2023

# A deep learning-based approach for identifying unresolved questions on Stack
Exchange Q&A communities through graph-based communication modelling

Hassan Abedi Firouzjaei<EMAIL_ADDRESS>Department of Computer Science,
Norwegian University of Science and Technology (NTNU), Trondheim, Norway

###### Abstract

In recent years, online question-answering (Q&A) platforms, such as Stack
Exchange (SE), have become increasingly popular for information and knowledge
sharing. Despite the vast amount of information available on these platforms,
many questions remain unresolved. In this work, we aim to address this issue
by proposing a novel approach to identify unresolved questions in SE Q&A
communities. Our approach utilises the graph structure of communication formed
around a question by users to model the communication network surrounding it.
We employ a property graph model and graph neural networks (GNNs), which can
effectively capture both the structure of communication and the content of
messages exchanged among users. By leveraging the power of graph
representation and GNNs, our approach can effectively identify unresolved
questions in SE communities. Experimental results on the complete historical
data from three distinct Q&A communities demonstrate the superiority of our
proposed approach over baseline methods that only consider the content of
questions. Finally, our work represents a first but important step towards
better understanding the factors that can affect questions becoming and
remaining unresolved in SE communities.

###### keywords:

Q&A communities, Graph neural networks, Few-shot learning, Stack Exchange,
Deep learning

##  1 Introduction

Stack Exchange (SE)111https://stackexchange.com is a large online platform
that hosts over 160 question-answering (Q&A) communities covering a plethora
of topics. At a basic level, on SE, a user asks a question which in turn is
answered by others aiming to give the question a satisfactory answer. A
question is considered open or unresolved until one of its answers is selected
as accepted. This simple process helps facilitate the flow of knowledge from
experts to people searching for high-quality information (Abedi Firouzjaei,
2022). Nevertheless, many questions on SE may not receive an accepted answer
and thus become unresolved. Moreover, a question can be unresolved due to many
factors, including the novelty of the topic of the question or its
specificity, or even its duplicity. For example, a question can be closed by
the community moderators shortly after it was posted because its topic is
deemed too narrow. A recent study shows that about half of the questions on
Stack Overflow, the largest Q&A community on SE, have yet to get an accepted
answer (Yazdaninia et al, 2021). And the percentage of unresolved questions
has been increasing for many communities hosted on the SE platform. Fig. 1
shows the same decreasing trend for another community, namely Computer Science
SE.

A graph neural network (GNN) is a deep learning model operating on graph-
structured data (Wu et al, 2022). GNNs have been successfully applied to many
tasks, such as node classification, link prediction, and graph classification.
They are designed to handle the unique characteristics of graph data, such as
variability in size and non-Euclidean structures, primarily by leveraging the
graph’s topology to propagate information throughout the network. GNNs
typically use a message-passing mechanism to aggregate information from
neighbouring nodes. This allows the model to learn representations of the
graph’s structure and node attributes. Moreover, recent advancements in GNNs
have resulted in the development of various architectures, such as graph
convolutional networks (GCNs) (Kipf and Welling, 2016) utilising convolutions
for graphs (Wu et al, 2022), graph attention networks (GATs) (Velickovic et
al, 2017) based on the attention mechanism (Chorowski et al, 2015; Vaswani et
al, 2017), and graph transformer networks (GTNs) (Yun et al, 2019) utilising
transformers (Vaswani et al, 2017). These architectures have achieved state-
of-the-art performance on various graph-based tasks such as graph and node
classification. They have been applied to problems in chemistry, social
network analysis, and computer vision (Wu et al, 2022). Furthermore, as graph-
structured data grows, GNNs are becoming increasingly crucial for various
applications.

The property graph model (PGM) is a flexible and powerful data model used to
represent and store graph-structured data (Bonifati et al, 2018). It is based
on a simple yet expressive set of concepts: nodes, edges, and properties.
Nodes represent entities in the graph, and edges represent relationships
between nodes. Properties are key-value pairs associated with nodes and edges,
allowing the graph to store rich, semi-structured data. Moreover, one of the
key advantages of the PGM is its ability to handle complex, multi-relational
data. For example, it can be used to model social networks, where nodes
represent people and edges represent relationships such as friends or family.
The properties of nodes and edges can include things like name, age, location,
and interests. The PGM also allows easy expression of complex queries and
traversals, which is helpful for data analysis. This, in turn, makes the PGM
popular for many applications, including graph databases and graph-based
machine learning.

In this work, we propose a novel and reliable approach utilising the PGM and
GNNs for identifying unresolved questions in Stack Exchange Q&A communities
with high accuracy. We aim to examine and study the possible causes that may
lead to unresolved questions; our approach is a first step towards that goal.
In our proposed approach, we first model the communication network surrounding
each question using the PGM, construct a communication graph for each
question, and then utilise state-of-the-art GNN-based techniques to accurately
detect unresolved questions. By communication network, we mean the network of
messages (mainly in the form of answers and comments) users exchange to
resolve a question. Our central hypothesis is that the expressive power of
GNNs makes them suitable tools for investigating the problem of unresolved
questions. Fig. 2 shows a high-level overview of the proposed approach.
Furthermore, we conduct thorough experiments to evaluate the effectiveness of
our approach in comparison to other state-of-the-art methods which do not
utilise the interconnected structure of the communication network of users.

In summary, the followings are the main contributions of our work:

  * •

We propose a method to model the communication network formed around a
question using the PGM, which can express both content (i.e., the text of
messages) and the structure of the communication (i.e., communication
patterns).

  * •

We propose a novel approach utilising GNNs that can reliably and accurately
identify and detect unresolved questions on Q&A communities on the SE
platform.

  * •

We experimentally evaluate the effectiveness of our approach against the
baselines on real-world historical data of three distinct Q&A communities.

  * •

We make the code and data used in our work publicly available so other
researchers can reproduce and build on the work described in this article more
easily (see Declarations).

In addition, broadly speaking, we believe that the work presented here can
contribute to the following subjects:

  * -

Academic Research. In the academic realm, understanding why certain questions
remain unresolved could provide insights into knowledge gaps in specific
areas. These insights, for example, could be used to guide curriculum
development, direct research efforts, or inspire the creation of new courses.

  * -

User Experience Enhancement. Identifying unresolved questions can be used to
improve the user experience by notifying users about unresolved questions in
their areas of expertise, encouraging them to contribute and enhancing their
engagement on the platform.

  * -

Question Quality Analysis. Over time, our model could be used to analyse the
quality of questions being asked. If a significant number of questions remain
unresolved, it could suggest that the questions are unclear or too broad. This
information could lead to the development of guidelines or resources to help
users ask better questions which in turn can improve user satisfaction.

The rest of this article is organised as follows: Sect. 2 describes and
discusses the related work. Sect. 3 introduces the essential concepts and
techniques needed to understand better the methodology used in this study.
Sect. 4 provides the details of the data used in the experiments. Sect. 5
provides information about the experiments, including the information about
the baselines and the evaluation metrics. Sect. 6 presents the results and
discusses the subsequent important findings and Sect. 7 discusses the
limitations of our work and suggests future work. Finally, Sect. 8 concludes
this article.

Figure 1: An illustrative example showing that for the Computer Science SE,
like many other communities on the SE platform, the percentage of resolved
questions (i.e., questions with accepted answers) followed a decreasing trend
over the years

Figure 2: Overview of the proposed approach

(a) The actual question

(b) The communication graph of the question (with id = 253); the blue nodes
represent users, the purple nodes represent answers, the orange nodes
represent comments, and the yellow node in the middle represents the question.
Note that the text property fields of the nodes are not shown

Figure 3: An illustrative example showing the question with id 253 from the
Data Science SE community; the picture of the actual question can be seen on
the left; the corresponding communication graph (which was created by
modelling the corresponding communication network as a property graph) is
shown on the right

##  2 Related work

In recent years, the interest in studying phenomena in Q&A platforms such as
Stack Exchange and Quora222https://www.quora.com has skyrocketed, as these
platforms offer a user-friendly setting for sharing accessible knowledge. For
example, authors in (Anderson et al, 2012) explored the foundations for
understanding community processes in Q&A platforms, using Stack Overflow as an
example. Their work highlights the importance of considering the best answer
to a question, the set of answers, and the processes that create them. They
used the temporal structure of Stack Overflow to identify properties of
questions and their answers that are likely to have lasting value and those
that require further community involvement. Furthermore, authors in
(Asaduzzaman et al, 2013) developed a taxonomy to understand the factors
involved in questions receiving no answers on Stack Overflow. They used the
taxonomy to build a classifier that predicts how long a question will remain
unanswered. They found that about 7.5% of the questions on Stack Overflow are
unanswered. And it suggests that identifying the quality of a question could
also help better determine whether a question will receive an answer.
Similarly, authors in (Goderie et al, 2015) investigated the possibility that
the time between a question is posted and when it gets a response on the Stack
Overflow Q&A community could be predicted with reasonable accuracy. They
achieved an accuracy of around 30% to 35%.

Furthermore, and more relevant to our work, authors in (Yazdaninia et al,
2021) developed a predictive model utilising XGBoost (Chen and Guestrin, 2016)
to identify unresolved questions on Stack Overflow. Their model requires
extensive feature engineering to work correctly. In contrast, our approach
uses state-of-the-art text embedding methods to minimise the need for hand-
crafted feature engineering, which is typically challenging.

A very much recent development in the area of automated Q&A is the emergence
of AI-powered conversational systems such as ChatGPT (Bubeck et al, 2023),
which benefited from the recent advancement in the field of natural language
processing (NLP). Although the effect of the use of systems like ChatGPT is
not still well-understood due to the recency of their emergence, nonetheless,
it has been a growth in the amount of attention poured into this area by
scholars recently. In (Sanatizadeh et al, 2023), authors explored the impact
of large language models (LLMs), like ChatGPT and Bard, on online Q&A
platforms, specifically focusing on the Stack Exchange platform. Their study
reveals two key trends: an increase in the quality and complexity of both
questions and answers and a decrease in platform engagement (visits, posts,
user activity), particularly in the Technology sector post-LLM introduction.
Their research suggests that while LLMs enhance content depth, they may reduce
overall user engagement, presenting a potential need for future research. Work
presented in (Xue et al, 2023) analysed the impact of LLMs, such as ChatGPT,
on Q&A communities, using Stack Overflow as a case study. Following ChatGPT’s
launch, a 2.64% average reduction in question-asking was observed, suggesting
that LLMs decrease search costs, leading to fewer but potentially more
engaging and higher-quality questions. However, while questions became 2.7%
longer (indicating sophistication), they were found to be less readable and
cognitively challenging, possibly making them difficult for LLMs to understand
and process. Further, no significant change in viewer scores suggested no
improvement in question quality and decreased engagement across the platform.
The study concludes that LLMs might pose a risk to the survival of Q&A
communities, potentially impacting LLMs’ sustainable learning and long-term
improvement, with new users being the most affected. Finally, (Sohail et al,
2023) conducted a comprehensive survey of over 100 recent publications on
ChatGPT. It has demonstrated impressive achievements since its inception in
November 2022 but still grapples with biases and a lack of trust. The authors
propose a taxonomy for ChatGPT research, identify common methodologies, and
explore its application areas and critical issues. The paper also outlines
future research directions, suggests solutions to current challenges, and
speculates on future advancements. It is presented as the first comprehensive
review of ChatGPT and emphasises the vast potential for further research and
development across diverse application areas. Overall, the emergence of
ChatGPT and LLM-related technologies seems like an exciting direction for
future research, which can open up new rays of insight into the working of Q&A
platforms, especially concerning user engagement and satisfaction.

Overall, recent advancements in natural language processing have led to the
development of AI-powered conversational systems, such as ChatGPT. While their
impact is still being studied, research suggests that they enhance the quality
and complexity of questions and answers, but may decrease user engagement. For
example, a reduction in question-asking was observed following ChatGPT’s
launch, potentially impacting the sustainability of Q&A communities. However,
the technology’s potential for future research and development across diverse
application areas is vast. Overall, the emergence of ChatGPT and LLM-related
technologies offer exciting possibilities for gaining new insights into Q&A
platforms and user satisfaction.

##  3 Preamble

###  3.1 Property graph model

In this work, we used the property graph model to represent the communication
network of users. We created the corresponding communication graph of a
communication network by carefully designing a general schema that could
express both the content (i.e., messages exchanged by users) and the structure
of communication taking place between users. Formally, the communication graph
is a quintuple $G=(V,E,\mu,\lambda,\theta)$ where $V$ is the set of nodes. A
node $v\in V$ can be of the following items: a question, an answer, a comment,
or a user. Moreover, $E$ is the set of edges and
$\mu\mathrel{\mathop{\mathchar 58\relax}}E\to V\times V$ is a function that
assigns an ordered pair of nodes to each edge $e\in E$. And
$\lambda\mathrel{\mathop{\mathchar 58\relax}}V\cup E\to L$ is a function that
assigns each node or edge a label from the label set $L$. And
$\theta\mathrel{\mathop{\mathchar 58\relax}}(V\cup E)\times K\to N$ is a
function that assigns values to the properties assigned to each node or edge;
$K$ is the set of properties that a node or an edge can have, and $N$ is the
set of values that the properties can accept. Tab. 1 includes the information
about the nodes, and Tab. 2 presents the information about the edges in the
communication graph. Furthermore, Fig. 4 shows the schema of the communication
graph. And Tab. 5 presents the statistical characteristics of the set of
communication graphs in each of the datasets introduced in Sect. 4.

Table 1: Information about the nodes including their labels and the respective properties and values Node type | Label | Property | Value
---|---|---|---
Question | Question | id | Unique question id
text | Texts of the title and body of the question (concatenated)
Answer | Answer | id | Unique answer id
text | Text of the answer
Comment | Comment | id | Unique comment id
text | Text of the comment
User | User | id | Unique user id
text |  Text from AboutMe field of the user’s profile
Table 2: The information of the edge labels. Notice that edges do not have any properties Edge label | Source node label | Destination node label | Descritpion
---|---|---|---
POSTS | User | Question | User posted the question
Answer | User posted the answer
Comment | User posted the comment
ANSWERS | Answer | Question | Answer answered the question
COMMENTS | Comment | Question | Comment commented the question
Answer | Comment commented the answer

Figure 4: The figure shows the schema of the communication graph, including
the different nodes and edges distinguishable by their respective labels

###  3.2 Graph neural networks

Graph neural networks (GNNs) are deep neural networks that utilise the linked
structure of the data they operate on. In principle, GNNs work by iteratively
updating node representations. They achieve this by combining every node’s
representation with its neighbouring nodes’ representations in each iteration.
Formally, given graph $G=(V,E)$ where $V$ is the set of nodes, and $E$ is the
set of edges, let $H^{0}$ be the set of initial node representations where
$H^{0}_{v}$ is the initial representation of node $v$. Then, to implement a
GNN over $G$, we need to devise a neural network with the following two
important functions at each layer: an aggregation function and a combination
function. More specifically, starting with initial node representations
$H^{0}$, and number $k\in\\{1,2,3,...,K\\}$ which indicates the $k$th
iteration, the aggregation and combination functions will have the following
signatures:

  * •

Aggregation: $a^{k}_{v}=Aggregate(H^{k-1}_{u})$ where $u\in N(v)$ is a
neighbouring node of $v$ (i.e., $N(v)$ is the set of all nodes adjacent to
$v$). And $a^{k}_{v}$ is an aggregate representation of the representations of
all of the neighbours of $v$ in the $(k-1)$th iteration.

  * •

Combination: $H^{k}_{v}=Combine(H^{k-1}_{v},a^{k}_{v})$.

As you can see, at iteration $k$, for each node $v\in V$, the aggregation
function produces an aggregate representation (i.e., $a^{k}_{v}$) for the
neighbours of $v$ at iteration $k-1$. Then the combination function combines
this aggregate representation with the representation of $v$ at iteration
$k-1$ in order to generate the updated representation for $v$. Finally, after
$K$ iterations, $H^{K}$ would be the final node representations which can be
utilised for downstream machine learning tasks such as the graph or node
classification.

Since the introduction of the GNNs by authors of (Scarselli et al, 2009),
there has been a growing interest in the GNNs from the research community,
which has led to the invention of many different architectures and types of
GNNs. One significant difference between various GNN architectures is how they
define and implement their aggregation and combination functions. In this
study, we utilised two types of GNN architectures, namely, graph convolutional
neural networks (GCNs) (Kipf and Welling, 2016), arguably the most popular
graph neural network architecture due to its simplicity and effectiveness for
solving various tasks and applications (Wu et al, 2022). And general neural
networks (GGNNs) (You et al, 2020), a more recent architecture that has shown
good performance for graph classification.

###  3.3 Few-shot learning

Few-shot learning is a machine learning task aiming to train a classifier
using a small number of labelled samples. More specifically, given $c$ classes
and $k$ samples, the objective is to use $m$ samples per class, where $m<<k$,
to train the classifier.

In this work, we utilised SetFit few-shot learning framework (Tunstall et al,
2022) from HuggingFace333https://huggingface.co. SetFit is a state-of-the-art
few-shot learning framework that uses a contrastive learning approach to fine-
tune an already-trained sentence transformer model (Reimers and Gurevych,
2019). Sentence transformers are modifications of pre-trained transformer
models that use Siamese and triplet network structures to derive semantically
meaningful dense sentence embeddings for input text sequences (Reimers and
Gurevych, 2019).

Formally, for a binary classification task, the training sample set for
contrastive learning is created as follows: given a small dataset of samples
$D=\\{(x_{i},y_{i})\\}$, where $x_{i}$ is a sample, and $y_{i}\in\\{0,1\\}$ is
the corresponding label, two sets of paired samples are created. Namely, the
set of positive samples $R^{c}$ and the set of negative samples $N^{c}$ where
$c\in\\{0,1\\}$ indicates class label. A triplet $(x_{i},x_{j},1)\in R^{c}$
has the property that $y_{i}=y_{j}=c$ (the labels are the same and equal to
$c$), and for $(x_{i},x_{j},0)\in N^{c}$, respectively, $y_{i}=c$ and
$y_{j}\neq c$ (the labels are different and only the first sample belongs to
the class with label $c$). Moreover, $R^{c}$ and $N^{c}$ are constructed by
randomly picking samples from each class, and $\mathinner{\\!\left\lvert
R\right\rvert}=\mathinner{\\!\left\lvert N\right\rvert}$. Then the training
set $T$ is made by concatenating the corresponding subsets of samples from the
two sets as $T=\\{(R^{0},N^{0}),(R^{1},N^{1})\\}$. Using this method, given a
small sample size, $m$, the size of the training set $T$ can be as large as
${m\times(m-1)}\over{2}$ samples. Consequently, even a few labelled initial
examples can generate enough samples for contrastive learning.

The SetFit framework consists of two major components: a (model) fine-tuner
and a classification head. The fine-tuner uses contrastive learning to fine-
tune an existing sentence transformer model. The classification head uses
embeddings from the fine-tuned model to classify the samples using a logistic
regression learner (Yu et al, 2011).

In this work, we opted to use a few-shot learning-based approach, more
specifically SetFit framework, as baselines since transformer-based neural
networks, which power SetFit, such as BERT (Devlin et al, 2018) have shown
excellent performance in applications such as text classification and
embedding.

##  4 Data

###  4.1 Data description

In the work described in this article, we used the data from (Abedi
Firouzjaei, 2022). The data include the complete historical data of three SE
communities, namely, Computer Science (CS) SE, Data Science (DS) SE, and
Political Science (PS) SE, from their inception up until May 2021. The main
reason we chose these datasets was that they represent distinct communities of
users with different interests, which allowed us to more reliably evaluate and
compare the performance of our approach with other methods. Tab. 3 includes
the information about the datasets used in this work. For more extensive
information about the datasets, including the detailed characteristics of the
communities these datasets represent, please see (Abedi Firouzjaei, 2022).

Table 3: Information about the datasets Characteristic | Dataset
---|---
Pol | DS | CS
#Questions | 11,853 | 28,768 | 39,794
#Users | 31,242 | 100,582 | 113,434
#Answers | 24,712 | 32,107 | 45,517
#Comments | 126,838 | 63,677 | 136,085
% of resolved questions | 52% | 34% | 46%
Foundation year | 2012 | 2014 | 2008

###  4.2 Node representations

We used a state-of-the-art sentence transformer model, namely all-
MiniLM-L6-v2444https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2
to transform user-generated text (i.e., posts, comments, etc.) into
384-dimensional semantic embedding vectors. We used the semantic embedding
vectors as features for one of the baseline methods (i.e., the logistic
regression learner) and also to create node representations. In addition,
specifically, to be used in our approach, we embedded the information about
the node type in the communication graphs as a sparse one-shot vector.
Furthermore, we used a binary coding scheme to represent the types of the
nodes. Tab. 4 shows the information about the representation of the features
attached to nodes in the communication graphs used in our experiments. Please,
see the implementation files for more detail (see Declarations).

Table 4: The specification of features assigned to each type of the nodes in the communication graph Representation |  | Format | Description
---|---|---|---
Text embedding |  |  $[a_{0},a_{2},...,a_{383}]$, $0\leq a_{i}\leq 1$ | Semantic embedding vector of the text
Node type | Question | [0, 0, 0, 1] | Vector representation of a question node
Answer | [0, 0, 1, 0] | Vector representation of an answer node
Comment | [0, 1, 0, 0] | Vector representation of a comment node
User | [1, 0, 0, 0] | Vector representation of a user node
Table 5: Statistical characteristics of the communication graphs in each dataset Dataset | Characteristic | #Nodes | #Edges | #Answers | #Comments | #Users
---|---|---|---|---|---|---
Pol SE | Mean | 21.35 | 26.57 | 2.08 | 10.70 | 7.56
Mode | 4 | 7 | 1 | 2 | 3
Median | 15 | 17 | 1 | 7 | 6
First Quartile | 8 | 9 | 1 | 3 | 3
Third Quartile | 26 | 33 | 3 | 13 | 9
Standard Deviation | 21.65 | 30.24 | 2.05 | 13.48 | 6.91
Maximum | 276 | 399 | 27 | 179 | 76
Minimum | 2 | 1 | 0 | 0 | 1
DS SE | Mean | 7.03 | 7.68 | 1.13 | 2.21 | 2.69
Mode | 4 | 3 | 1 | 0 | 2
Median | 6 | 5 | 1 | 1 | 2
First Quartile | 4 | 3 | 1 | 0 | 2
Third Quartile | 9 | 9 | 1 | 3 | 3
Standard Deviation | 4.95 | 7.11 | 1.09 | 3.02 | 1.72
Maximum | 104 | 127 | 35 | 52 | 44
Minimum | 2 | 1 | 0 | 0 | 1
CS SE | Mean | 9.55 | 11.31 | 1.13 | 4.02 | 3.40
Mode | 6 | 5 | 1 | 0 | 2
Median | 8 | 9 | 1 | 3 | 3
First Quartile | 6 | 5 | 1 | 1 | 2
Third Quartile | 12 | 15 | 1 | 5 | 4
Standard Deviation | 6.81 | 10.18 | 1.01 | 4.57 | 2.09
Maximum | 137 | 183 | 13 | 81 | 49
Minimum | 2 | 1 | 0 | 0 | 1

##  5 Experiments

###  5.1 Methods

As mentioned earlier, we used two GNN architectures (or methods) in our
approach: the GCN and GGNN. Furthermore, for each of the two methods, we
experimented with three distinct types of node representations, including the
following:

  * •

The semantic embedding vector of the text property of the node, plus the
information related to the type of the node. From now on, we refer to this
type of representation as text embeddings plus node type.

  * •

Only semantic embedding vector of the text property of the node. From now on,
we refer to this type of representation as (only) text embeddings.

  * •

Only, the information related to the type of the node. From now on, we refer
to this type of representation as (only) node type.

Furthermore, we compared the performance of the methods used in our approach
with the performance of the following baseline learners:

  * •

Logistic regression: we used a logistic regression learner to predict whether
a question is unresolved. We used the semantic embedding of the text property
of the question nodes in communication graphs as features. We used the all-
MiniLM-L6-v2 sentence transformer model to generate the embeddings.

  * •

Few-shot learning: as mentioned earlier, we used SetFit (with all-MiniLM-L6-v2
as the base model) to train three few-shot learning models to detect the
unresolved questions. More specifically, we used 5, 10, and 20 shots (or
samples) from each class to fine-tune each of the models as mentioned above,
respectively. We used the text property of the question nodes in communication
graphs as input to these models.

###  5.2 Evaluation metrics

For evaluating the performance of the methods used in our approach and the
baseline methods, we used the following metrics:

  * •

Accuracy, which is the ratio of the correct prediction to the total number of
predictions made. Arguably, accuracy is the most critical measure in this work
because it can express the predictive power of each method with clarity.

  * •

Recall, which is the number of true positive predictions (i.e., correctly
classified positive samples) out of all positive samples in the test dataset.

  * •

Precision, which is the number of true positive predictions out of all
positive predictions made by the model. A high precision value means the model
avoids false positive predictions (i.e., incorrectly classified positive
samples).

  * •

F1-score, which is the harmonic mean of precision and recall, balances both
precision and recall. F1-score is an excellent metric to use when seeking a
balance between precision and recall, as it considers both. Generally, a high
f1-score value suggests the model has achieved a good balance between
precision and recall.

###  5.3 Experimental settings

We used stratified 5-fold cross-validation to both find the optimal values of
the hyperparameters and evaluate each method’s performance. We trained the two
methods in our approach for 400 epochs with a batch size of 32 and a learning
rate of $10^{-3}$. Moreover, we trained the learners used for few-shot
learning for one epoch with a batch size of 16 and a learning rate of
$10^{-5}$ to fine-tune the base model. In addition, we have shared the code we
developed and used to run the experiments on GitHub.com (see Declarations).
This includes the implementation of each method and the complete information
about the hyperparameters used during the experiments. Furthermore, we
employed the 5x2cv paired t-test (Dietterich, 1998), setting a significance
threshold (p-value) of 0.005 to verify the statistical significance of
differences in accuracy between the proposed models and the baselines. The
null hypothesis of this test assumes that the two models under comparison have
identical performance. Lastly, we used a comprehensive suite of software and
libraries, including Neo4J Desktop555https://neo4j.com/,
Spektral666https://graphneural.network/, Scikit-learn777https://scikit-
learn.org/stable/, PyTorch888https://pytorch.org/,
TensorFlow999https://www.tensorflow.org/, and
SetFit101010https://github.com/huggingface/setfit, to preprocess the data and
implement the code used in the experiments.

Table 6: The GCN model is a sequential GNN model which mainly consists of three graph convolutional layers (GConv) (Kipf and Welling, 2016) followed by a global average pooling layer (GlobAvgPool) (Lin et al, 2013), excluding the input and the output layers. Furthermore, all the activation functions are of type rectified linear unit (ReLU) (Fukushima, 1969). Overall, the model has relative a small number of neurons, i.e., 4.5k Layer (type) | #Param
---|---
layer 1 (GConv) | 288
layer 2 (GConv) | 2080
layer 3 (GConv) | 2080
layer 4 (GlobAvgPool) | 0
Table 7: Similar to the GCN model, GGNN is also a sequential GNN model with all the activation functions being parametric ReLU (PReLU) (He et al, 2015). However, the model, with over a million neurons, is much more complex in comparison. Please see (You et al, 2020) for more information, including the architectural details. Layer (type) | #Param
---|---
layer 1 (Concat) | 0
layer 2 (GlobSum) | 0
layer 3 (GenConv) | 132,608
layer 4 (GenConv) | 263,680

##  6 Results and discussion

Tab. 8, Tab. 9, and Tab. 10 present the results of the experiments on Pol SE,
DS SE, and CS SE datasets, respectively. Each table reports the average values
of each performance metric with its corresponding dispersion (i.e., the
standard deviation value). Notice that larger values indicate better
performance; the best values are shown in bold.

Based on the results from Tab. 8, the GCN method (with text embeddings plus
the node type information) achieved the highest accuracy of 0.61, which is
noticeably higher than that of the GCN with other node representations, and
the baseline methods. Moreover, GGNN (with text embeddings plus node type
information) also performed well, achieving an accuracy of 0.60 and the
highest precision of 0.61 among all methods. Interestingly, GGNN (with only
node type information) achieved the highest recall of 0.92, but with lower
precision than the other GGNN representations. As for the baselines, the
logistic regression method achieved an accuracy of 0.53, which was a
relatively low accuracy, slightly higher than the majority class ratio, but it
had a relatively high recall of 0.69, indicating that it was able to identify
most of the positive samples. On the other hand, the few-shot learners with 5,
10, and 20 shot sizes achieved the same accuracy of 0.50 on average, which was
not as good as the other methods. However, it is still noteworthy considering
the low number of samples used for training. It suggests that few-shot
learning may have the potential to be effective for our problem. However,
larger shot sizes may be required to improve the accuracy.

Based on the results from Tab. 9, in terms of accuracy, the GGNN method (with
text embeddings plus node type information) achieved the highest score of
0.72, which is slightly better than the best-performing GCN method (with text
embeddings plus node type information). On the other hand, the worst-
performing methods in terms of accuracy were 10-shot and 20-shot learners,
which achieved an average accuracy of only 0.49. Regarding recall, GGNN (with
text embeddings plus node type information) achieved the highest score of
0.54. In contrast, logistic regression had the lowest recall score of 0.08,
indicating that it could not correctly identify many positive cases. As for
precision, GCN (with text embeddings) had the highest score of 0.62. In
contrast, few-shot learners with different shot sizes had the lowest precision
scores, indicating that they could not identify many positive cases
accurately. Finally, regarding the f1-score, the best-performing method was
GGNN (with text embeddings plus node type information), which scored 0.56. And
the logistic regression learner was the worst-performing method, with an
f1-score of only 0.14, indicating poor overall method performance.

Based on the results from Tab. 10, the best-performing method was GGNN (with
text embeddings plus node type information), achieving an accuracy of 0.70 and
an f1-score of 0.71. Interestingly, GGNN (with node type information as node
features) achieved a recall of 0.94, indicating that it was excellent at
identifying positive examples, but has a low precision of 0.59 which suggests
that although using node type information as node features could help GGNN to
identify positive examples, it also might result in more false positives. The
logistic regression learner achieved an accuracy of 0.56, lower than the top-
performing GCN and GGNN methods. The results also show that the few-shot
learners did not perform very well on the dataset, as their accuracies are
around 0.50, lower than the majority class ratio on the dataset.

Overall, the results indicate that the performance of the different methods
varied depending on the evaluation metric and the feature set used. However,
GGNN (with text embeddings plus node type information) generally performed
well across all metrics. Moreover, including text embeddings in the node
representations generally improved the performance of the GCN and GGNN methods
compared to using only node type information. Additionally, our approach
outperformed the baselines regarding accuracy on all three datasets (see Sect.
4). It is worth noting that the majority class ratios for the Pol SE, DS SE,
and CS SE datasets are 0.52, 0.66, and 0.54, respectively. Despite this, our
approach consistently achieved higher accuracy compared to these ratios. On
the other hand, the baseline methods’ accuracy values were below the majority
class ratio in most cases. Also, as stated earlier, between the two GNN
methods that we used in our approach, the GGNN’s performance was better in
general than GCN. This difference can be attributed to the fact that GGNN has
a more complex architecture comprising more than eight layers, enabling it to
capture the information of the underlying graph more effectively. In
comparison, the GCN used in our implementation only has three (convolutional)
layers.

In summary, the results indicate the following key findings: i) although the
performance of the different methods varied depending on the evaluation metric
and the specificity of the feature set, nevertheless, our approach
outperformed the baselines in terms of accuracy on all three datasets; ii)
even though the majority class ratios for the three datasets were relatively
high, our approach consistently achieved higher accuracy than these ratios;
iii) finally, the GGNN’s performance was generally better than GCN, which
could be attributed to the fact that GGNN has a more complex architecture,
enabling it to capture the information of the underlying graph more
effectively.

Table 8: Results on the Pol SE dataset (with majority class ratio 0.52) Method | Node represnetation | Metric | Mean | Dispersion
---|---|---|---|---
GCN | Text embeddings + node type | Accuracy | 0.61 | 0.01
Recall | 0.69 | 0.04
Precision | 0.60 | 0.01
F1-score | 0.64 | 0.02
Text embeddings | Accuracy | 0.58 | 0.01
Recall | 0.62 | 0.02
Precision | 0.58 | 0.01
F1-score | 0.61 | 0.03
Node type | Accuracy | 0.57 | 0.00
Recall | 0.65 | 0.05
Precision | 0.58 | 0.01
F1-score | 0.61 | 0.02
GGNN | Text embeddings + node type | Accuracy | 0.60 | 0.01
Recall | 0.66 | 0.10
Precision | 0.61 | 0.01
F1-score | 0.63 | 0.04
Text embeddings | Accuracy | 0.60 | 0.01
Recall | 0.59 | 0.07
Precision | 0.61 | 0.01
F1-score | 0.60 | 0.03
Node type | Accuracy | 0.59 | 0.01
Recall | 0.92 | 0.01
Precision | 0.56 | 0.01
F1-score | 0.70 | 0.00
Logistic regression |  | Accuracy | 0.53 | 0.00
Recall | 0.69 | 0.02
Precision | 0.53 | 0.00
F1-score | 0.60 | 0.01
Few-shot(5) | Accuracy | 0.50 | 0.01
Recall | 0.53 | 0.13
Precision | 0.51 | 0.01
F1-score | 0.52 | 0.05
Few-shot(10) | Accuracy | 0.50 | 0.01
Recall | 0.54 | 0.10
Precision | 0.51 | 0.01
F1-score | 0.52 | 0.05
Few-shot(20) | Accuracy | 0.50 | 0.01
Recall | 0.56 | 0.01
Precision | 0.52 | 0.01
F1-score | 0.54 | 0.01
Table 9: Results on the DS SE dataset (with majority class ratio 0.66) Method | Node feature set | Metric | Mean | Dispersion
---|---|---|---|---
GCN | Text embeddings + node type | Accuracy | 0.71 | 0.00
Recall | 0.43 | 0.03
Precision | 0.60 | 0.01
F1-score | 0.50 | 0.01
Text embeddings | Accuracy | 0.70 | 0.00
Recall | 0.31 | 0.05
Precision | 0.62 | 0.02
F1-score | 0.41 | 0.04
Node type | Accuracy | 0.66 | 0.01
Recall | 0.33 | 0.01
Precision | 0.49 | 0.01
F1-score | 0.40 | 0.01
GGNN | Text embeddings + node type | Accuracy | 0.72 | 0.00
Recall | 0.54 | 0.09
Precision | 0.59 | 0.03
F1-score | 0.56 | 0.05
Text embeddings | Accuracy | 0.71 | 0.01
Recall | 0.46 | 0.02
Precision | 0.59 | 0.02
F1-score | 0.52 | 0.01
Node type | Accuracy | 0.69 | 0.01
Recall | 0.34 | 0.04
Precision | 0.56 | 0.01
F1-score | 0.42 | 0.03
Logistic regression |  | Accuracy | 0.66 | 0.00
Recall | 0.08 | 0.01
Precision | 0.50 | 0.02
F1-score | 0.14 | 0.01
Few-shot(5) | Accuracy | 0.52 | 0.04
Recall | 0.47 | 0.11
Precision | 0.34 | 0.01
F1-score | 0.39 | 0.04
Few-shot(10) | Accuracy | 0.49 | 0.03
Recall | 0.53 | 0.07
Precision | 0.34 | 0.01
F1-score | 0.41 | 0.02
Few-shot(20) | Accuracy | 0.49 | 0.00
Recall | 0.49 | 0.00
Precision | 0.33 | 0.00
F1-score | 0.39 | 0.00
Table 10: Results on the CS SE dataset (with majority class ratio 0.54) Method | Node feature set | Metric | Mean | Dispersion
---|---|---|---|---
GCN | Text embeddings + node type | Accuracy | 0.68 | 0.01
Recall | 0.64 | 0.04
Precision | 0.65 | 0.01
F1-score | 0.65 | 0.02
Text embeddings | Accuracy | 0.64 | 0.00
Recall | 0.54 | 0.04
Precision | 0.62 | 0.01
F1-score | 0.58 | 0.02
Node type | Accuracy | 0.65 | 0.00
Recall | 0.67 | 0.01
Precision | 0.61 | 0.00
F1-score | 0.64 | 0.01
GGNN | Text embeddings + node type | Accuracy | 0.70 | 0.01
Recall | 0.81 | 0.05
Precision | 0.63 | 0.02
F1-score | 0.71 | 0.01
Text embeddings | Accuracy | 0.68 | 0.01
Recall | 0.73 | 0.03
Precision | 0.63 | 0.01
F1-score | 0.68 | 0.01
Node type | Accuracy | 0.68 | 0.00
Recall | 0.94 | 0.02
Precision | 0.59 | 0.00
F1-score | 0.73 | 0.01
Logistic regression |  | Accuracy | 0.56 | 0.00
Recall | 0.32 | 0.01
Precision | 0.54 | 0.01
F1-score | 0.40 | 0.01
Few-shot(5) | Accuracy | 0.50 | 0.02
Recall | 0.44 | 0.19
Precision | 0.46 | 0.01
F1-score | 0.43 | 0.11
Few-shot(10) | Accuracy | 0.50 | 0.01
Recall | 0.47 | 0.06
Precision | 0.45 | 0.01
F1-score | 0.46 | 0.03
Few-shot(20) | Accuracy | 0.49 | 0.00
Recall | 0.53 | 0.07
Precision | 0.45 | 0.00
F1-score | 0.49 | 0.03

##  7 Limitations and future work

In this paper, we presented a GNN-based approach for predicting unresolved
questions in SE Q&A communities. However, we must acknowledge our work’s
primary limitation: the lack of absolute forecasting utility. More
specifically, when a question is posted in an online community, the
preliminary information available is its content, such as the title, body, and
associated tags. In order to use a GNN-based approach, information about the
structure of the communication network surrounding the question is also
required. This limitation can be partially addressed using GNN architectures
that operate on evolving graphs, for example, as described in (Pareja et al,
2020). However, a content-based approach, such as the baselines used in our
experiments, only requires information about the questions, making it more
flexible for forecasting purposes.

Another relevant problem is: given an unresolved question and its answer, and
other information, including the structure of the communication network formed
around the question, how could we utilise an ML-based approach in order to
rank answers and recommend promising answers to the user who asked the
question to get the question resolved.

Despite the abovementioned limitation, our results suggest that a GNN-based
approach can outperform a content-based approach for predicting unresolved
questions. Further research is needed to fully explore the potential of GNNs
for this task, address the limitations of our current approach, and
investigate the scalability and robustness of our approach on larger datasets
and in different domains.

##  8 Conclusion

In this work, we proposed a novel approach to identify unresolved questions on
Stack Exchange question-answering communities utilising the graph structure of
user communication formed around a question. Our approach models the
communication network encompassing a question using the property graph model.
It uses graph neural networks, which can work both on the structure of
communication and the content (i.e., messages exchanged among users) to
identify unresolved questions. The results of our experiments show the
effectiveness of the proposed approach compared to baseline methods, which
only utilise the content of questions. We believe our work is a first step
towards better understanding the factors that can affect questions being
unresolved in Stack Exchange communities, utilising state-of-the-art graph
neural network methods.

## Declarations

Funding

This work was carried out as part of the Trondheim Analytica
project111111https://www.ntnu.edu/trondheimanalytica, supported by NTNU’s
Digital Transformation programme.

Conflict of interest I do not have any conflicts or competing interests to
declare.

Availability of data and materials The data used in this study are publicly
available from Archive.org under Creative Commons licences. Furthermore, for
reproducibility, the code and other related artefacts, such as the
preprocessed version of the data used in the experiments, are also available
on GitHub.com 121212https://github.com/habedi/GNNforUnresolvedQuestions.

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# Magnetoresistance of edge states of a two-dimensional topological insulator

Leonid Braginsky<EMAIL_ADDRESS>Institute of Semiconductor Physics, 630090
Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia
M. V. Entin<EMAIL_ADDRESS>Institute of Semiconductor Physics, 630090
Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia

###### Abstract

The theory of magnetoresistance of the edge state of a two-dimensional
topological insulator is developed. The magnetic field violates the time
reversal invariance. Magnetoresistance arises due to the energy gap opened by
a magnetic field parallel to the sample surface. The combined action of
impurities and the magnetic field causes backscattering of edge electrons.
Although impurities are necessary for scattering, a sufficiently strong
interaction with impurities leads to the suppression of backscattering.

##  I Introduction

The edge states of 2D topological insulator is one of most inspiring problem
of the modern solid state physics. It was shown that these states possess so
called topological protection, preventing the electron from backscattering. At
a moment there is a large and rapidly growing number of publications on this
topic, (see, i.g., [1, 2, 3, 4]). The study of the conductance of the edge
states of a two-dimensional topological insulator assumes their topological
protection, this makes backscattering prohibited or it is extremely weak.
Therefore, the one-dimensional states are collisionless and the conductance of
the electrons on them is $e^{2}/h$.

There were some attempts to implement mechanisms violating the time-
reversibility and, therefore, causing backscattering. In particular, Ref. [5,
6] ascribe backscattering to the transitions between edges with opposite
direction of travel of the same-spin electrons. The transition between
opposite edges of a TI strip were considered also in Ref. [7].

The present paper was stimulated by the experimental finding [8] of the strong
magnetoresistence of the edge state electron conductance. Important
experimental fact that has to be explained is the presence of
magnetoresistance, as well as its gigantic value and fluctuations with the
Fermi level. What is unusual, is the sensitivity of 1D edge states to the
magnetic field, which is absent in other 1D systems. The results of [8] can
not be explained by means of interedge transitions [7] due to large width of
the TI strip in the experimental conditions.

The model is based on the one-dimensional Hamiltonian

$\displaystyle H=\left(\begin{array}[]{cc}vp+V(x)&\Delta_{1}\\\
\Delta_{1}^{*}&-vp+V(x)\\\ \end{array}\right).$ (3)

Here $p$ and $v$ are the electron momentum and velocity, and $V(x)$ is the
impurity potential. The Hamiltonian (3) is the simplest that violates the time
reversal invariance in the magnetic field and leads to the gap $2|\Delta_{1}|$
at $p=0$.

The gap in the electron spectrum of the 2D topological insulator arises in the
magnetic field ${\bf B}$ directed along the specimen plane.[9]

$\Delta_{1}=\mu_{B}(g_{xx}B_{x}+ig_{yy}B_{y})$

for the (0,0,1) orientation. Here $\mu_{B}$ is the Bohr magneton, and
$g_{xx},~{}g_{yy}$ are the $g$-factor components in the specimen plane.

The gap is

$2|\Delta_{1}|=\mu_{B}\sqrt{g_{xx}^{2}B_{x}^{2}+g_{yy}^{2}B_{y}^{2}}.$

At a weak magnetic field the magnetoresistance is due to the backscattering
and localization of previously delocalized states by the magnetic field.
Indeed, the presence of off-diagonal terms in the Hamiltonian leads to
backscattering of electrons. The potential itself does not cause transitions
between the states. The scattering occurs due to the off-diagonal part
$H_{nd}$ of the Hamiltonian (3).

Two approaches can be utilized. First is the classical approach. In the
absence of localization the 1D conductivity is of the Drude form:

$\sigma=\frac{e^{2}\tau v}{\pi}.$ (4)

Since the electron velocity in the linear spectrum is constant, the
conductivity is determined by the backscattering time $\tau$. The classical
approach is valid in the absence the phase coherence at a finite temperature
(similar problem has been considered in connection to the TI strip [7]). This
means that during the forward and backward travel at the distance $v\tau$ the
phase of the electron changes by $\pi$ due to, e. g., interaction with
phonons.

Second, the coherence of the electron states results in their localization in
the infinite volume and, therefore, exponential size dependence of the
conductivity at a zero temperature. In this case the 1D conductivity is not
established. Instead, the conductance $G$ of the system drops exponentially
with the system length $L$: $\log G\propto-L/v\tau$.

In this paper we find the backscattering time $\tau$.

### Magnetoinduced backscattering

The selfwave functions of the electrons of the Hamiltonian (3) with energies
$\epsilon_{\pm}=\pm vp$ at $B=0$ are

$\displaystyle\psi_{+}=\frac{1}{\sqrt{L}}\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)e^{ipx+i\int dxV(x)/v}$
$\displaystyle\psi_{-}=\frac{1}{\sqrt{L}}\left(\begin{array}[]{c}0\\\ 1\\\
\end{array}\right)e^{ipx-i\int dxV(x)/v}.$

The amplitude of the backscattering is determined by the first order
perturbation on the non-diagonal part of the Hamiltonian
$H_{nd}\propto\Delta_{1}$, i.e., the matrix element between the
$|\psi_{-p,-\sigma}\rangle$ and $|\psi_{p,\sigma}\rangle$ states of the same
energy $\epsilon_{p,\sigma}$:

$\displaystyle A=\langle\psi_{-p,-}|H_{nd}|\psi_{p,+}\rangle$
$\displaystyle=\frac{\Delta_{1}}{L}\int_{-L}^{L}\exp{\left(2ipx+\frac{2i}{v}\int
V(x)\,dx\right)\,dx}.$

##  II Weak impurity potential. The perturbation theory

In the first order with regard to the potential $V(x)$ we find

$\displaystyle
A=2i\langle\psi_{-p,-}|H_{nd}|\psi_{p,+}\rangle=\frac{2i\Delta_{1}}{Lpv}\tilde{V}(2p),$
$\displaystyle\tilde{V}(p)=\int_{-\infty}^{\infty}V(x)e^{ipx}dx.$

Consider the electron scattering at the impurity potential $V({\bf
r})=\sum_{n}u({\bf r}-{\bf r}_{n})$, where $r_{n}$ is the position of n-th
impurity and $u({\bf r}-{\bf r}_{n})$ is its potential. The probability of
backscattering has to be averaged over the random impurity positions in 2D
layer, so that

$1/\tau=\frac{L}{v}\langle|A|^{2}\rangle,$ (7)

The 2D Fourier transform of the Coulomb impurity is

$u(q)=\frac{2\pi e^{2}}{\kappa q},$

therefore

$\langle|V(q_{x})|^{2}\rangle=n_{i}\int\frac{dq_{y}}{2\pi}u^{2}(q)=2\pi^{2}n_{i}\frac{e^{4}}{q_{x}\kappa^{2}},~{}~{}~{}q^{2}=q_{x}^{2}+q_{y}^{2}.$

Here $n_{i}$ is the impurity density. For the backscattering probability we
find

$1/\tau_{0}=\frac{8\pi^{2}e^{4}n_{i}|\Delta_{1}|^{2}}{\hbar\kappa^{2}(vp_{F})^{3}}=\frac{8e^{4}n_{i}|\Delta_{1}|^{2}}{\hbar^{4}\kappa^{2}v^{3}\pi
n_{e}^{3}}.$ (8)

Here $n_{e}$ is the 1D electron density.

The probability (8) essentially increases at a small electron density.
Therefore, screening of the Coulomb potential by the field electrode has to be
taken into account. Then

$u(q)=\frac{2\pi e^{2}}{\kappa q}(1-e^{-2qd}).$ (9)

Here $d$ is the distance to the field electrode. The last factor in Eq. (9)
restricts the impurity matrix elements at small $q$. For $1/\tau$ value we
obtain

$\displaystyle\frac{1}{\tau}=\frac{8\pi^{2}e^{4}n_{i}|\Delta_{1}|^{2}}{\hbar\kappa^{2}(vp_{F})^{3}}\phi(p_{F}d),$
(10) $\displaystyle\mbox{
where}~{}~{}~{}\phi(y)=\frac{2}{\pi}\int_{0}^{\infty}\frac{(1-e^{-4y\cosh
t})^{2}}{\cosh t}dt.$

$\phi(y)\approx 16y\ln 2/\pi$ at $y\ll 1$ and $\phi(y)\to 1$ at $y\to\infty$.
Thus, screening reduces the divergence of $1/\tau$ value at small electron
density: $\tau\propto n_{e}^{2}$, instead of $\tau\propto n_{e}^{3}$ for
$n_{e}\to 0$.

It should be noted that the obtained in this section expressions hold only for
the small impurity potentials. This means applicability of the perturbation
theory for this potential as well as for the gap value.

##  III Strong impurity potential

The conductivity of electronic gas with quadratic spectrum is too complicated,
if it considered outside the frames of the perturbation theory. This is not
the case for the electronic 1D gas with the linear spectrum, because this
problem has an exact solution.[10] This allows us to obtain the result at
$n_{e}\to 0$.

Consider the problem of small gap, but arbitrary impurity potential. Assume
the random impurity potential obeys the Gaussian distribution. This is
correct, if $|u({\bm{r}-r_{n}}|\ll vp_{F}$. Then in Gauss approximation from
Eq. (I) we obtain:

$\displaystyle\langle|A|^{2}\rangle=L^{-2}|\Delta_{1}|^{2}\int\int\exp\Bigg{[}2ip(x-x^{\prime})-$
$\displaystyle\frac{4}{v^{2}}\int\int_{x^{\prime}}^{x}dx_{1}dx_{2}W(x_{1}-x_{2})\Bigg{]}dxdx^{\prime},$

where

$W(x-x^{\prime})\equiv\int_{-\infty}^{\infty}dk(2\pi)^{-1}e^{-ik(x-x^{\prime})}\tilde{W}(k)=\langle
V(x)V(x^{\prime})\rangle$

is the potential correlation function. Its Fourier transform is

$\tilde{W}(k)=\frac{4\pi^{2}e^{4}}{\kappa^{2}}n_{i}\frac{\phi(kd/2)}{k}.$

Then using Eq. (7) we obtain

$\displaystyle\frac{1}{\tau}=2(d/v)|\Delta_{1}|^{2}$ (11)
$\displaystyle\times\int_{-\infty}^{\infty}dz\cos(sz)\exp\left(-\beta\int\frac{dy}{y^{3}}\phi(y/2)\sin^{2}(yz)\right),$
$\displaystyle s=4p_{F}d,\beta=\frac{2\pi e^{4}}{\kappa^{2}v^{2}}n_{i}d^{2}.$

The behavior of $1/\tau$ vs $\beta$ and $s$ is presented in Fig. 1. The
dependence on $\beta$ is linear at small $\beta$ [or at small impurity
density, in agreement with Eqs. (8, 10)]] followed by an exponential decrease.
The latter is due to the random potential that results in fluctuating increase
of the local Fermi momentum reducing the scattering.

Figure 1: (Coloronline) Dependence of $1/\tau$ on the parameter $s$. Values of
$\beta$ are shown on the curves.

### Conductance of the finit-size electron system at zero temperature

Without backscattering the zero temperature conductance of the edge state $G$
is equal to the conductance quantum $G_{0}$ for any edge length. Presence of
the backscattering leads to the localization of the edge states and,
therefore, to the exponential drop of the conductance, if the edge length
exceeds the backscattering length: $\ln{G/G_{0}}\propto-L/v\tau$. Or
$\ln{G/G_{0}}\propto-B^{2}$ in agreement with the obtained behavior of $\tau$.

## Discussions

To estimate the edge state conductivity at a low temperature, consider the 1D
electron gas with the linear spectrum. The electron-phonon interaction affects
the phase of the electron wave function, however, this phase is not important
for the scattering time. This means that the conductivity obeys the Drude
expression Eq. (4) with the relaxation time found in the present paper.

Thus, we found that a magnetic field parallel to the plane of the sample
results in elastic backscattering of the edge states electrons with the linear
spectrum. This scattering leads to localization of states in an infinite-size
system and, therefore, to finite conductivity at a non-zero temperature. The
localization is due to magnetic field. The transition between the ballistic
and localization behavior of conductance occurs under magnetic field in a
finite-size sample. This transition depends on the relation between the
magnetic localization length and sample length. It is shown that the
probability of backscattering increases with the impurity concentration at a
low concentration followed by decrease at a high concentration of impurities.

Note also that our results correct, if only the Fermi level is far apart from
the gap $|E_{F}|\gg|\Delta_{1}|$. This is necessary for the expansion over the
magnetic field holds.

Another note concerns non-magnetic mechanisms of backscattering. The paper
[11] showed that in a TI with smooth edge the overlapping of linear topology-
protected edge states with the Dirac gapped branches leads to the elastic
backscattering. In this case the magnetoinduced backscattering complements the
latter in the energy domains where they coexist.

The work is supported by the RFBR grant No.20-02-00622.

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