019; Kohno et al., 2021). An LVD
along the Galactic plane at $b=0^{\circ}$ has been published by Sofue (2021)
in order to derive the rotation curve. In this paper, we use FOT to cover the
entire galactic disk, including the distant region outside the solar circle
$\sim 20$ kpc away. Since the integrated LVD in the $b$ direction weights near
and far clouds equally, not only does the face-on map give more weight to the
far side disc, but also the treated disc thickness as a function of distance
also increases. To avoid this inconvenience, and for the convenience of data
processing, we use the LVD at $b=0^{\circ}$ here. This eliminates the need to
consider the height distribution of the gas, but does not allow us to discuss
the 3D structure of the disk.

The full beam width at half maximum of the 45-m telescope was
$15^{\prime\prime}$ at the 12CO ($J=1-0$)-line frequency, and the velocity
coverage and resolution were $\pm 250$ km s-1and 1.3 km s-1, respectively, and
the rms noise levels were $\sim 1$ K. The original pixel size was $(\Delta
l,\Delta b,\Delta V_{\rm lsr})=$ (8”.5, 8”.5, 0.65 km s-1). The FUGIN LVD is
then combined with that in the Galactic Centre between $l=-1^{\circ}$ and
$1^{\circ}.4$ with the same angular and spectral resolutions from the GC CO
line survey (Tokuyama et al., 2019). The thus combined 45-m LVD was resized to
$1000\times 1000$ pixel sized LVD between $l=-10^{\circ}$ and $+50^{\circ}$,
having $0^{\circ}.06$ and 0.5 km s-1pixel sizes. Thus obtained LVDs are shown
in Fig. 1, and are used individually to obtain face-on maps using FOT.

The resulting face-on maps from the three data sets (Columbia, FUGIN and GC)
will be merged on the $(X,Y)$ plane, where $X$ and $Y$ are the Cartesian
coordinate axes with the origin at the Galactic Centre and are positive toward
$l=90^{\circ}$ direction and toward the Sun, respectively. The distance to the
Galactic center from the Sun ($R_{0}$) is assumed to be $R_{0}=8.0$ kpc (VERA
Collaboration et al., 2020; Reid et al., 2019).

##  3 Rotation curve

The rotation curve (RC) is the most important and sensitive parameter for the
conversion of radial velocity to kinematic distance. As mentioned in the
previous section, the current studies have used fairly simple analytical
models that approximate the overall behavior of the observed rotation
velocities. While acceptable, thought not accurate enough, in the mid to outer
disc, the currently adopted rotation curves do not necessarily represent the
Galactic rotation with sufficient accuracy in the inner disc and the Galactic
Centre (GC) region.

Deviations between the assumed RC and the actual rotation speed will
artificially increase or decrease the calculated density. If the model RC is
underestimated compared to the true velocity, FOT produces an overestimated
density near the tangent circle. This occurs because gas observed at higher
velocities than the model collects gas from the surrounding region, and,
furthermore, gas that does not satisfy the equation is ”abandoned” (forgotten)
from solution. This means that the total mass of the gas is not conserved
during the FOT.

Conversely, if the RC is overestimated, the gas is ”swept” away from the right
position to the surrounding area and the density is underestimated. As the
consequence it creates an artificial ”hole” on the tangent circle. In fact, a
flat rotation curve fixed at $V=240$ km s-1, about $R\sim 40$ km s-1faster
than the actual rotation at $R\sim 2-4$ kpc, has yielded an artificial hole
along the tangent circle (Fujita et al., 2023).

To avoid or minimize such artifacts, we adopt here the most accurate rotation
curve obtained so far for the Milky Way. Figure 2 shows the rotation curve,
where the black dots represent the internal rotation curve obtained by
terminal velocity fitting using the FUGIN CO-line LVD (Sofue, 2021). Triangles
are taken from the grand rotation curve from the central black hole of the GC
to the galactic outskirts, which was constructed by compiling currently
published RC data and fitting to the central CO line LVDs (Sofue, 2013). The
red line shows the empirically fitted curve adopted for the FOT in this paper.

Figure 2: Rotation curves from FUGIN (black dots: (Sofue, 2021)), from
compilation for the entire Galaxy from the nucleus to the halo (triangles:
(Sofue, 2013), and a model curve by the red line used for the distance
determination which approximately fits the observations.

##  4 Face-on map for circular rotation by FOT-C

We first apply the face-on map transformation assuming the circular rotation,
which we call the FOT-C method. The radial velocity is related to the rotation
velocity by

$v_{\rm r}=\left(V_{\rm rot}\frac{R_{0}}{R}-V_{0}\right)\sin\ l.$ (1)

The distance $s$ of the object from the Sun is given by

$s=R_{0}\cos\ l\pm\sqrt{R^{2}-R_{0}^{2}\sin^{2}l}.$ (2)

The $+$ and $-$ signs stand for the far- and near-side solutions,
respectively. At the distance $s$, the volume density of the H2 gas is
calculated by

$n_{\rm H_{2}}=dN_{\rm H_{2}}/ds=X_{\rm CO}T_{\rm B}dv/ds,$ (3)

where $N_{\rm H_{2}}=X_{\rm CO}\int T_{\rm B}dv$, $T_{\rm B}$ is the 12CO
brightness temperature, $X_{\rm CO}=2\times 10^{20}$ H2 cm -2 [K km
s${}^{-1}]^{-1}$is the CO-to-H2 conversion factor, and the velocity gradient
along the line of sight, $dv/ds$, is calculated by

$dv/ds=\sqrt{(dv_{1}/ds)^{2}+(dv_{2}/ds)^{2}},$ (4)

where

$\frac{dv_{1}}{ds}=\left|\frac{R_{0}}{R}\left(\frac{dV_{\rm
rot}}{dR}-\frac{V_{\rm rot}}{R}\right)\sin\ l~{}\cos\ (l+\theta)\right|$ (5)

is the gradient due to Galactic rotation, and

$dv_{2}/ds=v_{\sigma}/\Delta s\sim 5\ {\rm km\ s^{-1}kpc^{-1}}$ (6)

is the internal gradient due to turbulent motion of the gas of the order of
$v_{\sigma}\sim 5$ km s-1which gives the minimum value of gradient along each
line of sight. Here, $R$ and $R_{0}=8$ kpc are the Galacto-centric radius and
solar distance from the GC, respectively, $l$ and $\theta$ are the Galactic
longitude and Galacto-centric longitude, $V_{\rm rot}$ and $V_{0}=238$ km
s-1are the rotation velocity at $R$ and $R_{0}$, as explained in Fig. 3.

Figure 3: Illustration of the velocities in rotation and expansion, and LVD
from Columbia survey (Dame et al., 2001) with explanation of the structures
discussed in this paper.

Figure 4 shows the face-on maps of the volume density of H2 gas obtained by
FOT-C using the rotation curve in Fig. 2 and LVDs in Fig. 1 under an
assumption of pure circular rotation. The results from the both surveys are
merged according to their longitudinal coverage. The near and far emissions
are duplicated in this map, so that the map is not appropriate to discuss such
non-axisymmetric structure like spiral arms. The circular assumption results
in such a duplicated artifact of the nearby features such as the sharp ring-
like arm along the far-side solar circle, running coherently from $l\sim
20^{\circ}$ to $\sim 50^{\circ}$, which is the consequence of erroneous
location of the structure associated with the Aquila Rift. However, it still
provides with some basic information about the axi-symmetric structures such
as the 4-kpc molecular ring and some arms as identified in the earlier works
(Reid et al., 2016).

Due to the high accuracy of the rotation curve in the 1st and 4th quadrant
(northern disc of the Milky Way), the artifact ”hole” does not appear any more
on the tangent circle in the right side of the GC in this map. However, hole-
like regions still remains in the left side in the 2nd and 3rd quadrants
(southern disc). Such lopsidedness of the holes are due to the asymmetric
rotation curves between the northern and southern Milky Way, which are
represented here by a single RC model for convenience.

Another notable artifact feature in this map is the two long bow-shaped arms
symmetric about the $Y$ axis, as indicated by the two dashed white lines. The
positive $X$ side bow runs towards $l\sim 20^{\circ}$ near the Sun, crosses
the 4 kpc molecular ring at $(X,Y)\sim(2,+4)$ kpc, returns almost parallel to
the $Y$ axis, crosses the 4kpc ring in the far side again at $(3,-4)$ kpc, and
further extends to $(2,-9)$ kpc, finally crossing the distant solar circle.
This bow feature is also visible in the map by Fujita et al. (2023) as a
massive ”leading arm” across the circle beyond the solar circle. The bow on
the negative $X$ side is also traced symmetrically.

Such a bow structure extending for $\sim 10-20$ kpc across the inner disc and
the solar circle is difficult to explain by any astrophysical mechanism in the
Galactic disc. The bow cannot be removed even using the sophisticated near-far
deconstruction procedure, as demonstrated in the recent detailed study using
the Nobeyama high-resolution data (Fujita et al., 2023). In the next section,
we argue that the bow structures are artifacts due to the assumption of
circular rotation applied to the 3-kpc expanding ring in a highly non-circular
motion.

We also mention that the narrow fan region around the $Y$ axis (Sun-GC line)
filled with straight features extending from the Sun in the GC direction is
composed mainly of artifacts of strongly deformed central molecular zone (CMZ)
due to the insufficient resolution and too much crowded equal-velocity lines
(Fig. 5). So, this central fan region is out of the analysis hereafter.

Figure 4: [Top left] FOT-C maps of H2 density in H2 cm-3 from the Columbia and
[top right] Nobeyama CO-line LV diagrams for circular rotation. Bottom panel
shows composite, where the region of panel (a) which is observed at Nobeyama
has been replaced with map (b). Near and far emissions are duplicated in these
maps. The white dashed lines trace artifact ”bows” corresponding to the 3-kpc
expanding ring caused by the assumption of circular rotation. The far-side
massive ”leading” arm is due to the (erroneous) location of the ”forbidden” LV
belt (Fig. 3), which is actually the near-side expanding (approaching) ring.

##  5 Expanding ring with near-far separation by FOT-Ex

In addition to the accuracy of rotation curve, non-circular motion also
affects the face-on transformation. In this section we propose a new method to
solve the degeneracy of near-far distances incorporating the expanding motion,
which we call the FOT-Ex method. The radial velocity $v_{\rm r}$ superposed by
an expanding motion of $V_{\rm e}$ (positive outward) is given by

$v_{\rm r}=\left(V_{\rm rot}\frac{R_{0}}{R}-V_{0}\right)\sin\ l\pm V_{\rm
e}^{0}\cos\ (l+\theta).$ (7)

In Fig. 5 we show the distribution of $v_{\rm r}$, or a velocity field, in the
Galactic plane in the $(X,Y)$ coordinates (Fig. 3) as calculated for a flat
rotation curve superposed by an expanding ring of radius 3 kpc, width 1 kpc,
and expanding velocity $V_{\rm e}=50$ km s-1. The velocity field is
significantly deformed from the general symmetric butterfly pattern with
respect to the Sun-GC line ($Y$ axis).

The complex velocity field shows that FOT-C, assuming simple circular
rotation, leads to highly deformed maps that are quite different from the
actual distribution. This complex behavior of the velocity field offers a
unique opportunity to resolve the degeneracy of near-far solutions on the same
line of sight. However, it is impractical to transform the measured LVD into a
face-on map without knowing the object’s motion, whether it follows a pure
circular rotation or a combination of circular and extensional motions. So we
perform a separation on the LV diagram, where the 3 kpc ring is recognized and
traced as a tilted LV ellipse as in Fig. 3.

(a)

Figure 5: (a) Radial velocity field of the Milky Way with an expanding ring of
a radius 3 kpc, and (b) close up in the inner region. The complicated velocity
field demonstrates that the radial-velocity to space transformation on an
assumption of simple circular rotation results in a considerably deformed map
from the true distribution.

The 3-kpc expanding ring shows up as a tilted ellipse in the LV diagrams in
Fig 6 as reproduced from the Columbia CO survey Dame et al. (2001). Panel (a)
shows the ellipse by a red oval, and (b) shows the LVD after subtracting the
emission along the ellipse. Panel (c) shows an LVD of the subtracted
elliptical component.

Figure 6: [Top] LVD at $b=0^{\circ}$ from Columbia CO survey (Dame et al.,
2001) at longitude within $\pm 50^{\circ}$ (green) superposed by a tilted
ellipse representing the 3-kpc expanding ring (red). [Middle] Same, but the LV
ellipse has been removed. [Bottom] Same, but only the elliptical region
representing the 3-kpc ring.

We first apply the simple FOT-C with circular rotation to the LVD in Fig. 6(b)
using Eq. (1), where the ellipse component (3 kpc ring) has been subtracted.
The result is shown in Fig 7(a), where the dark bows represent the eliminated
disc gas corresponding to the LV ellipse caused by the 3-kpc expanding ring.

We then apply FOT-Ex using Eq. (7) to the 3 kpc ring component represented by
the tilted LV ellipse in Fig. 6(c). This solves the degeneracy problem by
choosing the corresponding side of the ellipse depending on the approaching
(near) and the receding (far) arms. The results are shown in Fig. 7(b) for the
Columbia data and in (c) for the Nobeyama data. Figure 8 enlarges the
resulting face-on maps of the 3 kpc ring and adds descriptions of the visible
components. Considering the radial motion of the gas on the expanding ring,
the emission of the lower half of the tilted LV ellipse is assigned to the
near side in FOT-Ex, and the upper half to the far side. This separation
eliminates near-far degeneracy. On the other hand, the resulting map shows a
clear cut along the tangent circles, as seen in the resulting map as indicated
by the dashed half circle in Fig. 8.

Figure 7: [Top] Face-on projection of $n_{\rm H_{2}}$, where the LV ellipse
representing the 3-kpc expanding ring (Fig. 6) has been removed. [Middle]
3-kpc expanding ring corresponding to LV ellipse in Fig. 6 middle panel. Near-
far degeneracy has been resolved because of the expanding motion. [Bottom]
Same, but using Nobeyama 45-m data.  Figure 8:  [Top] Close up of the 3-kpc
expanding ring from Columbia survey. The clear-cut gap on the tangent point
circle is artifact. The triangular region near the $Y$ axis (Sun-GC line) is
ambiguous due to the $V_{\rm lsr}$ degeneracy and is avoided from the
analysis. Yellow circles are maser sources associated with the 3-kpc expanding
arms with known trigonometric distances (Reid et al., 2019), which lye on a
circle slightly ($\sim 0.5$ kpc) displaced from the molecular ring to the
direction of the Sun. [Bottom] Same, but using the 45-m CO data.

##  6 Discussion

###  6.1 Accuracy of the maps

The accuracy of the FOT-C method has been discussed in detail by Sofue (2011),
where the rotation curve has been assumed to be valid, while the measurement
of the radial velocity includes errors. In the present study, the accuracy of
velocity measurement of individual molecular clouds is on the order of the
resolution of the 12CO spectrometer, $\sim 1$ km s-1, sufficiently small
compared to the observed velocity $V_{\rm lsr}$ of $\sim 100$ km s-1.

The uncertainty of the FOT map in this study arises mainly from the
uncertainty of the assumed rotation velocity $V_{\rm rot}$ and expansion
velocity of the 3-kpc ring $V_{\rm exp}^{0}$. We here examine the effects of
these quantities, $\delta V_{\rm rot}$ and $\delta V_{\rm exp}^{0}$,
respectively, on the distance determination whose error (uncertainty) is
denoted by $\delta s$. Using equations 7 and 2, the uncertainties propagate to
$\delta s$ as follows111Small deviation of a multi variable function
$f(x_{i})$ due to small deviation $\delta x_{i}$ is assumed to be expressed by
$\delta f^{2}\simeq\Sigma_{i}\left(\frac{\partial f}{\partial x_{i}}\delta
x_{i}\right)^{2}$ .

$\delta s\simeq A\left[\left(\frac{\delta V_{\rm
rot}}{B}\right)^{2}+\left(\frac{V_{\rm rot}\delta V_{\rm
exp}^{0}}{B^{2}}\right)^{2}\right]^{1/2},$ (8)

where

$A=\frac{R\ {\rm sin}\ l}{\sqrt{(R/R_{0})^{2}-{\rm sin}^{2}l}}$ (9)

($R\geq R_{0}{\rm sin}\ l$) and

$B=v_{\rm r}-V_{0}\ {\rm sin}\ l\pm V_{\rm exp}^{0}{\rm cos}(l+\theta).$ (10)

Equation 9 indicates that the error (uncertainty) attains maximum along the
tangent-point circle with $R=R_{0}{\rm sin}\ l$, along which the result is
most sensitive to the rotation curve error. On the other hand, the effect of
the rotation curve is minimized as $\propto{\rm sin}\ l$ near the Sun-GC line,
where the radial velocity is degenerated to zero.

We then estimate a typical value of distance error (uncertainty) in the FOT
map in a representative region around $R\sim 3$ kpc, $l\sim 20^{\circ}$, and
$v_{\rm r}\sim 80$ km s-1, assuming that the rotation curve has uncertainty of
$\delta V_{\rm rot}=10$ km s-1, as read from Fig. 2.

First, we consider a case of purely circular rotation, so that the second term
of Eq. 8 is ignored, or $\delta s\sim A\delta V_{\rm rot}/B$. Then we obtain
$\delta s\sim 0.41$ kpc, which is proportional to the uncertainty $\delta
V_{\rm rot}$. If the rotation curve is sufficiently accurate with $\delta
V_{\rm rot}\lesssim 5$ km s-1, which is the case in the present study in the
1st and 2nd quadrants ($l=0^{\circ}$ to $90^{\circ}$) of the disc, where the
RC determination was made. The distance uncertainty is only $\delta s\sim 0.2$
kpc here, and the FOT map is reasonably accurate. However, in the 3rd and 4th
quadrants ($l=270^{\circ}$ to $360^{\circ}$), the RC may not be accurate
enough, which causes the hole-like dark region in the left side of the GC and
around $(X,Y)\sim(-4,+3)$ kpc in Fig. 4.

If we adopt a flat rotation curve which overestimates the rotation velocity by
$\delta V_{\rm rot}\sim+30$, the distance error is as large as $\delta s\sim
1.3$ km s-1, yielding significant under/over estimation of the distance in the
near/far side. This is the reason for the artifact ’hole’ in the FOT map in
the inner disc obtained for the flat rotation curve (Fujita et al., 2023),
where the gas has been swept away from the right position for $\pm 1.3$ kpc,
making a hole of diameter $\sim 2.6$ kpc.

We next consider the case including the expanding ring, which is assumed to
have expanding velocity of $V_{\rm exp}^{0}=50$ km s-1. The distance
uncertainty is estimated to be $\delta s\sim 0.55$ for $\delta V_{\rm
exp}^{0}\sim 2$ km s-1, and $0.76$ kpc for $\sim 4$, for the same uncertainty
of rotation curve $\delta V_{\rm rot}=10$ km s-1as above, while it varies
according to Eq. 8.

To summarize, the uncertainty of the distance determination by FOT-Ex is about
$\sim 0.5$ kpc for $\delta V_{\rm rot}\sim 10$ km s-1around the 3-kpc ring. It
increases toward the tangent circle attaining the maximum there, and is
minimum along the Sun-GC line. However, it must be recalled that the distance
determination has a maximum uncertainty along the Sun-GC line for another
reason that the radial velocity is degenerated to zero, so that the error due
to $\delta V_{\rm lsr}$ increases to infinity (Sofue, 2021). For this reason,
the plot near the Sun-GC line ($l\sim 0^{\circ}$) is avoided in Fig. 8.

###  6.2 Comparison with trigonometric parallax distances of star forming
regions associated with the 3-kpc expanding arms

There have been detected a number of maser sources along the tilted ellipse on
the LVD corresponding the 3-kpc expanding ring (Green et al., 2011). It is
reported that the trigonometric parallaxes of maser sources associated with
the star-forming region G9.62+0.20 correspond to a distance of $5.2\pm 0.6$
kpc, while its LSR velocity is $\sim 2$ km s-1(Sanna et al., 2009). This
velocity corresponds to kinematic distances of 0.5 or 16 kpc, if a circular
rotation is assumed, which contradicts the trigonometric distance. The
trigonometirc distance is consistent with the here derived distance of the
molecular 3-kpc arm at $l\sim 10^{\circ}$ in Fig. 8, where the LSR velocity is
$\sim 2$ km s-1. Therefore, we may reasonably consider that G9.62+0.20 is
associated with the 3-kpc expanding molecular ring.

Several more maser sources with trigonometric distances are located at $R\sim
3$ kpc ($s\sim 5$ kpc) associated with the near 3-kpc expanding arm, and one
source near the far-side arm (Reid et al., 2019), as plotted by yellow circles
in Fig. 8. For their circular alignment aong the 3-kpc ring the maser sources
are considered to be physically associated with the molecular ring.

However, a closer look at the figure suggests that the maser sources are
systematically displaced from the molecular ring toward the Sun by about $\sim
0.5$ kpc, whish is also demonstrated by an off-center circle fitted to the
sources by Reid et al. (2019). If we rely on the maser parallax distances and
consider that the sources are physically associated with the 3-kpc molecular
ring, we need a modification of the ring model, so that it produces an off-
center ring and asymmetric expansion. In fact, a slightly lopsided expansion
is observed as the faster expanding velocity of the far side ring at $\sim 60$
km s-1(Dame & Thaddeus, 2008) than the near side ring at 50 km s-1. However,
this pauses a puzzling problem why the galacto-centric radius of the far-side
arm is smaller ($\sim 2.5$ kpc) than the near side arm ($\sim 3.2-3.5$ kpc)
despite the faster expansion in the far side.

###  6.3 Expanding ring vs non-circular motion by a bar

The present result does not exclude the possibility that the expanding ”ring-
like” feature on the LVD is due to a non-circular motion induced by the oval
orbits in a barred potential (Binney et al., 1991; Athanassoula & Bureau,
1999; Li et al., 2022). The FOT-Ex method assumes that the 3-kpc ring is
expanding at $\sim 50$ km s-1. If the same amount of radial motion can be
assumed, any model, such as the bar model, would be able to lead to a similar
face-on map as here. However, the nearly perfect elliptical nature of the
3-kpc ring on the LVD without nodal crossing near $l\sim 0^{\circ}$ seems to
be in favor of the expanding ring model. The bar potential model predicts that
the arms on the LVD have either nodes around $l\sim 0$ drawing a tilted 8
shape (not oval) or a parallelogram structure, as often observed in barred
spiral galaxies including the 4-kpc molecular ring of the Milky Way. In
anyway, the present analysis is not accurate enough to distinguish a ring from
an ellipse. Namely, we cannot exclude the possibility that the 3-kpc expanding
”ring” is due to a bar potential, and, vise versa, the bar theory cannot rule
out the expanding ring model in the present accuracy.

##  7 Summary

We have shown that face-on transformations (FOT) of molecular-line radial
velocities yield maps showing circular structures representing molecular arms
when suitable rotation curves are used. However, when the model RC deviates
from the true rotational velocity, an artifact hole along the tangent circle
occurs for an overestimated RC, and anomalous gas concentrations for
underestimated RC.

Even if the RC is sufficiently accurate, the non-circular motion due to the 3
kpc expanding ring causes considerable artifacts in the resulting face-on-map
as long as circular rotation is assumed. Such artifacts are seen as giant
leading arms across the far side of the solar circle. This is a result of the
erroneous positioning of the nearby 3 kpc expanding ring, which includes the
”forbidden velocity” belt with negative velocity at $l\sim 0^{\circ}$ to
$+12^{\circ}$ as shown in Fig. 3. The approaching LV ridge is transformed to a
near-side 3kpc ring when we extract the lower half of the tilted elliptical LV
component and apply FOT-Ex considering the expanding motion. The heavy leading
arm on the far side, which resulted from the assumption of circular rotation,
has disappeared in this FOT-Ex map. The 3-kpc ring on the far-side (Dame &
Thaddeus, 2008) is also clearly visible on the face-on map as the opposite
ring beyond the tangent circle by the transformation of the upper half of the
LV ellipse.

Finally, we comment on the limitations and future prospects of this study.
This paper is the first to point out, mainly through the analysis of large-
scale LVDs in the Columbia survey, that the problem of near-far degeneracy
around the tangent circle can be solved by using non-circular motions of the
arm and ring. The LVD from Nobeyama’s high-resolution observations has been
smoothed to an angular resolution of $0^{\circ}.06$, so the original
resolution ($20^{\prime\prime}$) is not properly incorporated into the current
analysis. The Nobeyama CO LVD at the Galactic Centre is also significantly
smoothed and thus did not provide useful information about gas distribution in
the CMZ. Since we used LVD only in the galactic plane at $b=0^{\circ}$, we
were unable to touch on the vertical (3D) structure of the disk and ring.
Higher-resolution and 3D analyses of the Nobeyama LVDs inside the Milky Way
galaxy as well as in CMZ, which was avoided here as the artifact fan region,
would be a subject for future work.

## Acknowledgments

The author is indebted to the FUGIN team (Prof. T. Umemoto, et al.) for the CO
data observed with the Nobeyama 45-m telescope operated by the NAOJ (National
Astronomical Observatory of Japan). The data analysis was carried out on the
computer system at the Astronomy Data Center of the National Astronomical
Observatory of Japan. The author is indebted to the anonymous referee for the
valuable comments to improve the paper.

## Data availability

The FUGIN data were retrieved from the JVO portal at
http://jvo.nao.ac.jp/portal.

## Conflict of interest

The author declares that there is no conflict of interest.

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players. Each player $i$’s action space $\mathcal{A}^{i}$ is specified by the
5 shortest routes and any route that more than three times longer than the
shortest route is further removed from $\mathcal{A}^{i}$. To exploit the
correlations among actions $(a^{i},a^{-i})$ in the reward function, the
composite kernel proposed in (Sessa et al., 2019) is used. For player $i$, let
$a^{i},b^{i}\in\mathcal{A}^{i}$ and $a^{-i},b^{-i}\in\mathcal{A}^{-i}$, then
the kernel function associating $(a^{i},a^{-i})$ and $(b^{i},b^{-i})$ is

$K^{i}((a^{i},a^{-i}),(b^{i},b^{-i}))=k_{\rm L}(a^{i},b^{i})k_{\rm
P}(a^{i}+g(a^{-i}),b^{i}+g(b^{-i})),$

where $k_{\rm L}(\cdot,\cdot)$ and $k_{\rm P}(\cdot,\cdot)$ are linear and
polynomial kernels respectively. The hyperparameters of kernels are set the
same as in (Sessa et al., 2019). Details on GP for estimating reward functions
can be found in Section 2 of (Sessa et al., 2019) and we do not repeated them
here333 The implementation refers the code released by authors of (Sessa et
al., 2019) at https://github.com/sessap/stackelucb. Except the regret, the
congestion of edge $e$ is also included as a performance metric for a joint
action $\bm{a}$, which is computed as
$0.15(\sum_{i=1}^{N}[a^{i}]_{e}/C_{e})^{4}$. The averaged congestion is
obtained by averaging congestion over all edges.