# Position-aware Structure Learning for Graph Topology-imbalance by Relieving
Under-reaching and Over-squashing

Qingyun Sun  Beihang UniversityBeijingChina<EMAIL_ADDRESS>,  Jianxin Li
Beihang UniversityBeijingChina<EMAIL_ADDRESS>,  Haonan Yuan  Beihang
UniversityBeijingChina<EMAIL_ADDRESS>,  Xingcheng Fu  Beihang
UniversityBeijingChina<EMAIL_ADDRESS>,  Hao Peng  Beihang
UniversityBeijingChina<EMAIL_ADDRESS>,  Cheng Ji  Beihang
UniversityBeijingChina78229<EMAIL_ADDRESS>,  Qian Li  Beihang
UniversityBeijingChina<EMAIL_ADDRESS>and  Philip S. Yu  University of
Illinois ChicagoChicagoIL 60607USA<EMAIL_ADDRESS>
(2022)

###### Abstract.

Topology-imbalance is a graph-specific imbalance problem caused by the uneven
topology positions of labeled nodes, which significantly damages the
performance of GNNs. What topology-imbalance means and how to measure its
impact on graph learning remain under-explored. In this paper, we provide a
new understanding of topology-imbalance from a global view of the supervision
information distribution in terms of under-reaching and over-squashing, which
motivates two quantitative metrics as measurements. In light of our analysis,
we propose a novel position-aware graph structure learning framework named
PASTEL, which directly optimizes the information propagation path and solves
the topology-imbalance issue in essence. Our key insight is to enhance the
connectivity of nodes within the same class for more supervision information,
thereby relieving the under-reaching and over-squashing phenomena.
Specifically, we design an anchor-based position encoding mechanism, which
better incorporates relative topology position and enhances the intra-class
inductive bias by maximizing the label influence. We further propose a class-
wise conflict measure as the edge weights, which benefits the separation of
different node classes. Extensive experiments demonstrate the superior
potential and adaptability of PASTEL in enhancing GNNs’ power in different
data annotation scenarios.

graph representation learning, graph neural networks, imbalance learning,
graph structure learning, node classification

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
31st ACM International Conference on Information and Knowledge Management;
October 17–21, 2022; Atlanta, GA, USA††booktitle: Proceedings of the 31st ACM
International Conference on Information and Knowledge Management (CIKM ’22),
October 17–21, 2022, Atlanta, GA, USA††price: 15.00††doi:
10.1145/3511808.3557419††isbn: 978-1-4503-9236-5/22/10††ccs: Computing
methodologies Neural networks††ccs: Computing methodologies Learning latent
representations

##  1\. Introduction

Graph learning (Zhang et al., 2020; Li et al., 2022; Fu et al., 2021) has
gained popularity over the past years due to its versatility and success in
representing graph data across a wide range of domains (Nastase et al., 2015;
Sun et al., 2020; Gao et al., 2021; Yu et al., 2022; Chen et al., 2022). Graph
Neural Networks (GNNs) (Wu et al., 2020; Sun et al., 2021) have been the
“battle horse” of graph learning, which propagate the features on the graph by
exchanging information between neighbors in a message-passing paradigm (Gilmer
et al., 2017). Due to the asymmetric and uneven topology, learning on graphs
by GNNs suffers a specific imbalance problem, i.e., topology-imbalance.
Topology-imbalance (Chen et al., 2021) is caused by the uneven position
distribution of labeled nodes in the topology space, which is inevitable in
real-world applications due to data availability and the labeling costs. For
example, we may only have information for a small group of users within a
local community in social networks, resulting in a serious imbalance of
labeled node positions. The uneven position distribution of labeled nodes
leads to uneven information propagation, resulting in the poor quality of
learned representations.

Although the imbalance learning on graphs has attracted many research
interests in recent years, most of them focus on the class-imbalance issue
(Park et al., 2021; Wang et al., 2021), i.e., the imbalanced number of labeled
nodes of each class. The topology-imbalance issue is proposed recently and is
still under-explored. The only existing work, ReNode (Chen et al., 2021),
provides an understanding of the topology-imbalance issue from the perspective
of label propagation and proposes a sample re-weighting method. However,
ReNode takes the node topological boundaries as decision boundaries based on a
homophily assumption, which does not work with real-world graphs. The strong
assumption leads to poor generalization and unsatisfied performance of ReNode
(see Section 5.2.1). There are two remaining questions: (1) Why does topology-
imbalance affect the performance of graph representation learning? and (2)
What kind of graphs are susceptible to topology-imbalance? To answer the above
two questions, how to measure the influence of labeled nodes is the key
challenge in handling topology-imbalance due to the complex graph connections
and the unknown class labels for most nodes in the graph.

Figure 1. Schematic diagram of under-reaching and over-squashing in the
topology-imbalance issue.

New understanding for topology-imbalance.  In this work, we provide a new
understanding of the topology-imbalance issue from a global view of the
supervision information distribution in terms of under-reaching and over-
squashing: (1) Under-reaching: the influence of labeled nodes decays with the
topology distance (Buchnik and Cohen, 2018), resulting in the nodes far away
from labeled nodes lack of supervision information. In Figure 1, the node
$v_{a}$ cannot reach the valuable labeled node $v_{c}$ within the receptive
field of the GNN model, resulting in the quantity of information it received
is limited. (2) Over-squashing: the supervision information of valuable
labeled nodes is squashed when passing across the narrow path together with
other useless information. In Figure 1, the valuable supervision information
of $v_{b}$ to $v_{a}$ is compressed into a vector together with the
information of many nodes belonging to other classes, resulting in the quality
of supervision information that $v_{a}$ received being poor. Then we introduce
two metrics (reaching coefficient and squashing coefficient) to give a
quantitative analysis of the relation between the learning performance, label
positions, and graph structure properties. We further draw a conclusion that
better reachability and lower squashing to labeled nodes lead to better
classification performance for GNN models.

Present work.  In light of the above analysis, we propose a Position-Aware
STructurE Learning method named PASTEL, which directly optimizes the
information propagation path and solves the problem of topology-imbalance
issue in essence. The key insight of PASTEL is to enable nodes within the same
class to connect more closely with each other for more supervision
information. Specifically, we design a novel anchor-based position encoding
mechanism to capture the relative position between nodes and incorporate the
position information into structure learning. Then we design a class-wise
conflict measure based on the Group PageRank, which measures the influence
from labeled nodes of each class and acts as a guide to increase the intra-
class connectivity via adjusting edge weight. The main contributions are as
follows:

  * •

We provide a new understanding of the topology-imbalance issue from the
perspective of supervision information distribution in terms of under-reaching
and over-squashing and provide two new quantitative metrics for them.

  * •

Equipped with the proposed position encodings and class-wise conflict measure,
PASTEL can better model the relationships of node pairs and enhance the intra-
class inductive bias by maximizing the label influence.

  * •

Experimental results demonstrate that the proposed PASTEL enjoys superior
effectiveness and indeed enhances the GNN model’s power for in-the-wild
extrapolation.

##  2\. Related Work

###  2.1. Imbalance Learning

Imbalanced classification problems (Sun et al., 2009; He and Garcia, 2009)
have attracted extensive research attention. Most existing works (Haixiang et
al., 2017; Lin et al., 2017) focus on the class-imbalance problem, where the
model performance is dominated by the majority class. The class-imbalance
learning methods can be roughly divided into two types: data-level re-sampling
and algorithm-level re-weighting. Re-sampling methods re-sample (Chawla et
al., 2002; Ando and Huang, 2017; Xiaolong et al., 2019) or augment data (Park
et al., 2021) to balance the number of data for each class during the data
selection phase. Re-weighting methods (Cao et al., 2019; Cui et al., 2019; Ren
et al., 2018) adjust different weights to different data samples according to
the number of data during the training phase.

For the graph-specific topology-imbalance issue as mentioned in Section 1,
directly applying these methods to the graph data fails to take the special
topology properties into consideration. ReNode (Chen et al., 2021) is the
first work for the graph topology-imbalance issue, which follows the paradigm
of classical re-weighting methods. Specifically, ReNode defines an influence
conflict detection based metric and re-weights the labeled nodes based on
their relative positions to class boundaries. However, ReNode is limited by
its homophily assumption and only has a slight performance improvement. In
this paper, PASTEL alleviates topology-imbalance by learning a new structure
that maximizes the intra-class label influence, which can be seen as “label
re-distribution” in the topology space.

###  2.2. Graph Structure Learning

Graph structure learning (Zhu et al., 2022) learns an optimized graph
structure for representation learning and most of them aim to improve the
robustness (Jin et al., 2020; Zheng et al., 2020) of GNN models. There are
also some works (Franceschi et al., 2019; Chen et al., 2020b; Topping et al.,
2022; Chen et al., 2020c; Sun et al., 2022) that utilize the structure
learning to improve the graph representation quality. As for the over-
squashing problem, (Wang and Leskovec, 2021) assigns different weights to
edges connected to two nodes of the same class for better representations.
However, (Wang and Leskovec, 2021) still fails with the issue of under-
reaching. SDRF (Topping et al., 2022) rewires edges according to the Ricci
curvatures to solve the over-squashing problem by only considering topology
properties.

Multiple measurements in existing structure learning works are leveraged for
modeling node relations, including node features (Zhao et al., 2021), node
degrees (Jin et al., 2020), node encodings (Zhang et al., 2019) and edge
attributes (Zheng et al., 2020). The node positions play an important role in
generating discriminative representations (You et al., 2019) and are seldom
considered in structure learning. In this work, we advance the structure
learning strategy for the graph topology-imbalance issue and introduce a
position-aware framework to better capture the nodes’ underlying relations.

##  3\. Understanding Topology-Imbalance

In this section, we provide a new understanding of the topology-imbalance
issue in terms of under-reaching and over-squashing. Then we perform a
quantitative analysis of the relations between them to answer two questions:

Q1: Why does topology-imbalance affect the performance of graph representation
learning?

Q2: What kind of graphs are susceptible to topology-imbalance?

###  3.1. Notations and Preliminaries

Consider a graph $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, where
$\mathcal{V}$ is the set of $N$ nodes and $\mathcal{E}$ is the edge set. Let
$\mathbf{A}\in\mathbb{R}^{N\times N}$ be the adjacency matrix and
$\mathbf{X}\in\mathbb{R}^{N\times d_{0}}$ be the node attribute matrix, where
$d_{0}$ denotes the dimension of node attributes. The diagonal degree matrix
is denoted as $\mathbf{D}\in\mathbb{R}^{N\times N}$ where
$\mathbf{D}_{ii}=\sum^{N}_{j=1}\mathbf{A}_{ij}$. The graph diameter is denoted
as $D_{\mathcal{G}}$. Given the labeled node set $\mathcal{V}_{L}$and their
labels $\mathcal{Y}_{L}$ where each node $v_{i}$ is associated with a label
$y_{i}$, semi-supervised node classification aims to train a node classifier
$f_{\theta}:v\rightarrow\mathbb{R}^{C}$ to predict the labels
$\mathcal{Y}_{U}$ of remaining nodes
$\mathcal{V}_{U}=\mathcal{V}\setminus\mathcal{V}_{L}$, where $C$ denotes the
number of classes. we separate the labeled node set $\mathcal{V}_{L}$ into
$\\{\mathcal{V}^{1}_{L},\mathcal{V}^{2}_{L},\cdots,\mathcal{V}^{C}_{L}\\}$,
where $\mathcal{V}^{i}_{L}$ is the nodes of class $i$ in $\mathcal{V}_{L}$.

###  3.2. Understanding Topology-Imbalance via Under-reaching and Over-
squashing

In GNNs, node representations are learned by aggregating information from
valuable neighbors. The quantity and quality of the information received by
the nodes decide the expressiveness of their representations. We perceive the
imbalance of the labeled node positions affects the performance of GNNs for
two reasons:

(1) Under-reaching: The influence from labeled nodes decays with the topology
distance (Buchnik and Cohen, 2018), resulting in that the nodes far away from
labeled nodes lack supervision information. When the node can’t reach enough
valuable labeled nodes within the receptive field of the model, the quantity
of information it received is limited.

(2) Over-squashing: The receptive field of GNNs is exponentially-growing and
all information is compressed into fixed-length vectors (Alon and Yahav,
2021). The supervision information of valuable labeled nodes is squashed when
passing across the narrow path together with other useless information.

###  3.3. Quantitative Analysis

To provide quantitative analysis for topology-imbalance, we propose two
metrics for reachability and squashing. First, we define a reaching
coefficient based on the shortest path, which determines the minimum layers of
GNNs to obtain supervision information:

###### Definition 1 (Reaching coefficient).

Given a graph $\mathcal{G}$ and labeled node set $\mathcal{V}_{L}$, the
reaching coefficient $RC$ of $\mathcal{G}$ is the mean length of the shortest
path from unlabeled nodes to the labeled nodes of their corresponding classes:

(1)
$RC=\frac{1}{|\mathcal{V}_{U}|}\sum_{v_{i}\in\mathcal{V}_{U}}\frac{1}{|\mathcal{V}^{y_{i}}_{L}|}\sum_{v_{j}\in\mathcal{V}^{y_{i}}_{L}}\left(1-\frac{\log|\mathcal{P}_{sp}(v_{i},v_{j})|}{\log
D_{\mathcal{G}}}\right),$

where $\mathcal{V}^{y_{i}}_{L}$ denotes the nodes in $\mathcal{V}_{L}$ whose
label is $y_{i}$, $\mathcal{P}_{sp}(v_{i},v_{j})$ denotes the shortest path
between $v_{i}$ and $v_{j}$, and $|\mathcal{P}_{sp}(v_{i},v_{j})|$ denotes its
length, and $D_{\mathcal{G}}$ is the diameter of graph $\mathcal{G}$.
Specifically, for the unconnected $v_{i}$ and $v_{j}$, we set the length of
their shortest path as $D_{\mathcal{G}}$.

The reaching coefficient reflects how long the the distance when the GNNs
passes the valuable information to the unlabeled nodes. Note that $RC\in[0,1)$
and larger $RC$ means better reachability.

For the quantitative metric of over-squashing, we define a squashing
coefficient using the Ricci curvature to formulate it from a geometric
perspective. The Ricci curvature (Ollivier, 2009) reflects the change of
topology properties of the two endpoints of an edge, where the negative
$Ric(v_{i},v_{j})$ means that the edge behaves locally as a shortcut or bridge
and positive $Ric(v_{i},v_{j})$ indicates that locally there are more
triangles in the neighborhood of $v_{i}$ and $v_{j}$ (Ni et al., 2015; Topping
et al., 2022).

###### Definition 2 (Squashing coefficient).

Given a graph $\mathcal{G}$, the squashing coefficient $SC$ of $\mathcal{G}$
is the mean Ricci curvature of edges on the shortest path from unlabeled nodes
to the labeled nodes of their corresponding classes:

(2)
$SC=\frac{1}{|\mathcal{V}_{U}|}\sum_{v_{i}\in\mathcal{V}_{U}}\frac{1}{|\mathcal{N}_{y_{i}}(v_{i})|}\sum_{v_{j}\in\mathcal{N}_{y_{i}}(v_{i})}\frac{\sum_{e_{kt}\in\mathcal{P}_{sp}(v_{i},v_{j})}Ric(v_{k},v_{t})}{|\mathcal{P}_{sp}(v_{i},v_{j})|},$

where $\mathcal{N}_{y_{i}}(v_{i})$ denotes the labeled nodes of class $y_{i}$
that can reach $v_{i}$, $Ric(\cdot,\cdot)$ denotes the Ricci curvature, and
$|\mathcal{P}_{sp}(v_{i},v_{j})|$ denotes the length of shortest path between
$v_{i}$ and $v_{j}$.

We leverage the Ollivier-Ricci curvature (Ollivier, 2009) as
$Ric(\cdot,\cdot)$ here:

(3)
$Ric(v_{k},v_{t})=\frac{Wasserstein(mass_{k},mass_{t})}{d_{geo}(v_{k},v_{t})},$

where $Wasserstein(\cdot,\cdot)$ is the Wasserstein distance,
$d_{geo}(\cdot,\cdot)$ is the geodesic distance function, and $mass_{k}$ is
the mass distribution (Ollivier, 2009) of node $v_{k}$. Note that $SC$ can be
either positive or negative and larger $SC$ means lower squashing because the
ring structures are more friendly for information sharing.

Figure 2. Predictions of GCN with the same graph structure and different
labeled nodes.

In Figure 2 and Figure 3, we show the relation between the reaching
coefficient $RC$, the squashing coefficient $SC$, and the classification
accuracy. The higher the accuracy, the darker and larger the corresponding
scatter. First, we analyze the performance of GCN when trained on the same
graph structure but with different labeled nodes. In Figure 2, we generate a
synthetic graph by the Stochastic Block Model (SBM) (Holland et al., 1983)
with 4 classes and 3,000 nodes. We randomly sample some nodes as the labeled
nodes 10 times and scatter the classification accuracy in Figure 2. We can
observe that even for the same graph structure, the difference in positions of
labeled nodes may bring up to 15.42% difference in accuracy. There is a
significant positive correlation between the reaching coefficient, the
squashing coefficient, and the model performance.

Then we analyze the performance of GCN when trained with the same labeled
nodes but on different graph structures. In Figure 3, we set the labeled nodes
to be the same and generate different structures between them by controlling
the edge probability between communities in the SBM model. We can observe that
with the same supervision information, there is up to a 26.26% difference in
accuracy because of the difference in graph structures. There is also a
significant positive correlation between the reaching coefficient, the
squashing coefficient, and the model performance. When the graph shows better
community structure among nodes of the same class, the node representations
can be learned better.

Therefore, we make the following conclusions:  (1) Topology-imbalance hurts
the performance of graph learning in the way of under-reaching and over-
squashing. (for Q1) (2) The proposed two quantitative metrics can effectively
reflect the degree of topology-imbalance. Graph with poor reachability (i.e.,
smaller $RC$) and stronger squashing (i.e., smaller $SC$) is more susceptible
to topology-imbalance. (for Q2) (3) Optimizing the graph structure can
effectively solve the topology-imbalance issue. The above conclusions provide
the guideline for designing the framework of PASTEL, i.e., balance the
supervision information distribution by learning a structure with better
reachability and lower squashing.

##  4\. Alleviate Topology-Imbalance by Structure Learning

In this section, we introduce PASTEL, a Position-Aware STructurE Learning
framework, to optimize the information propagation path directly and address
the topology-imbalance issue in essence. In light of the analysis in Section
3.2, PASTEL aims to learn a better structure that increases the intra-class
label influence for each class and thus relieves the under-reaching and over-
squashing phenomena. The overall architecture of PASTEL is shown in Figure 4.

###  4.1. Position-aware Structure Learning

To form structure with better intra-class connectivity, we use an anchor-based
position encoding method to capture the topology distance between unlabeled
nodes to labeled nodes. Then we incorporate both the merits of feature
information as well as topology information to learn the refined structure.

Anchor-based Position Encoding.  Inspired by the position in transformer
(Vaswani et al., 2017; Shaw et al., 2018), we use an anchor-based position
encoding method to capture the relative position of unlabeled nodes with
respect to all the labeled nodes of the graph. Since we focus on maximizing
the reachability between unlabeled nodes and labeled nodes within the same
class, we directly separate the labeled node set $\mathcal{V}_{L}$ into $C$
anchor sets
$\\{\mathcal{V}^{1}_{L},\mathcal{V}^{2}_{L},\cdots,\mathcal{V}^{C}_{L}\\}$,
where each subset $\mathcal{V}^{c}_{L}$ denotes the labeled nodes whose labels
are $c$. The class-wise anchor sets help distinguish the information from
different classes rather than treating all the anchor nodes the same and
ignoring the class difference as in (You et al., 2019). Concretely, for any
node $v_{i}$, we consider a function $\phi(\cdot,\cdot)$ which measures the
position relations between $v_{i}$ and the anchor sets in graph $\mathcal{G}$.
The function can be defined by the connectivity between the nodes in the
graph.

(4)
$\mathbf{p}_{i}=\left(\phi\left(v_{i},\mathcal{V}^{1}_{L}\right),\phi\left(v_{i},\mathcal{V}^{2}_{L}\right),\cdots,\phi\left(v_{i},\mathcal{V}^{C}_{L}\right)\right),$

where $\phi(v_{i},\mathcal{V}^{c}_{L})$ is the position encoding function
defined by the connectivity between the node $v_{i}$ and the anchor set
$\mathcal{V}^{c}_{L}$ in graph. Here we choose
$\phi(v_{i},\mathcal{V}^{c}_{L})$ to be the mean length of shortest path
between $v_{i}$ and nodes in $\mathcal{V}^{c}_{L}$ if two nodes are connected:

(5)
$\phi(v_{i},\mathcal{V}^{c}_{L})=\frac{\sum_{v_{j}\in\mathcal{N}_{c}(v_{i})}|\mathcal{P}_{sp}(v_{i},v_{j})|}{|\mathcal{N}_{c}(v_{i})|},$

where $\mathcal{N}_{c}(v_{i})$ is the nodes connected with $v_{i}$ in
$\mathcal{V}^{c}_{L}$ and $|\mathcal{P}_{sp}(v_{i},v_{j})|$ is the length of
shortest path between $v_{i}$ and $v_{j}$. Then we transform the position
encoding into the $d_{0}$ dimensional space:

(6) $\mathbf{h}^{p}_{i}=\mathbf{W}_{\phi}\cdot\mathbf{p}_{i},$

where $\mathbf{W}_{\phi}$ is a trainable vector. If two nodes have similar
shortest paths to the anchor sets, their position encodings are similar.

Figure 3.  Predictions of GCN with and the same labeled nodes and different
graph structures.  Figure 4.  Overall architecture of PASTEL. PASTEL encodes
the relative position between nodes with the labeled nodes as anchor sets
{$\mathcal{S}$} and incorporates the position information with node features
for structure learning. For each pair of nodes, PASTEL uses the class-wise
conflict measure as the edge weights to learn a graph with better intra-class
connectivity.

Position-aware Metric Learning.  After obtaining the position encoding, we use
a metric function that accounts for both node feature information and the
position-based similarities to measure the possibility of edge existence.
PASTEL is agnostic to various similarity metric functions and we choose the
widely used multi-head cosine similarity function here:

(7)
$a^{P}_{ij}=\frac{1}{m}\sum^{m}_{h=1}\cos\left(\mathbf{W}_{h}\cdot\left(\mathbf{z}_{i}||\mathbf{h}^{p}_{i}\right),\mathbf{W}_{h}\cdot\left(\mathbf{z}_{j}||\mathbf{h}^{p}_{j}\right)\right),$

where $m$ is the number of heads, $\mathbf{W}_{h}$ is the weight matrix of the
$h$-th head, $\mathbf{z}_{i}$ denotes the representation vector of node
$v_{i}$ and $||$ denotes concatenation. The effectiveness of the position-
aware structure learning is evaluated in Section 5.3.1.

###  4.2. Class-wise Conflict Measure

We aim to increase the intra-class connectivity among nodes, thereby
increasing the supervision information they received and their influence on
each other. Here we propose a class-wise conflict measure to guide what nodes
should be more closely connected. According to the inherent relation of GNNs
with Label Propagation (Wang and Leskovec, 2021; Chen et al., 2021), we use
the Group PageRank (Chen et al., 2020a) as a conflict measure between nodes.
Group PageRank (GPR) extends the traditional PageRank(Page et al., 1999) into
a label-aware version to measure the supervision information from labeled
nodes of each class. Specifically, for class $c\in\\{1,2,\cdots,C\\}$, the
corresponding GPR matrix is

(8)
$\mathbf{P}^{gpr}(c)=(1-\alpha)\mathbf{A}^{\prime}\mathbf{P}^{gpr}(c)+\alpha\mathbf{I}_{c},$

where $\mathbf{A}^{\prime}=\mathbf{A}\mathbf{D}^{-1}$, $\alpha$ is the random
walk restart probability at a random node in the group and
$\mathbf{I}_{c}\in\mathbb{R}^{n}$ is the teleport vector:

(9)
$\mathbf{I}^{i}_{c}=\left\\{\begin{matrix}\frac{1}{|\mathcal{V}^{c}_{L}|},&if~{}y_{i}=c\\\
0,&otherwise\end{matrix}\right.$

where $|\mathcal{V}^{c}_{L}|$ is the number of labeled nodes with class $c$.
We calculate the GPR for each group individually and then concatenate all the
GPR vectors to form a final GPR matrix $\mathbf{P}^{gpr}\in\mathbb{R}^{N\times
C}$ as in (Chen et al., 2020a):

(10)
$\mathbf{P}^{gpr}=\alpha\left(\mathbf{E}-\left(1-\alpha\right)\mathbf{A}^{\prime}\right)^{-1}\mathbf{I}^{*},$

where $\mathbf{E}$ is the unit matrix of nodes and $\mathbf{I}^{*}$ is the
concatenation of $\\{\mathbf{I}_{c},c=1,2,\cdots,C\\}$. Under
$\mathbf{P}^{gpr}$, node $v_{i}$ corresponds to a GPR vector
$\mathbf{P}^{gpr}_{i}$ (the $i$-th row of $\mathbf{P}^{gpr}$), where the
$c$-th dimension represents the the supervision influence of labeled nodes of
class $c$ on node $v_{i}$. The GPR value contains not only the global topology
information but also the annotation information.

For each node pair nodes $v_{i}$ and $v_{j}$, we use the Kullback Leiber (KL)
divergence of their GPR vectors to measure their conflict when forming an
edge:

(11)
$\kappa_{ij}=\operatorname{KL}\left(\mathbf{P}^{gpr}_{i},\mathbf{P}^{gpr}_{j}\right).$

The distance of GPR vectors reflects the influence conflict of different
classes when exchanging information. We use a cosine annealing mechanism to
calculate the edge weights by the relative ranking of the conflict measure:

(12)
$w_{ij}=\frac{1}{2}\left[-\cos{\frac{\operatorname{Rank}(\kappa_{ij})}{|\mathcal{V}|\times|\mathcal{V}|}*\pi}+1\right],$

where $Rank(\cdot)$ is the ranking function according to the magnitude. The
more conflicting the edge is, the less weight is assigned to it. With the
class-wise conflict measure, we aim to learn a graph structure that makes the
GPR vectors of nodes have “sharp” distributions focusing on their ground-truth
classes. Then $w_{ij}$ is used as the connection strength of edge $e_{ij}$,
with the corresponding element $\tilde{a}^{P}_{ij}$ in the adjacency matrix
being:

(13) $\tilde{a}^{P}_{ij}=w_{ij}\cdot a^{P}_{ij}.$

The effectiveness of the class-wise conflict measure is evaluated in Section
5.3.2 and the change of GPR vectors is shown in Section 5.4.3.

Input: Graph $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$ with node labels
$\mathcal{Y}$; Number of heads $m$; Number of training epochs $E$; Structure
fusing coefficients $\lambda_{1},\lambda_{2}$; Loss coefficients
$\beta_{1},\beta_{2},\beta_{3}$

Output: Optimized graph $\mathcal{G}^{*}=(\mathbf{A}^{*},\mathbf{X})$,
predicted label $\hat{\mathcal{Y}}$

1 Parameter initialization;

2 for  _$e=1,2,\cdots,E$_ do

// Learn position-aware graph structure

3 Learn position encodings $\mathbf{h}^{p}_{i}$ $\leftarrow$ Eq. (6);

4 Learn edge possibility $a^{P}_{ij}$ $\leftarrow$ Eq. (7);

5 Calculate the Group PageRank matrix $\mathbf{P}^{gpr}\leftarrow$ Eq. (10);

6 Calculate the class-wise conflict measure $w_{ij}\leftarrow$ Eq. (12);

7 Obtain position-aware structure $\mathbf{A}_{P}$ $\leftarrow$ Eq. (14);

// Learn node representations

8 Obtain the optimized structure $\mathbf{A}^{*}$ $\leftarrow$ Eq. (16);

9 Calculate representations and labels $\mathbf{Z}$, $\hat{\mathcal{Y}}$
$\leftarrow$ Eq. (20);

// Optimize

10 Calculate the losses $\mathcal{L}_{cls}\leftarrow$ Eq. (21),
$\mathcal{L}_{smooth}\leftarrow$Eq. (17), $\mathcal{L}_{con}\leftarrow$Eq.
(18), and $\mathcal{L}_{spar}\leftarrow$Eq. (19);

11 Update model parameters to minimize $\mathcal{L}$ $\leftarrow$ Eq. (22).

12 end for

Algorithm 1 The overall process of PASTEL

###  4.3. Learning with the Optimized Structure

With the above structure learning strategy, we can obtain a position-aware
adjacency $\mathbf{A}_{P}$ with maximum intra-class connectivities:

(14) $\mathbf{A}_{P}=\\{\tilde{a}^{P}_{ij},i,j\in\\{1,2,\cdots,N\\}\\}.$

The input graph structure determines the learning performance to a certain
extent. Since the structure learned at the beginning is of poor quality,
directly using it may lead to non-convergence or unstable training of the
whole framework. We hence incorporate the original graph structure
$\mathbf{A}$ and a structure in a node feature view $\mathbf{A}_{N}$ as
supplementary to formulate an optimized graph structure $\mathbf{A}^{*}$.
Specifically, we also learn a graph structure
$\mathbf{A}_{N}=\\{a^{N}_{ij},i,j\in\\{1,2,\cdots,N\\}\\}$ in a node feature
view with each element being:

(15)
$a^{N}_{ij}=\frac{1}{m}\sum^{m}_{h=1}\cos\left(\mathbf{W}_{h}\cdot\left(\mathbf{x}_{i}||\mathbf{h}^{p_{0}}_{i}\right),\mathbf{W}_{h}\cdot\left(\mathbf{x}_{j}||\mathbf{h}^{p_{0}}_{j}\right)\right),$

where $\mathbf{x}_{i}$ is the feature vector of node $v_{i}$ and
$\mathbf{h}^{p_{0}}_{i}$ is the position encoding with the original structure.
Then we can formulate an optimized graph structure $\mathbf{A}^{*}$ with
respect to the downstream task:

(16)
$\mathbf{A}^{*}=\lambda_{1}\mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}+\left(1-\lambda_{1}\right)\left(\lambda_{2}f\left(\mathbf{A}_{N}\right)+\left(1-\lambda_{2}\right)f\left(\mathbf{A}_{P}\right)\right),$

where $f(\cdot)$ denotes the row-wise normalization function, $\lambda_{1}$
and $\lambda_{2}$ are two constants that control the contributions of original
structure and feature view structure, respectively. Here we use a dynamic
decay mechanism for $\lambda_{1}$ and $\lambda_{2}$ to enable the position-
aware structure $\mathbf{A}_{P}$ to play a more and more important role during
training.

To control the quality of learned graph structure, we impose additional
constraints on it following (Kalofolias, 2016; Chen et al., 2020b) in terms of
smoothness, connectivity, and sparsity:

(17) $\displaystyle\mathcal{L}_{smooth}$ $\displaystyle=\frac{1}{N^{2}}{\rm
tr}\left(\mathbf{X}^{T}\mathbf{L}^{*}\mathbf{X}\right),$ (18)
$\displaystyle\mathcal{L}_{con}$
$\displaystyle=\frac{1}{N}\mathbf{1}^{T}\log(\mathbf{A}^{*}\mathbf{1}),$ (19)
$\displaystyle\mathcal{L}_{spar}$
$\displaystyle=\frac{1}{N^{2}}||\mathbf{A}^{*}||^{2}_{F},$

where $\mathbf{L}^{*}=\mathbf{D}^{*}-\mathbf{A}^{*}$ is the Laplacian of
$\mathbf{A}^{*}$ and $\mathbf{D}^{*}$ is the degree matrix of
$\mathbf{A}^{*}$. To speed up the computation, we extract a symmetric sparse
non-negative adjacency matrix by masking off (i.e., set to zero) those
elements in $\mathbf{A}^{*}$ which are smaller than a predefined non-negative
threshold $a_{0}$. Then $\mathcal{G}^{*}=(\mathbf{A}^{*},\mathbf{X})$ is input
into the $\operatorname{GNN-Encoder}$ for the node representations
$\mathbf{Z}\in\mathbb{R}^{N\times d}$, predicted labels $\hat{y}$ and
classification loss $\mathcal{L}_{cls}$:

(20) $\mathbf{Z}=\operatorname{GNN-
Encoder}(\mathbf{A}^{*},\mathbf{X}),\hat{\mathcal{Y}}=\operatorname{Classifier}(\mathbf{Z}),$
(21) $\mathcal{L}_{cls}=\operatorname{Cross-
Entropy}(\mathcal{Y},\hat{\mathcal{Y}}).$

The overall loss is defined as the combination of the node classification loss
and graph regularization loss:

(22)
$\mathcal{L}=\mathcal{L}_{cls}+\beta_{1}\mathcal{L}_{smooth}+\beta_{2}\mathcal{L}_{con}+\beta_{3}\mathcal{L}_{spar}.$

The overall process of PASTEL is shown in Algorithm 1.

##  5\. Experiment

In this section, we first evaluate PASTEL111The code of PASTEL is available at
https://github.com/RingBDStack/PASTEL. on both real-world graphs and synthetic
graphs. Then we analyze the main mechanisms of PASTEL and the learned
structure. We mainly focus on the following research questions:

  * •

RQ1. How does PASTEL perform in the node classification task? (Section 5.2)

  * •

RQ2. How does the position encoding and the class-wise conflict measure
influence the performance of PASTEL? (Section 5.3)

  * •

RQ3. What graph structure PASTEL tend to learn? (Section 5.4)

###  5.1. Experimental Setups

Table 1. Weighted-F1 score and Macro-F1 score (% ± standard deviation) of node
classification on real-world graph datasets.

|  | Cora | Citeseer | Photo | Actor | Chameleon | Squirrel
---|---|---|---|---|---|---|---
Backbone | Model | W-F1 | M-F1 | W-F1 | M-F1 | W-F1 | M-F1 | W-F1 | M-F1 | W-F1 | M-F1 | W-F1 | M-F1
GCN | original | 79.4±0.9 | 77.5±1.5 | 66.3±1.3 | 62.2±1.2 | 85.4±2.8 | 84.6±1.3 | 21.8±1.3 | 20.9±1.4 | 30.5±3.4 | 30.5±3.3 | 21.9±1.2 | 21.9±1.2
ReNode | 80.0±0.7 | 78.4±1.3 | 66.4±1.0 | 62.4±1.1 | 86.2±2.4 | 85.3±1.6 | 21.2±1.2 | 20.2±1.6 | 30.3±3.2 | 30.4±2.8 | 22.4±1.1 | 22.4±1.1
AddEdge | 79.0±0.9 | 77.0±1.4 | 66.2±1.3 | 62.2±1.3 | 85.5±1.5 | 86.1±1.8 | 21.2±1.3 | 20.3±1.5 | 30.6±1.6 | 30.4±1.7 | 21.7±1.5 | 21.7±1.5
DropEdge | 79.8±0.8 | 77.8±1.0 | 66.6±1.4 | 63.4±1.6 | 86.8±1.7 | 85.4±1.3 | 22.4±1.0 | 21.4±1.3 | 30.6±3.5 | 30.6±3.3 | 22.8±1.2 | 22.8±1.2
SDRF | 82.1±0.8 | 80.6±0.8 | 69.6±0.4 | 66.6±0.3 | ¿ 5 days | ¿ 5 days | ¿ 5 days | ¿ 5 days | 39.1±1.2 | 39.0±1.2 | ¿ 5 days | ¿ 5 days
NeuralSparse | 81.7±1.4 | 80.9±1.4 |  71.8±1.2 |  69.0±1.0 |  89.7±1.9 | 88.7±1.8 | 24.4±1.5 |  23.6±1.6 | 44.9±3.0 | 44.9±2.8 | 28.1±1.8 | 28.1±1.8
IDGL |  82.3±0.6 |  81.0±0.9 | 71.7±1.0 | 68.0±1.3 | 88.6±2.3 |  88.8±1.4 |  24.9±0.8 | 22.0±0.7 |  55.4±1.8 |  55.0±1.7 |  28.8±2.3 |  28.9±2.2
PASTEL | 82.5±0.3 | 81.2±0.3 | 72.9±0.8 | 69.3±0.9 | 91.4±2.7 | 91.3±2.2 | 26.4±1.0 | 24.4±1.2 | 57.8±2.4 | 57.3±2.4 | 37.5±0.6 | 37.5±0.7
GAT | original | 78.3±1.5 | 76.4±1.7 | 64.4±1.7 | 60.6±1.7 | 88.2±2.9 | 86.2±2.6 | 21.8±1.2 | 20.9±1.1 | 29.9±3.5 | 29.9±3.1 | 20.5±1.4 | 20.5±1.4
ReNode | 78.9±1.2 | 77.2±1.5 | 64.9±1.6 | 61.0±1.5 | 89.1±2.4 | 87.1±2.6 | 21.5±1.2 | 20.5±1.1 | 29.2±2.3 | 29.1±2.0 | 20.4±1.8 | 20.4±1.8
AddEdge | 78.0±1.6 | 76.2±1.6 | 64.0±1.3 | 60.2±1.3 | 88.2±2.4 | 86.2±2.5 | 21.3±1.2 | 20.3±1.1 | 29.8±1.7 | 29.6±1.5 | 20.7±1.6 | 20.7±1.6
DropEdge | 78.7±1.3 | 76.9±1.5 | 64.5±1.4 | 60.5±1.3 | 88.9±1.9 | 87.1±2.1 | 22.9±1.2 | 21.8±1.1 | 30.3±1.6 | 30.2±1.2 | 21.2±1.5 | 21.2±1.5
SDRF | 77.9±0.7 | 75.9±0.9 | 64.9±0.6 |  61.9±0.9 | ¿ 5 days | ¿ 5 days | ¿ 5 days | ¿ 5 days | 43.0±1.9 | 42.5±1.9 | ¿ 5 days | ¿ 5 days
NerualSparse |  81.4±4.8 | 79.4±4.8 | 64.8±1.5 |  61.9±1.3 |  90.2±2.5 |  88.0±2.3 |  23.4±1.7 | 22.4±1.5 | 45.6±2.1 | 45.5±1.8 |  28.8±1.3 |  28.8±1.3
IDGL | 80.6±1.0 |  79.7±0.9 |  66.5±1.5 |  61.9±1.9 | 89.9±3.1 | 87.7±2.6 | 22.4±1.5 | 21.8±1.2 |  48.4±4.0 |  47.8±3.1 | 27.0±2.6 | 27.0±2.6
PASTEL | 81.9±1.4 | 80.7±1.2 | 66.6±1.9 | 62.0±1.7 | 91.8±3.2 | 89.4±2.9 | 24.4±2.6 |  22.1±2.6 | 52.1±2.7 | 52.5±2.8 | 35.3±0.9 | 35.3±0.8
APPNP | original | 80.6±1.6 | 79.3±1.2 | 66.5±1.5 | 62.3±1.5 | 89.3±1.6 | 86.3±1.7 | 21.1±1.5 | 20.7±1.1 | 35.3±4.0 | 35.0±3.8 | 23.1±1.6 | 23.1±1.6
ReNode | 81.1±0.9 | 79.9±0.9 | 66.6±1.7 | 62.4±1.6 | 89.6±1.4 | 87.2±1.3 | 20.2±2.0 | 20.0±1.7 | 33.5±2.5 | 33.3±2.3 | 23.9±2.0 | 23.9±2.0
AddEdge | 80.3±1.3 | 78.8±1.1 | 66.6±2.1 | 62.5±2.1 | 89.3±1.2 | 86.4±1.2 | 21.5±1.3 | 20.7±1.4 | 35.7±1.7 | 35.4±1.2 | 23.1±1.6 | 23.2±1.7
DropEdge | 80.9±1.4 | 79.4±1.2 | 66.7±2.0 | 63.0±1.9 | 90.0±1.2 | 87.0±1.2 |  21.8±1.8 | 20.8±1.4 | 36.0±1.7 | 35.7±1.6 | 23.3±1.7 | 23.3±1.7
SDRF | 80.7±0.9 | 79.1±0.8 |  67.1±0.6 |  63.1±0.8 | ¿ 5 days | ¿ 5 days | ¿ 5 days | ¿ 5 days | 36.5±2.1 | 35.8±2.1 | ¿ 5 days | ¿ 5 days
NerualSparse | 81.1±1.4 | 79.9±1.2 | 66.8±1.9 | 62.7±1.9 | 91.3±1.8 |  89.4±1.6 |  21.8±1.9 | 21.4±1.5 | 39.1±2.9 | 38.7±2.8 | 28.3±1.5 | 28.3±1.5
IDGL |  81.3±0.9 | 80.2±0.9 | 67.0±1.3 | 62.9±1.3 |  91.6±1.3 | 88.6±2.2 | 21.4±2.4 | 20.1±2.4 |  41.2±2.2 |  40.6±2.6 |  29.6±2.3 |  29.7±2.2
PASTEL | 82.0±1.0 |  80.0±0.9 | 67.3±1.3 | 63.2±1.5 | 92.3±3.1 | 89.9±2.5 | 22.5±2.0 |  20.9±2.1 | 44.2±3.2 | 43.8±3.4 | 34.6±1.6 | 34.6±1.6
GraphSAGE | original | 75.4±1.6 | 74.1±1.6 | 64.8±1.6 | 60.7±1.6 | 86.1±2.5 | 83.3±2.4 | 24.0±1.2 | 23.2±1.0 | 36.5±1.6 | 36.2±1.6 | 27.2±1.7 | 27.2±1.7
ReNode | 76.4±0.9 | 75.0±1.1 | 65.4±1.7 | 61.2±1.7 | 86.5±1.7 | 84.1±1.7 | 23.7±1.2 | 22.8±1.0 | 36.4±1.9 | 36.1±1.9 | 27.7±1.8 | 27.7±1.8
AddEdge | 75.2±1.2 | 73.7±1.2 | 65.0±1.4 | 60.9±1.3 | 86.1±2.8 | 83.4±2.6 | 23.8±1.7 | 23.2±1.6 | 36.5±1.5 | 36.2±1.3 | 26.9±2.1 | 26.9±2.1
DropEdge | 76.0±1.6 | 74.5±1.6 | 65.1±1.4 | 60.9±1.4 | 86.2±1.6 | 83.5±1.4 | 24.1±1.0 | 23.3±0.9 | 37.5±1.4 | 37.2±1.4 | 27.5±1.8 | 27.5±1.8
SDRF | 75.7±0.8 | 74.6±0.8 | 65.3±0.6 | 61.4±0.6 | ¿ 5 days | ¿ 5 days | ¿ 5 days | ¿ 5 days | 41.5±2.6 | 41.6±2.7 | ¿ 5 days | ¿ 5 days
NerualSparse |  79.7±1.8 | 77.8±1.6 | 64.7±1.4 | 61.1±1.3 | 89.1±5.4 |  86.7±5.5 |  25.1±1.2 | 24.4±1.1 | 39.1±1.9 | 39.0±1.9 | 32.2±2.4 | 32.2±2.4
IDGL | 79.2±0.9 |  78.4±0.8 |  65.6±0.9 |  61.3±1.2 |  90.0±1.0 | 86.3±1.3 | 24.0±2.6 | 22.4±2.7 |  43.8±3.4 |  43.0±3.2 |  33.9±0.9 |  33.9±0.8
PASTEL | 81.1±0.8 | 79.8±0.7 | 65.7±1.1 | 61.4±1.4 | 92.0±0.6 | 89.0±1.0 | 26.0±2.4 |  23.6±2.7 | 47.7±0.9 | 46.9±0.9 | 35.5±1.4 | 35.5±1.4

####  5.1.1. Datasets

We conduct experiments on synthetic and real-world datasets to analyze the
model’s capabilities in terms of both graph theory and real-world scenarios.
The real-word datasets include various networks with different heterophily
degrees to demonstrate the generalization of PASTEL. Cora and Citeseer (Sen et
al., 2008) are citation networks. Photo (Shchur et al., 2018) and and Actor
(Pei et al., 2020) are co-occurrence network. Chameleon and Squirrel
(Rozemberczki et al., 2021) are page-page networks in Wikipedia. Since we
focus on the topology-imbalance issue in this work, we set the number of
labeled nodes in each class to be 20.

####  5.1.2. Baselines

We choose representative GNNs as backbones including GCN (Kipf and Welling,
2017), GAT (Veličković et al., 2017), APPNP (Klicpera et al., 2019), and
GraphSAGE (Hamilton et al., 2017). The most important baseline is ReNode (Chen
et al., 2021), which is the only existing work for the topology-imbalance
issue. We also include some graph structure learning baselines to illustrate
the specific effectiveness of PASTEL for the topology-imbalance issue.
DropEdge (Rong et al., 2019) randomly removes edges at each epoch as structure
augmentation. To evaluate the effect of increasing the reachability randomly,
we use a adding edges method named AddEdge, whose adding strategy is similar
to DropEdge. SDRF (Topping et al., 2022) rewires edges according to their
curvatures for the over-squashing issue. NeuralSparse (Zheng et al., 2020)
removes potentially task-irrelevant edges for clearer class boundaries. IDGL
(Chen et al., 2020b) updates the node representations and structure based on
these representations iteratively.

####  5.1.3. Parameter Settings

For the GNN backbones, we set their depth to be 2 layers and adopt the
implementations from the PyTorch Geometric Library in all experiments. We set
the representation dimension of all baselines and PASTEL to be 256. We re-
implement the NeuralSparse (Zheng et al., 2020) and SDRF (Topping et al.,
2022) and the parameters of baseline methods are set as the suggested value in
their papers or carefully tuned for fairness. For DropEdge and AddEdge, we set
the edge dropping/adding probability to 10%. For PASTEL, we set the number of
heads $m=4$ and the random walk restart probability $\alpha=0.15$. The
structure fusing coefficients ($\lambda_{1}$ and $\lambda_{2}$) and the loss
coefficients ($\beta_{1}$, $\beta_{2}$ and $\beta_{3}$) are tuned for each
dataset.

###  5.2. Evaluation (RQ1)

Table 2. Weighted-F1 scores and improvements on graphs with different levels
of topology-imbalance.

| Cora-L | Cora-M | Cora-H
---|---|---|---
|  | $RC$
---
0.4130
| $SC$
---
-0.6183
| $RC$
---
0.4100
| $SC$
---
-0.6204
| $RC$
---
0.4060
| $SC$
---
-0.6302
| W-F1 (%) |  $\Delta$ (%) | W-F1 (%) |  $\Delta$ (%) | W-F1 (%) |  $\Delta$ (%)
GCN | 80.9±0.9 | — | 78.8±0.8 | — | 77.5±1.0 | —
ReNode | 81.3±0.7 |  $\uparrow$0.4 | 79.3±0.8 |  $\uparrow$0.5 | 78.3±1.1 |  $\uparrow$0.8
SDRF | 81.0±0.7 |  $\uparrow$0.1 | 78.9±0.8 |  $\uparrow$0.1 | 77.9±0.7 |  $\uparrow$0.4
IDGL | 82.5±1.0 |  $\uparrow$1.6 | 80.4±1.0 |  $\uparrow$1.6 | 81.6±1.1 |  $\uparrow$4.1
PASTEL | 82.7±0.9 |  $\uparrow$1.8 | 81.0±0.9 |  $\uparrow$2.2 | 81.9±1.1 |  $\uparrow$4.4

####  5.2.1. PASTEL for Real-world Graphs

We compare PASTEL with the baselines on several datasets on node
classification. The overall Weighted-F1 (W-F1) scores and the class-balance
Macro-F1 (M-F1) scores on different backbones are shown in Table 1. The best
results are shown in bold and the runner-ups are underlined. PASTEL shows
overwhelming superiority in improving the performance of backbones on all
datasets. It demonstrates that PASTEL is capable of learning better structures
with a more balanced label distribution that reinforces the GNN models. ReNode
(Chen et al., 2021) achieves fewer improvements on datasets of poor
connectivity (e.g., CiteSeer) and even damages the performance of backbones on
heterophilic datasets (e.g., Chameleon and Actor). We think it’s because
ReNode (Chen et al., 2021) detects conflicts by Personalized PageRank and
fails to reflect the node topological position well when the graph
connectivity is poor. Besides, ReNode takes the topology boundary as the
decision boundary, which is not applicable for heterophilic graphs. AddEdge
doesn’t work in most cases, demonstrating that randomly adding edge is not
effective in boosting the reachability. The structure augmentation strategy
should be carefully designed considering the node relations. SDRF (Topping et
al., 2022) can improve the performance, supporting our intuition that
relieving over-squashing helps graph learning. But SDRF is still less
effective than PASTEL because it only considers the topological properties
rather than the supervision information. Both NeuralSparse (Zheng et al.,
2020) and IDGL (Chen et al., 2020b) show good performance among the baselines,
showing the effectiveness of learning better structures for downstream tasks.
However, they are still less effective than PASTEL which takes the supervision
information distribution into consideration.

Table 3. Weighted-F1 scores (%) and improvements ($\Delta$) on synthetic SBM
graphs with different community structures.

| SBM-1 | SBM-2 | SBM-3 | SBM-4 | SBM-5 | SBM-6 | SBM-7
---|---|---|---|---|---|---|---
$p$ | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000
$q$ | 0.0300 | 0.0100 | 0.0083 | 0.0071 | 0.0063 | 0.0056 | 0.0050
$RC$ | 0.4979 | 0.4984 | 0.4990 | 0.4994 | 0.5002 | 0.5004 | 0.5009
$SC$ | 0.0998 | 0.0999 | 0.1000 | 0.1001 | 0.1007 | 0.1017 | 0.1144
| W-F1 | $\Delta$ | W-F1 | $\Delta$ | W-F1 | $\Delta$ | W-F1 | $\Delta$ | W-F1 | $\Delta$ | W-F1 | $\Delta$ | W-F1 | $\Delta$
GCN | 40.29 | — | 42.37 | — | 42.99 | — | 44.13 | — | 45.19 | — | 45.21 | — | 45.22 | —
ReNode | 41.33 |  $\uparrow$1.04 | 42.40 |  $\uparrow$0.03 | 43.21 |  $\uparrow$0.22 | 44.56 |  $\uparrow$0.43 | 45.20 |  $\uparrow$0.01 | 45.08 |  $\downarrow$0.13 | 44.89 |  $\downarrow$0.33
PASTEL | 45.67 |  $\uparrow$5.38 | 57.61 |  $\uparrow$15.24 | 58.33 |  $\uparrow$15.34 | 60.29 |  $\uparrow$16.16 | 66.41 |  $\uparrow$21.22 | 66.45 |  $\uparrow$21.24 | 66.57 |  $\uparrow$21.35

####  5.2.2. PASTEL under Different Levels of Topology-imbalance

To further analyze PASTEL’s ability in alleviating the topology-imbalance
issue, we verify the PASTEL under different levels of topology-imbalance. We
randomly sampled 1,000 training sets and calculate the reaching coefficient
$RC$ and squashing coefficient $SC$ as introduced in Section 3.2. Then we
choose 3 training sets with different levels of topology-imbalance according
to the conclusion in Section 3.3 and we denote them as Cora-L, Cora-M, and
Cora-H, according to the degree of topology imbalance. Note that larger $RC$
means better reachability and larger $SC$ means lower squashing. We evaluate
PASTEL and several baselines with the GCN as the backbone and show the dataset
information, the Weighted-F1 scores, and their improvements ($\Delta$) over
the backbones in Table 2. The performance of node representation learning
generally gets worse with the increase of the topology-imbalance degree of the
dataset. Both the node re-weighting method (i.e., ReNode (Chen et al., 2021))
and the structure learning methods (i.e., IDGL (Chen et al., 2020b), SDRF
(Topping et al., 2022) and PASTEL) can achieve more improvement with the
increase of dataset topology-imbalance. PASTEL performs best on all datasets
with different degrees of topology-imbalance and it can achieve up to 4.4%
improvement on the highly topology-imbalance dataset.

####  5.2.3. PASTEL for Synthetic Graphs

Figure 5. The impact of position encoding.

We generate 7 synthetic graph datasets with different community structures
using the Stochastic Block Model (SBM) $\mathcal{G}(N,C,p,q)$ (Holland et al.,
1983), where the number of nodes $N=3000$, the number of community $C=6$, $p$
denotes the edge probability within a community and $q$ denotes the edge
probability between communities. We show the classification Weighted-F1 scores
and improvements are shown in Table 3. With a more clear community structure,
the reaching coefficient $RC$ increases and the squashing coefficient $SC$
also increases, leading to the increase of GCN’s performance, which agrees
with the conclusion obtained in Section 3.3. ReNode shows unsatisfied
performance in boosting the node classification. PASTEL can increase the
classification weighted-F1 score by 5.38%-21.35% on SBM graphs with different
community structures, showing superior effectiveness.

###  5.3. Analysis of PASTEL (RQ2)

We conduct ablation studies for the two main mechanisms of PASTEL, position
encoding and class-wise conflict measure.

####  5.3.1. Impact of the Position Encoding

We design an anchor-based position encoding mechanism in Section 4.1, which
reflects the relative topological position to labeled nodes and further
maximizes the label influence within a class. To evaluate the effectiveness of
position encoding, we compare PASTEL with a variant PASTEL (w/o PE), which
removes the position encoding and directly take the node features for metric
learning in Eq. (7).

Figure 6. The impact of class-wise conflict measure.

Here we use the GCN as the backbone. As shown in Figure 5, the structure
learning strategy of PASTEL contributes the most, which can achieve at most
25.5% improvement in terms of Weighted-F1 score with only node features.
Although PASTEL (w/o PE) effectively improves the performance of backbones to
some extent, the position encoding still benefits learning better structure to
relieve the topology-imbalance with 1.0%-1.8% improvements than PASTEL (w/o
PE).

(a) Original Graph.

(b) ReNode.

(c) SDRF.

(d) IDGL.

(e) PASTEL.

Figure 7. Structure visualization. (a) Original graph of Cora and learned
graphs by (b) ReNode, (c) SDRF, (d) IDGL and (e) PASTEL.

####  5.3.2. Impact of the Class-wise Conflict Measure

We designed a class-wise conflict measure in Section 4.2 as edge weights to
guide learning structures with better intra-class connectivity. Here, we
compare PASTEL with its two variants to analyze the impact of the class-wise
conflict measure: (1) PASTEL (w/o CCM), which removes the class-wise conflict
measure and directly takes the learned edge possibilities in Eq. (7) as the
edge weights. (2) PASTEL (Totoro), which takes the Totoro metric introduced in
ReNode (Chen et al., 2021) as the conflict measure of nodes in Eq. (13). Here
we use the GCN as the backbone. The comparison results are shown in Figure 6.
On four datasets, PASTEL consistently outperforms the other two variants. Even
without the conflict measure, PASTEL (w/o CCM) still shows better performance
than PASTEL (Totoro), indicating the limitation of ReNode when capturing the
relative topology positions without clear homophily structures.

###  5.4. Analysis of Learned Structure (RQ3)

We analyze the learned graph by PASTEL in terms of visualization and
structural properties.

####  5.4.1. Structure Visualization

In Figure 7, we visualize the original graph of Cora and the graphs learned by
ReNode (Chen et al., 2021), SDRF (Topping et al., 2022), IDGL (Chen et al.,
2020b) and PASTEL using networkx. For clarity, the edges are not shown. The
solid points denote the labeled nodes, the hollow points denote the unlabeled
nodes, and the layout of nodes denotes their connectivities. The node size in
Figure 7(b) denotes the learned node weight in ReNode, and the solid lines and
dashed lines in Figure 7(c) denote the added and deleted edges by SDRF,
respectively. As we can observe, ReNode gives more weights to nodes in the
topology center of each class and SDRF tends to build connections between
distant or isolated nodes. Even though the structure learned by IDGL can make
the nodes of a class close, there are still some overlapping and entangled
areas between classes. Benefiting from the position encoding and class-wise
conflict measure, PASTEL can obtain graph structure with clearer class
boundaries.

Table 4. Properties and performance of the original graph and learned graphs
of Cora.

| Original Graph | ReNode | SDRF | IDGL | PASTEL
---|---|---|---|---|---
$RC$ | 0.4022 | 0.4022 | 0.4686 | 0.5028 | 0.5475
$SC$ | -0.6299 | -0.6299 | -0.4942 | -0.4069 | -0.3389
W-F1 (%) | 79.44 | 80.34 | 82.01 | 82.38 | 82.86

####  5.4.2. Change of $RC$ and $SC$

We also show the reaching coefficient $RC$ and the squashing coefficient $SC$
of the above graphs in Figure 7 and the Weighted-F1 score learned on them in
Table 4. Here we choose the GCN as the model backbone. All of the structure
learning methods (SDRF (Topping et al., 2022), IDGL (Chen et al., 2020b) and
PASTEL) learn structures with larger reaching coefficient and larger squashing
coefficient, leading the performance improvement of node classification. This
phenomenon supports our propositions in Section 3.3 again.

####  5.4.3. Change of GPR Vector

The class-wise conflict measure is calculated by the Group PageRank (GPR),
which reflects the label influence of each class. We randomly choose 10 nodes
for each class in Cora and show their GPR vectors $\mathbf{P}^{gpr}_{i}$ in
the original graph in Figure 8(a) and the learned graph in Figure 8(b),
respectively, where the color shade denotes the magnitude, $V_{i}$ denotes 10
nodes of class $i$ and $C_{i}$ denotes the $i$-th class. In Figure 8(a), the
off-diagonal color blocks are also dark, indicating that the label influence
of each class that nodes obtained from the original graph is still entangled
to some extent, which could bring difficulties to the GNN optimization. After
the structure learning guided by the proposed class-wise conflict measure,
Figure 8(b) exhibits 7 clear diagonal blocks and the gaps between the diagonal
and off-diagonal block are widened, indicating that nodes can receive more
supervision information of its ground-truth class. We can further make a
conclusion that the class-wise conflict measure plays an important role on
giving guidance for more class connectivity orthogonality.

(a) Original Graph.

(b) Learned Graph.

Figure 8. Heat maps for the Group PageRank value of (a) the original graph and
(b) the learned graph by PASTEL.

##  6\. Conclusion

We proposed a novel framework named PASTEL for the graph topology-imbalance
issue. We provide a new understanding and two quantitative analysis metrics of
topology-imbalance in the perspective of under-reaching and over-squashing,
answering the questions that how topology-imbalance affects GNN’s performance
as well as what graphs are susceptible to it. PASTEL designs an anchor-based
position encoding mechanism and a class-wise conflict measure to obtain
structures with better in-class connectivity. Comprehensive experiments
demonstrate the potential and adaptability of PASTEL. An interesting future
direction is to incorporate the proposed two quantitative metrics into the
learning process to address topology-imbalance more directly.

###### Acknowledgements.

The corresponding author is Jianxin Li. The authors of this paper were
supported by the NSFC through grant 6187202 and the Academic Excellence
Foundation of BUAA for PhD Students. Philip S. Yu was supported by NSF under
grants III-1763325, III-1909323, III-2106758, and SaTC-1930941.

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# Maxwell-Scalar device based on the electric dipole

D. Bazeia  Departamento de Física, Universidade Federal da Paraíba, 58051-970
João Pessoa, PB, Brazil  M.A. Marques  Departamento de Biotecnologia,
Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil
Departamento de Física, Universidade Federal da Paraíba, 58051-970 João
Pessoa, PB, Brazil  R. Menezes  Departamento de Ciências Exatas, Universidade
Federal da Paraíba, 58297-000 Rio Tinto, PB, Brazil  Departamento de Física,
Universidade Federal da Paraíba, 58051-970 João Pessoa, PB, Brazil

###### Abstract

In this work we study the electric field of a dipole immersed in a medium with
permittivity controlled by a real scalar field which is non-minimally coupled
to the Maxwell field. We model the system with an interesting function, which
allows the presence of exact solutions, describing the possibility of the
permittivity to encapsulate the charges at very high values, giving rise to an
effect that is not present in the standard situation. The results are of
direct interest to applications for emission and absorption of radiation, and
may motivate new studies concerning binary stars and black holes in gravity
scenarios of current interest.

160 years ago, Maxwell considered results described by Ampère, Coulomb,
Faraday and Gauss (and others) to write down a set of four equations that
changed the physics in some important ways. These equations and the work of
Hertz helped to set the basics for the generation and detection of
electromagnetic waves. And more, with Lorentz and Einstein, they paved the way
to develop the special theory of relativity. Nowadays, electromagnetism is
seen as an Abelian $U(1)$ gauge theory, described by the gauge field $A_{\mu}$
and the gauge-invariant tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Its Lagrange density
is very well-known in physics, but in the present work we consider a much less
known system, described by the addition of a real scalar field $\phi$ which is
non-minimally coupled to the gauge field in the form

${\mathcal{L}}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{\varepsilon(\phi)}{4}F_{\mu\nu}F^{\mu\nu}-A_{\mu}j^{\mu},$
(1)

in the presence of the (conserved) external current $j_{\mu}$. We notice that
the Maxwell coupling to the scalar field is driven by the non-negative
function $\varepsilon(\phi)$, which governs the electric permittivity of the
system. This was used before in lee ; L  to describe a bag similar to the MIT
mit  and SLAC slac  bag models. It was also considered in several distinct
scenarios, in particular, in Refs. V1 ; V2 ; V3  to describe specific
properties of vortices, in Refs. H1 ; H2 ; H22 ; H3  in connection with the
AdS/CFT correspondence, to model insulators and metals within the holographic
setup and in the context of the gauge/gravity duality, to investigate
hydrodynamic behavior in a hot and dense strongly coupled quark-gluon plasma.
And also in S1 ; S2 ; S3 ; S4 , to study dielectric Skyrme models and possible
connections with the binding energies of nuclei and other related issues. In
the present work, we go beyond the case of a single charge recently
investigated in electric  and consider the presence of an electric dipole
characterized by charge $e$ and distance $d=2a$.

In a way similar to V1 ; V2 , other works have investigated the Maxwell-Scalar
coupling in the magnetic context in some recent investigations; see, e.g.,
Refs. M1 ; M2 ; M3 ; M4 , where the magnetic permeability is considered to
induce internal structure to planar configurations of the vortex type. The
above non-minimal coupling has also been considered in other scenarios, for
instance, in Maxwell-Scalar models in the presence of nontrivial geometric
backgrounds and also in Einstein-Maxwell-Scalar systems, with the scalar field
giving rise to specific effects such as spontaneous scalarization, among
others; see, e.g., G00 ; G0 ; G1 ; G12 ; G2 ; G3 ; G4  and references therein.
Under specific coupling, it is of current interest to enlarge the system to
construct black hole solutions with so-called spin-induced scalarization SI1 ;
SI2 .

In the present work, owing to applications in condensed matter, we suppose the
system lives in the simplest case of a flat spacetime with $(2,1)$ dimensions.
The spatial coordinates are described by $(x,y)$, and an electric dipole is
symmetrically located in the $x$ axis, with the $y$ coordinate added to
describe the plane of reference of the dipole. We start varying the action
associated to the above Lagrange density to get the following equations of
motion

$\displaystyle\partial_{\mu}\partial^{\mu}\phi+\frac{1}{4}\varepsilon_{\phi}F_{\mu\nu}F^{\mu\nu}=0,$
(2a) $\displaystyle\partial_{\mu}\left(\varepsilon F^{\mu\nu}\right)=j^{\nu},$
(2b)

where $\varepsilon_{\phi}=d\varepsilon/d\phi$. We then consider static
configurations with time-independent $j^{0}$ and $j^{i}=0$, so there is no
magnetic field involved in the system. The components of the electric field E
are defined as $E^{i}=(E_{x},E_{y})=F^{i0}$. In this case, the above equations
of motion become

$\displaystyle\nabla^{2}\phi+\frac{1}{2}\varepsilon_{\phi}|\textbf{E}|^{2}=0,$
(3a) $\displaystyle\nabla\cdot(\varepsilon\,\textbf{E})=j^{0}.$ (3b)

Eq. (3b) is the Gauss’ law in the presence of a generalized electric
permittivity. Since we are dealing with static configurations, we can
calculate a conserved energy density standardly, which has the form

$\rho=\frac{1}{2}\left(\nabla\phi\right)^{2}+\frac{1}{2}\varepsilon(\phi)|{\bf
E}|^{2}+A_{0}j^{0}.$ (4)

The charge density of the dipole has the following form $j^{0}=2\pi
e\left(\delta(x-a)\delta(y)-\delta(x+a)\delta(y)\right)$, and the electric
field is obtained from Eq. (3b), which leads us to the expression

$\textbf{E}=\frac{e}{\varepsilon(\phi)}\left(\frac{\hat{r}_{+}}{|\textbf{r}_{+}|}-\frac{\hat{r}_{-}}{|\textbf{r}_{-}|}\right),$
(5)

where $\hat{r}_{\pm}$ identify the unit radial vectors in the direction that
connects the charges $\pm e$ to the point of reference, given by
$\hat{r}_{\pm}=\textbf{r}_{\pm}/|\textbf{r}_{\pm}|$, with
$\textbf{r}_{\pm}=\left(x\mp a\right)\hat{x}\,+\,y\,\hat{y}$. We can see from
Eq. (5) that the electric field is modified by the function
$\varepsilon(\phi)$, which drives the electric permittivity of the medium in
which the charges are placed.

The presence of the dipole suggests that we change the Cartesian $(x,y)$
coordinates to the bipolar $(\xi,u)$ coordinates, in the form
$x=a\sinh\xi/(\cosh\xi-\cos u)$ and $y=a\sin u/(\cosh\xi-\cos u),$ where
$\xi\in\mathbb{R}$ and $u\in[-\pi,\pi)$ are now dimensionless coordinates. The
line element is given by

$ds^{2}=dt^{2}-\frac{a^{2}}{\left(\cosh\xi-\cos
u\right)^{2}}\left(d\xi^{2}+du^{2}\right),$ (6)

and the electric field now reads

$\textbf{E}=-\frac{e}{\varepsilon(\phi)}\,\frac{\cosh\xi-\cos
u}{a}\,\hat{\xi},$ (7)

always pointing in the $\xi$ direction. In this expression, one notices that
the $\cosh\xi$ in the numerator diverges as $\xi\to\pm\infty$. Nevertheless,
the presence of the generalized electric permittivity $\varepsilon(\phi)$ may
be used to modify this behavior. In bipolar coordinates, the equation of
motion of the scalar field (3a) becomes

$\frac{\partial^{2}\phi}{\partial\xi^{2}}+\frac{\partial^{2}\phi}{\partial
u^{2}}+\frac{a^{2}}{2\left(\cosh\xi-\cos
u\right)^{2}}\,\varepsilon_{\phi}|\textbf{E}|^{2}=0.$ (8)

The above Eqs. (7) and (8) allow that we take $\phi=\phi(\xi)$, that is, the
scalar field may not depend on $u$. In this case, for $\phi(\xi)$ one gets

$\frac{d^{2}\phi}{d\xi^{2}}=\frac{d}{d\phi}\left(\frac{e^{2}}{2\varepsilon}\right).$
(9)

Comparing this with the usual equation of motion for kinks in $(1,1)$
dimensions, which has the form $d^{2}\phi/dx^{2}=dV/d\phi$ (see, e.g., Ref.
bazeia ), one can show that the electric permittivity may play the role of a
potential. We define

$V(\phi)=\frac{e^{2}}{2\varepsilon(\phi)},$ (10)

and rewrite Eq. (9) as

$\frac{d^{2}\phi}{d\xi^{2}}=\frac{dV}{d\phi}.$ (11)

To bring to light novel information on the subject, let us now search for
kinklike configuration for the scalar field. We then suppose the potential
engenders two neighbour minima, say $\bar{\phi}_{\pm}$ such that
$V(\bar{\phi}_{\pm})=0$, which may be connected asymptotically by the field
configuration $\phi(\xi)$. In this situation, since the potential and the
permittivity are related by Eq. (10), one can see that the permittivity
diverges at the two zeroes of the potential, and this feature may be used to
compensate the divergence caused by the $\cosh\xi$ in the electric field, as
we have commented below Eq. (7). As usual, the above equation may be
integrated to a first order one, in the form

$\frac{1}{2}\left(\frac{d\phi}{d\xi}\right)^{2}=V(\phi).$ (12)

To get the solution of the gauge field, $A_{0}$, we must take into account
that $\textbf{E}=-\nabla A_{0}$, where the gradient is now given by

$\nabla=\frac{\cosh\xi-\cos
u}{a}\left(\hat{\xi}\,\frac{\partial}{\partial\xi}+\hat{u}\,\frac{\partial}{\partial
u}\right).$ (13)

Using the above expression in Eq. (7) combined with Eq. (10), we get that
$\partial A_{0}/\partial u=0$ and $\partial A_{0}/\partial\xi=2V/e$. So, we
must have $A_{0}=A_{0}(\xi)$, which is given by

$\begin{split}A_{0}(\xi)&=\frac{2}{e}\int\,d\xi\,V(\phi(\xi))=\frac{1}{e}\int\,d\xi\left(\frac{d\phi}{d\xi}\right)^{2}.\end{split}$
(14)

Thus, by knowing the solution $\phi(\xi)$ obtained from Eq. (12), one can
calculate $A_{0}(\xi)$ and then the electric field. The corresponding energy
density is calculated from Eqs. (4); it reads

$\rho=\frac{\left(\cosh\xi\\!-\\!\cos
u\right)^{2}}{2a^{2}}\left(\frac{d\phi}{d\xi}\right)^{2}\\!\\!+\frac{1}{2}\varepsilon(\phi)|{\bf
E}|^{2}+A_{0}j^{0},$ (15)

The electric field may be written in terms of the potential (10) as in Eq.
(7), and this allows to write $\rho=\rho_{f}+\rho_{c}$, representing the
contributions due to the scalar field and the charges, respectively. The two
contributions are

$\displaystyle\rho_{f}$ $\displaystyle=\frac{\left(\cosh\xi-\cos
u\right)^{2}}{a^{2}}\left(\frac{1}{2}\left(\frac{d\phi}{d\xi}\right)^{2}+V(\phi)\right),$
(16a) $\displaystyle\rho_{c}$ $\displaystyle=2\pi
eA_{0}\left(\delta(x-a)\delta(y)-\delta(x+a)\delta(y)\right).$ (16b)

To calculate the energy, we now have to use the equality
$dx\,dy=a^{2}du\,d\xi/(\cosh\xi-\cos u)^{2}$. From Eq. (16b), we get

$\displaystyle E_{c}$ $\displaystyle=2\pi
e\left(A_{0}(x=a,y=0)-A_{0}(x=-a,y=0)\right)$ (17) $\displaystyle=2\pi
e\left(A_{0}(\xi\to\infty)-A_{0}(\xi\to-\infty)\right).$

The contribution due to the scalar field also depends on the specific form of
$\phi(\xi)$. Here we dive further into the construction of stable structures
attaining topological feature; we follow the Bogomol’nyi procedure bogo  and
introduce another function, $W=W(\phi)$, such that

$V(\phi)=\frac{1}{2}\left(\frac{dW}{d\phi}\right)^{2}.$ (18)

This allows that we rewrite $\rho_{f}$ in the form

$\displaystyle\rho_{f}=\frac{\left(\cosh\xi-\cos
u\right)^{2}}{a^{2}}\Bigg{(}\frac{1}{2}\left(\frac{d\phi}{d\xi}\mp\frac{dW}{d\phi}\right)^{2}\\!\pm\frac{dW}{d\xi}\Bigg{)},$
(19)

such that

$E_{f}=2\pi\int_{-\infty}^{\infty}d\xi\;\Bigg{(}\frac{1}{2}\left(\frac{d\phi}{d\xi}\mp\frac{dW}{d\phi}\right)^{2}\\!\pm\frac{dW}{d\xi}\Bigg{)}.$
(20)

This energy can then be minimized to the value

$2\pi|W(\phi(\xi\to\infty))-W(\phi(\xi\to-\infty))|,\\\ $ (21)

when the field configuration $\phi(\xi)$ obeys one of the first order
equations

$\frac{d\phi}{d\xi}=\pm\frac{dW}{d\phi}.$ (22)

One can show that solutions of the above first order equations also solve the
equation of motion (11). Moreover, we follow the lines of Refs. prl ; derrick
; hobart  and make a rescale in the argument of the solution, in the form
$\phi\to\phi_{\kappa}=\phi(\kappa\xi)$, to show that the solutions are stable
under rescaling, so they evade Derrick’s theorem. An extra important gain in
the present case, is that we can write the function $A_{0}$ in Eq. (14) as

$A_{0}(\xi)=\pm\frac{1}{e}W(\phi(\xi)),$ (23)

which makes the energy $E_{c}$ in Eq. (17) to be

$E_{c}=2\pi\left|W(\phi(\xi\to\infty))-W(\phi(\xi\to-\infty))\right|,$ (24)

such that $E_{c}=E_{f}$. In Eq. (23), we have discarded a constant of
integration because it plays no role in the energy as one can see from the
above expression. In the case of a single charge that we investigated in Ref.
electric , the constant of integration can be seen as a reference for the
energy. Here, since we have two charges of opposite signs, it has no
importance and can be set to zero. The procedure naturally reveals how to
remove the singular behavior of the energy associated to the standard case
($\varepsilon=1$) due to the presence of the two charges. For simplicity, we
take $e=1$ from now on.

Figure 1: The solution in Eq. (29) is depicted in the plane $(x,y)$ for $a=1$
and $v=1$. In the top panels we take $\lambda=1$ and $\xi_{0}=0$, $0.2$, and
$0.4$, from left to right. In the bottom panels we take $\xi_{0}=0$ and
$\lambda=0.4$, $0.6$ and $0.8$, also from left to right. The red color stands
for $\phi=-1$, the white color represents $\phi=0$ and the blue denotes
$\phi=1$.

To illustrate the above procedure, we consider the model described by the
polynomial function

$W(\phi)=\lambda v\phi-\frac{\lambda}{3v}\phi^{3},$ (25)

where $\lambda$ and $v$ are positive parameters. In this case, the potential
in Eq. (18) and the electric permittivity in Eq. (10) change to

$\displaystyle V(\phi)$
$\displaystyle=\frac{\lambda^{2}v^{2}}{2}\\!\left(1-\frac{\phi^{2}}{v^{2}}\right)^{2},$
(26) $\displaystyle\varepsilon(\phi)$
$\displaystyle=\left(\lambda^{2}v^{2}\left(1-\frac{\phi^{2}}{v^{2}}\right)^{2}\right)^{-1}.$
(27)

The above $V(\phi)$ has minima at $\bar{\phi}_{\pm}=\pm v$ and a local maximum
at $\phi=0$, such that $V(0)=\lambda^{2}v^{2}/2$. Also, the permittivity is
constant at $\phi=0$ ($\varepsilon(0)=1/\lambda^{2}v^{2}$) and increases to
higher and higher values as the scalar field approaches the minima $\pm v$.
The presence of the double-well potential is mandatory to provide the kinklike
configuration to be used to regularize the behavior of the electric field at
the two charges. The model is also motivated by the Landau-Ginzburg-Devonshire
theory for ferroelectric materials, which may describe negative capacitance
and improve the energy efficiency of conventional electronics (see, e.g.,
nature  and references therein). The first order equation (12) with positive
sign now reads

$\frac{d\phi}{d\xi}=\lambda v\\!\left(1-\frac{\phi^{2}}{v^{2}}\right),$ (28)

and the solution which connects the aforementioned minima of $V(\phi)$ is

$\phi(\xi)=v\tanh\\!\left(\lambda(\xi-\xi_{0})\right),$ (29)

where $\xi_{0}$ determines the point in the $\xi$ axis where the solution
changes its sign. The situation is similar to the case of kinks in the real
line bazeia , and $\xi_{0}$ is an integration constant which does not modify
the energy of the solution. The behavior of this analytical result is depicted
in Fig. 1 in the $(x,y)$ plane for $a=1$ and $v=1$, and for several values of
$\xi_{0}$ and $\lambda$. In this figure, the white region represents $\phi=0$,
which requires $\xi=\xi_{0}$. In the top panel the circle appears in
accordance with $(x-a\coth\xi_{0})^{2}+y^{2}=a^{2}/\sinh^{2}\xi_{0}$, so it
has radius $a/\sinh\xi_{0}$ and is centered at the point
$(\tilde{x},\tilde{y})=(a\coth\xi_{0},0)$. In the present context, $\xi_{0}$
can be used to induce an asymmetry in the system, which also appears in the
permittivity and in the electric field, as we show below. The asymmetry can be
used for applications of practical interest.

Moreover, the contribution of the field for the energy density $\rho_{f}$ is
given by Eq. (16a). We have

$\rho_{f}=\frac{\left(\cosh\xi-\cos u\right)^{2}}{a^{2}}\,\lambda^{2}v^{2}{\rm
sech}^{4}(\lambda(\xi-\xi_{0})).$ (30)

We can use Eqs. (21) and (24) to show that $E_{f}=E_{c}=8\pi\lambda v^{2}/3$,
giving the total energy $E=16\pi\lambda v^{2}/3$.

The electric potential and the electric field are obtained from Eqs. (23) and
(7). The electric potential in Eq. (23) becomes

$A_{0}(\xi)=\lambda
v^{2}\\!\\!\left(\\!\tanh(\lambda(\xi\\!-\\!\xi_{0}))\\!-\\!\frac{1}{3}\tanh^{3}(\lambda(\xi\\!-\\!\xi_{0}))\\!\right)\\!,$
(31)

and has a smooth behavior as one approaches the electric charges:
$A_{0}(\xi\to\pm\infty)\to\pm 2\lambda v^{2}/3$. The electric field, however,
has the form

E $\displaystyle=-\frac{\lambda^{2}v^{2}}{a}(\cosh\xi-\cos u)\,{\rm
sech}^{4}(\lambda(\xi\\!-\\!\xi_{0}))\,\hat{\xi}.$ (32)

The presence of the generalized electric permittivity in Eq. (27), may now
contribute to modify the singular character of the standard Coulomb’s law at
the location of the charges. To see this, one notices that near the charges
one has $\xi_{\pm}\approx\pm\ln\left(2a/\sqrt{(x\mp a)^{2}+y^{2}}\right)$, so
we can expand the above expression of the electric field to get

$|\textbf{E}|\approx\frac{16^{(1-\lambda)}\lambda^{2}v^{2}}{a^{4\lambda}}\left((x\mp
a)^{2}+y^{2}\right)^{2\lambda-1/2}.$ (33)

The quantity $2\lambda-1/2$ appears from the interplay between the bipolar
coordinate system and the field configuration that solves the model.

We then see that the behavior of the electric field depends on $a$, $v$ and
$\lambda$, with $\lambda_{c}=1/4$ being a critical value, for which the
electric field is ill-defined at the location of the charges due to the
presence of the unit vector $\hat{\xi}$. At the charges, for
$\lambda<\lambda_{c}$ the electric field is divergent, as it is in the
standard case, and for $\lambda>\lambda_{c}$ the electric field vanishes.
Since the case of $\lambda$ greater than $1/4$ induces a novel scenario, we
then focus on $\lambda>1/4$ from now on.

Figure 2: The electric permittivity of Eq. (34), depicted in the $(x,y)$ plane
for $a=1$, $v=1$, $\lambda=1$ and for $\xi_{0}=0,0.2$ and $0.4$, from left to
right. Here, the white color represents the vacuum $(\varepsilon=1)$, and the
darkness of the brown color increases with the enlargement of the
permittivity.

Figure 3: The electric field in Eq. (32) is depicted in the $(x,y)$ plane for
$a=1$ and $v=1$. In the top panels, we take $\lambda=1$ and $\xi_{0}=0$,
$0.2$, and $0.4$, from left to right. In the bottom panels, we take
$\xi_{0}=0$ and $\lambda=1/2$, $1$ and $3/2$, also from left to right. The
color white represents vanishing electric field, and the shades of green are
used to describe its variation, with the darkest standing for its maximum.

In the above model, for the field configuration shown in Eq. (29), the
electric permittivity becomes

$\varepsilon(\xi)=\lambda^{-2}v^{-2}\cosh^{4}(\lambda(\xi-\xi_{0})).$ (34)

It also introduces an asymmetry for $\xi_{0}\neq 0$, so in Fig. 2 we depict
this permittivity using $a=1$, $v=1$ and $\lambda=1$, for some values of
$\xi_{0}$, remembering that $\varepsilon=1$ represents the permittivity of the
vacuum, which in the figure is identified with the color white. To illustrate
the behavior of the electric field, in Fig. 3 we depict the analytical results
in Eq. (32) in the $(x,y)$ plane for $a=1$, $v=1$, and for several values of
$\xi_{0}$ and $\lambda$.

In summary, in this work we have investigated a model in which a single real
scalar field is coupled to the Maxwell field through an electric permittivity.
We studied the case in which two point charges with opposite signs are
symmetrically fixed in the $x$ axis, representing an electric dipole, using an
interesting model of permittivity, inspired by the Landau-Ginzburg-Devonshire
theory for ferroelectric materials nature , which allows for the presence of
exact solutions. The study unveils an interesting behavior of the electric
field, showing that it may vanish at the positions of the electric charges.
This is very different from the standard situation, and is attained when the
two charges of the electric dipole are encapsulated inside regions with
permittivity that increases to larger and larger values, as we approach the
charges. The results suggest an interesting modification of the standard
electric dipole, which we believe is not hard to construct in the laboratory,
if one uses ceramic or other material with very high dielectric constants;
see, e.g., Refs. VH1 ; VH2 ; VH3 ; VH4 ; VH5  for more details on the use of
ceramic VH1 ; VH2 , dielectric film VH3 ; VH4  and gel VH5  elements. The
encapsulated dipole engenders new physics, so it may be used for practical
purposes of current interest in applications including emission and absorption
of radiation. Moreover, as one can see from Refs. electric ; G4 , in the
Maxwell-Scalar model we can trade off the electric charge for a nontrivial
gravitational background. In this sense, the above results are of direct
interest to study binary stars, binary black holes and other similar
astrophysical systems. In particular, it seems of current interest to
investigate how the process of scalarization works for binary black holes
systems G0 ; G1 ; SI1 ; SI2 ; mer1 ; mer2 .

The Maxwell-Scalar system considered in (1) may also be studied to unveil
other possibilities, for instance, the modification of the model considered in
Eq. (25) to describe different features of the field configuration that solve
the equation of motion, inducing novel possibilities of applications
concerning negative capacitance effects in ferroelectric materials nature ;
new . These and other related issues are presently under consideration, and we
hope to report on them in the near future.

###### Acknowledgements.

This work is supported by the Brazilian agencies Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), grants Nos. 404913/2018-0
(DB), 303469/2019-6 (DB) and 306504/2018-9 (RM), Paraiba State Research
Foundation (FAPESQ-PB) grants Nos. 0003/2019 (RM) and 0015/2019 (DB and MAM)
and by Federal University of Paraíba (PROPESQ/PRPG/UFPB) project code
PII13363-2020.

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