# Student Answer Forecasting: Transformer-Driven Answer Choice Prediction for
Language Learning

Elena Grazia Gado Equal contribution. Listing order is alphabetical. Tommaso
Martorella  Luca Zunino
Paola Mejia-Domenzain  Vinitra Swamy  Jibril Frej  Tanja Käser

EPFL  {elena.gado, paola.mejia<EMAIL_ADDRESS>
###### Abstract

Intelligent Tutoring Systems (ITS) enhance personalized learning by predicting
student answers to provide immediate and customized instruction. However,
recent research has primarily focused on the correctness of the answer rather
than the student’s performance on specific answer choices, limiting insights
into students’ thought processes and potential misconceptions. To address this
gap, we present MCQStudentBert, an answer forecasting model that leverages the
capabilities of Large Language Models (LLMs) to integrate contextual
understanding of students’ answering history along with the text of the
questions and answers. By predicting the specific answer choices students are
likely to make, practitioners can easily extend the model to new answer
choices or remove answer choices for the same multiple-choice question (MCQ)
without retraining the model. In particular, we compare MLP, LSTM, BERT, and
Mistral 7B architectures to generate embeddings from students’ past
interactions, which are then incorporated into a finetuned BERT’s answer-
forecasting mechanism. We apply our pipeline to a dataset of language learning
MCQ, gathered from an ITS with over 10,000 students to explore the predictive
accuracy of MCQStudentBert, which incorporates student interaction patterns,
in comparison to correct answer prediction and traditional mastery-learning
feature-based approaches. This work opens the door to more personalized
content, modularization, and granular support.

###### keywords:

LLMs, Student Models, Answer Forecasting

##  1 Introduction

Figure 1: We present a four-stage pipeline for answer forecasting integrating
student history: 1) we preprocess ITS data from Lernnavi, 2) we train student
embeddings from several models (MLP, LSTM, Mistral 7B, BERT), 3) we use the
ITS data and student embeddings to train several models (LernnaviBERT for
Domain Adaptation, MCQBert for Correct Answer Prediction, and MCQStudentBert
for Student Answer Forecasting) and 4) we evaluate the models using
qualitative and quantitative analyses (accuracy, F1 Score, and MCC).

The image is a flowchart depicting the answer forecasting pipeline which
includes several stages. The first stage is "Data Processing," represented by
a laptop icon, which prepares Q&A data for the model. Next is the "Model
Training," which involves three steps: "Domain Adaptation", "Correct Answer
Prediction", and "Student Answer Forecasting". The last stage is the
"Evaluation" stage, depicted by a bar chart, where the model’s performance is
assessed. The flowchart includes arrows indicating the flow of data between
these stages.

Intelligent tutoring systems (ITS) are powerful educational tools that
personalize the student’s learning experience through adaptive content [1, 2,
3]. Within these systems, the ability to predict student answers plays an
important role in tailoring the educational content to the student’s level of
understanding, knowledge gaps, and learning pace [4].

There is a large body of research modeling students’ learning [1, 2, 3, 4, 5,
6, 7, 8, 9]. This effort encompasses the development of probabilistic
frameworks such as Bayesian Knowledge Tracing [6] and Dynamic Bayesian
Networks [3], as well as deep learning approaches like Deep Knowledge Tracing
[7] and Graph-based Knowledge Tracing [8]. Other educational data mining (EDM)
approaches have developed statistical models such as Learning Factor Analysis
[10] and Performance Factor Analysis [11] to predict the probability of
correct student responses. Additionally, the EDM community has studied the
implementation of Machine Learning (ML) classifiers to predict learning
outcomes such as quiz answers [12].

Despite these advancements, the focus has predominantly been on predicting
whether a student’s answer will be correct or incorrect [1, 2, 3, 4, 6, 7,
12], rather than forecasting the specific answer the student would provide.
This could enrich the understanding of the student’s acquired knowledge. Thus,
enabling the development of more personalized content and hints [13].

Several works have tackled the challenge of analyzing Multiple Choice
Questions (MCQs) [14, 15, 16]. For example, [12] and [17] incorporated
temporal features, user history features, and subject features to train ML
classifiers, such as XGBoost, to predict question quality. Additionally, [18]
utilized a transformer model to fuse metadata and performance features for a
multiclass classification task. Another approach by [19] extended Binary
Knowledge Tracing using a BiLSTM with DAS3H features [20] and attention
mechanisms. Similarly, [14] proposed the Order-aware Cognitive Diagnosis (OCD)
model to predict students’ answers by considering question order effects,
without focusing on the question or answer text. However, a common limitation
in these studies is the lack of attention to the contextual richness in the
text of questions and answers, which could significantly influence human
cognition and decision-making processes.

In this regard, Large Language Models (LLMs) could be leveraged to incorporate
textual context into predictive models [21, 22, 23]. For example, [21] used
LLMs fine-tuned with personalization and contextualization to enhance early
forecasting of student performance in courses. Moreover, [22] proposed a
transformer-based knowledge tracing model using BERT to capture the sequential
knowledge states by randomly masking labels from the students’ answer
sequence.

While LLMs offer a promising solution to account for the content and context
of questions and answers, their application on student answer forecasting
remains underexplored. To forecast student answers, the inputs of the question
context, granular answer choices, and individual learning history becomes even
more relevant to the model than for general question answering tasks [24, 25].

To address this gap, we introduce a novel student answer forecasting pipeline
that leverages LLMs to understand the content and context of the question and
answer and the students’ history. We first compare four architectures (MLP,
LSTM, BERT, Mistral 7B) to compute student embeddings using a student’s
previous answering history. Then, we incorporate the student embedding into
the question-answering prediction, using a finetuned BERT architecture. We
focus on language learning MCQs from a real-world ITS used by 10499 students
consisting of 237 unique questions to answer the following research questions:
(RQ1) How can we design a performant embedding for student interactions in
German? (RQ2) How can we integrate these student interaction embeddings to
improve the performance of an answer forecasting model?

This work contributes a modeling pipeline for question-answer forecasting that
1) integrates student history into a transformer model and 2) focuses on
answer choice forecasting instead of correct answer forecasting. Unlike other
answer forecasting models, answer choice forecasting allows for independent
modularization of answer choices, enabling an educator to simply add a fifth
answer choice for an original four-answer MCQ question without retraining the
model. Importantly, we contribute to the literature in German EDM, presenting
a case study from over ten thousand students from a real-world ITS in a
language that is not often researched and therefore accompanied by several
biases due to data and model underrepresentation [26]. We only use open-source
models (including the recent Mistral 7B) and not API-based services, enabling
learning platforms to host their pipeline and data entirely on their servers.
Our code and models are provided open source at https://github.com/epfl-
ml4ed/answer-forecasting and https://go.epfl.ch/hf-answer-forecasting.

##  2 Methodology

The student answer forecasting pipeline depicted in Figure 1 is based on
students interactions with MCQs in an ITS named _Lernnavi_. The pipeline
predicts the likelihood of a student selecting a particular MCQ answer, based
on the question and answer text and a student embeddings generated from their
historical interaction data. In this section, we describe each step of the
pipeline.

###  2.1 Data Processing

Learning Context. We focus our analyses on data collected from Lernnavi, an
ITS for high-school students. Lernnavi offers adaptive learning and testing
sessions in mathematics and language learning.

Dataset. The dataset is characterized by the following three data
representations: 1) user-generated interactions, also referred to as
“transactions“ ($I_{u}^{s}={i_{1},\ldots,i_{K}}$ for each user $u$), 2)
Lernnavi questions also referred to as “documents”, representing the
associated questions ($\mathbb{Q}^{s}$), answer choices ($\mathbb{C}^{s}$),
and textual pages provided to students, and 3) the taxonomy of topics
($\mathbb{T}$) shown in the German and Math dashboards. We only consider
“documents” regarding German MCQs with at least one transaction from a user.
After filtering, the dataset is composed of 237 unique questions and 138,149
transactions. Moreover, the dataset consists of 10,499 users with at least one
transaction for German MCQs. The median number of MCQ answers from learners is
7 with some learners that answered up to 311 questions (including multiple
trials for the same question).

###  2.2 Problem Formulation

We analyze _users_ $U^{s}\subset\mathbb{U}$ engaging in _learning sessions_
$s$ within Lernnavi’s $\mathbb{S}$ offerings, focusing on sessions
$S={s_{1},\ldots,s_{M^{S}}}$, each a unique iteration within a broader topic
$t\subset\mathbb{T}$. These sessions are characterized by their interactive
quizzes, sourced from a question bank $\mathbb{Q}^{s}$ and designed to assess
user knowledge through multiple-choice formats. Interactions in these sessions
are represented as $I_{u}^{s}={i_{1},\ldots,i_{K}}$ for each user $u$, involve
selections ($c$) from the provided answer options for each question $q$.

These interactions are timestamped and detailed to capture the essence of user
engagement and learning behavior. To evaluate user trajectories, we introduce
_binary metrics_ for answer choices,
$\mathbb{C}^{s}={c_{q_{1}},\ldots,c_{q_{|Q^{s}|}}}$, allowing for an in-depth
analysis of user response selection. This design is to enable the multi-
response setting for question $q$, which can either have one correct answer
choice $c_{q_{i}}$ or multiple correct answer choices $\mathbb{C}_{q}$, of
which user $u$ chooses one answer $c_{q_{u}}$ or multiple
$\mathbb{C}_{q_{u}}$.

The answer-forecasting prediction task is to predict for a given user $u$ with
past interaction history $I_{u}^{s}$, which answer choices $\mathbb{C}_{q}$
are most likely to be chosen by the student.

###  2.3 Embedding Generation

To create student embeddings for a prediction model, we explored four
strategies to make a total of 22 different embeddings: one using the MLP
model, 16 using the LSTM models, one using LernnaviBERT, and four using
Mistral 7B Instruct models:
MLP Autoencoder Embedding: We utilized a Multilayer Perceptron (MLP)
autoencoder with specific architecture and feature engineering to encode
students’ previous performance, resulting in a size of 11907.
LSTM Autoencoder Embedding: We considered four stacked LSTM configurations
with varying sequence lengths and number of layers to balance computational
complexity and richness of student interaction history.
LernnaviBERT Embedding: We created embeddings using a finetuned German BERT
base model111https://huggingface.co/google-bert/bert-base-german-cased to MCQ-
specific language, LernnaviBERT, with a sequence length of 10 and mean pooling
strategy.
Mistral 7B Instruct Embedding: We used Mistral 7B Instruct to generate
embeddings with sequence lengths of 10, 20, 30, and 40 with mean pooling at
the penultimate layer.

Figure 2: MCQStudentBertSum (A) and MCQStudentBertCat (B) architectures. In
MCQStudentBertSum, the student embeddings are summed with LernnaviBERT
question embeddings at the input, before being passed to the MCQBert model and
classification layer. In MCQStudentBertCat, MCQ embeddings are generated with
LernnaviBERT, then passed to the MCQBERT model and concatenated with the
student embeddings just before the classification layer. German BERT image
taken from https://www.deepset.ai/german-bert

The image shows two architectures for multiple-choice question (MCQ) models:
MCQStudentBertSum (A) and MCQStudentBertCat (B). Both architectures include a
diagram of a puppet figure, with arrows indicating the flow of embeddings.

###  2.4 MCQStudentBert: Answer Forecasting

We initially train an MCQBert for the classification of correct/incorrect MCQ
answers. Learning the correct answers across all MCQs is necessary to ensure
that any failure in predicting a student’s response did not arise from a lack
of knowledge of the correct answer. Appendix 5.1 details the training and
evaluation strategy.

We then extend MCQBert to include the students’ history and predict students’
responses to MCQs. Inputs for this task include the text of the MCQs and
supplementary student-specific data encapsulated within embeddings. The
objective has changed; rather than pinpointing the correct answers from
available options, the emphasis is on predicting the actual responses provided
by students. This task continues to be treated as a binary classification
problem. Details are included in Appendix 5.2.

We explore the two models in Figure 2, differing in their handling of student
embeddings for integrating student information into the prediction process:
MCQStudentBertCat where the inputs are concatenated before the classification
layer and MCQStudentBertSum222MCQStudentBertCat and MCQStudentBertSum are
available at https://go.epfl.ch/hf-answer-forecasting where the embeddings are
summed at the input. Both models are based on MCQBert, finetuned in the
previous phase of the pipeline. These variants are augmented with a classifier
head, comprising two linear layers with a ReLU activation function, to predict
the likelihood of each potential answer being chosen by a student. Inset B of
Figure 2 illustrates MCQStudentBertCat strategy. The concatenation strategy
draws inspiration from context-aware embeddings [27, 28], where additional
features (like user or product embeddings) are appended before the final
classification layer to provide context. For this purpose, in our context, the
student embeddings are first transformed using a linear layer to match the
MCQBert’s hidden size. These transformed embeddings are then concatenated with
the output of the MCQBert model which is the representation of the first token
[CLS] token. This approach leaves the MCQBert processing unchanged and appends
additional information right before the final decision-making process (e.g.,
classification). It allows the classification model to consider both the
processed input representation and the student-specific information
distinctly.

In contrast, MCQStudentBertSum, depicted in Figure 2 (inset A), integrates the
student embeddings directly into the input embeddings of the MCQBert model.
This approach is similar to multimodal learning for LLMs to create combined
embeddings that represent both modalities at the input level (e.g. to create
visual-semantic embeddings) [29]. Specifically, the student embeddings are
first transformed to match the dimensionality of the MCQBert input embeddings
using a linear layer. These transformed student embeddings are then summed
with the original input embeddings. This approach alters the initial
representation that the MCQBert model processes. The student embeddings can be
seen as providing an initial bias or modification to the input embeddings,
potentially allowing the model to adapt more specifically to characteristics
represented by the student embeddings.

##  3 Experimental Evaluation

We finetune LernnaviBert to predict the correct MCQ answer, resulting in
MCQBert. Next, we integrate the student embeddings (RQ1) to forecast student
answers to produce variations of MCQStudentBert (RQ2). The experimental
evaluation of MCQBert for correct answer prediction can be found in Section
5.1

Table 1: MCQStudentBert Embedding Comparisons and Performance. For two models MCQStudentBertCat and MCQStudentBertSum, we evaluate the performance of different student embeddings (MLP, LSTM, LernnaviBERT, and Mistral 7B) against a baseline dummy classifier and MCQBert. We report MCC, F1 Score and Accuracy, with the highest values indicating the most performant embeddings (Appendix 5.2). * represents the best model determined by a hyperparameter search. | Embedding
---|---
| Dummy | MCQBert | MLP | LSTM∗ | LernnaviBERT | Mistral 7B∗
MCQStudentBertCat | MCC | 0 | 0.518 | 0.557 | 0.567 | 0.575 | 0.579
F1 Score | 0.305 | 0.740 | 0.772 | 0.777 | 0.780 | 0.782
Accuracy | 0.590 | 0.771 | 0.785 | 0.790 | 0.795 | 0.797
MCQStudentBertSum | MCC | 0 | 0.518 | 0.552 | 0.564 | 0.568 | 0.569
F1 Score | 0.305 | 0.740 | 0.767 | 0.774 | 0.777 | 0.778
Accuracy | 0.590 | 0.771 | 0.785 | 0.790 | 0.789 | 0.789

We evaluate the different embedding strategies and ways of integration to
predict student responses to MCQ. A key difference from MCQBert is the
incorporation of student embeddings, enabling models to use contextual
information.

Embedding Performance.  The training of the MLP autoencoder yielded a mean
validation loss of $1.3e^{-7}$. Further analysis highlighted a discrepancy in
the norm-2 distance between the input and output vectors across training,
validation, and test datasets, with an average input norm-2 ($\lVert
input\rVert_{2}$:) of 1.31 and an average discrepancy norm-2 ($\lVert input-
output\rVert_{2}$) of 0.03. The LSTM models yielded a mean validation loss of
$1.3e^{-2}$ with a mean input norm of 13.59 and an average reconstruction norm
of 10.94.

We did not find a trend in the number of hidden layers for the LSTM. For
subsequent analyses, the single-layer LSTMs are used. After tuning, for both
Mistral 7B and LSTMs, the best sequence length was 20.

Figure 3: t-SNE visualization of the LernnaviBERT embedding space. Highlighted
is the evolution of a single student’s embedding through time.

The image shows a light blue circle made out of smaller blue points. The t-SNE
visualization embedding of a single student are shown as colored points that
stand out from the small blue circles.

To examine the LernnaviBERT embeddings in detail, Figure 3 visualizes the
embedding space using t-SNE. Each point represents a student-embedding at a
given time. Students are represented by multiple points, reflecting the
evolution of their embeddings as they respond to successive MCQs. In Figure 3,
the trajectory of embeddings from an individual student is accentuated. The
temporal aspect of these embeddings is depicted through a color gradient,
transitioning from lighter shades for initial interactions to darker shades
for more recent activity. The highlighted trajectory suggests a discernible
shift in the student’s embedding space over time from the upper right to the
lower left corner.

Predictive Model Performance.  A total of 20 models were trained,
incorporating 10 distinct embeddings and 2 integration strategies, over three
epochs333Full results are available at https://go.epfl.ch/mcq-results. The
models were evaluated on a hold-out test set consisting of MCQs previously
encountered by the model, but not in the context of the specific student being
assessed.

The optimal performance for all models was recorded in either the second
(followed by a marginal decline in the third epoch) or third epoch. Table 1
presents the results from the best epoch, showcasing both integration
strategies (concatenation and addition) across four embedding types (MLP,
LSTM, LernnaviBERT, Mistral 7B), compared to a baseline Dummy Classifier and
MCQBert (no embedding). For brevity, only the results from the best-performing
LSTM, with 1 hidden layer and a sequence length of 20, and the highest-
achieving Mistral 7B model, with a sequence length of 20, are displayed3.

Integration Strategies. As seen in Table 1, the concatenation strategy
(MCQStudentBertCat) generally yields slightly better results compared to the
summation strategy (MCQStudent BertSum), particularly noticeable with
LernnaviBERT.

Embeddings. All embedding strategies show substantial improvements over the
Dummy Classifier across all metrics. Mistral 7B is the best-performing
embedding for both integration strategies (MCQStudentBertCat,
MCQStudentBertSum). When applied to MCQStudentBertCat, the Mistral 7B
embedding shows a notable increase in performance metrics: an improvement of
0.579 in MCC, 0.477 in F1 score, and 0.207 in accuracy compared to the Dummy
Classifier. For the second baseline (MCQBert), the Mistral 7B embedding showed
a 12% improvement.

Consistency is observed in the performance ranking of embeddings between the
two integration strategies. In the MCQStudentBertCat configuration,
LernnaviBERT ranks second with an $MCC=0.575$, followed by the LSTM
autoencoder with $MCC=0.567$, and the MLP autoencoder trailing with
$MCC=0.557$. Similarly, for the MCQStudentBertSum strategy, the LernnaviBERT
embedding is in the second position $MCC=0.575$ followed by the LSTM
autoencoder with an $MCC=0.564$, while the MLP autoencoder remains the least
effective, with an $MCC=0.552$.

The performance differentials between embeddings are marginal. For instance,
within the MCQStudentBertCat framework Mistral 7B, exhibits a modest 4%
improvement in MCC over the MLP autoencoder. Similarly, in the
MCQStudentBertSum framework, the margin is 3%.

##  4 Discussion and Conclusion

Our goal is to enhance the predictive capabilities of ITS by developing
embeddings that capture student interactions (RQ1) and integrating them into
an answer forecasting model (RQ2) to improve performance and personalization.

We explored various methods of encoding students’ interactions with the ITS
including using autoencoders with MLP and LSTM architectures, and LLMs
including LernnaviBERT model and Mistral 7B (RQ1). Our findings revealed that
the Mistral 7B embedding emerged as the best-performing method for our use-
case, demonstrating a 12% performance enhancement relative to MQCBert
(baseline with no embedding) and a 4% improvement over the least effective
embedding, the MLP autoencoder. The performance of Mistral 7B, closely
followed by LernnaviBERT, LSTM, and MLP autoencoder can be likely attributed
to the inherent capabilities of each embedding approach in capturing and
representing student interactions. For example, Mistral 7B has a sliding
window attention mechanism, facilitating a deeper understanding of contextual
relationships within student data [30]. This model’s efficacy is further
augmented by its fine-tuning on instructional datasets, potentially enhancing
its proficiency in interpreting question-answer pairs. Moreover, LernnaviBERT
also seemed to capture the contextual information of educational interactions
effectively, ranking it closely behind Mistral 7B. The slight difference in
performance between these two models may be due to Mistral 7B’s more advanced
mechanisms for handling long sequences and its ability to incorporate broader
contextual information. Notably, the optimal sequence length for Mistral 7B
was identified as 20, whereas the LernnaviBERT model was constrained to
sequence lengths of 10 due to its context limitations. This is further
supported by the Mistral 7B embedding visualizations that show a more
discernible trend with sequence lengths greater than 10. The LSTM autoencoder
underperforms compared to transformer-based models because it prioritizes
temporal dynamics over deep contextual understanding. While its sequential
processing is good for capturing learning progression, it may not handle
complex language structures as well as transformer models. The low performance
scores of the MLP autoencoder can be attributed to its simpler architecture,
which may not capture complex language-based information and temporal dynamics
effectively.

To study the integration of student interaction embeddings into an answer
forecasting model (RQ2), we used two approaches: MCQStudentBertCat, which
concatenates student embeddings with model outputs before classification, and
MCQStudentBertSum, which sums the embeddings at the input stage. The superior
performance of the MCQStudentBertCat model compared to the MCQStudentBertSum
model could be attributed to its ability to maintain a clear separation
between the question-answer information and the student-specific embeddings,
promoting distinct utilization of both sources of information in the
prediction process. Future research could further explore the optimization of
such integration techniques, potentially investigating the impact of varying
the point of concatenation.

One limitation of our study is the interpretability of the embeddings
generated by the models we explored, such as LernnaviBERT and Mistral 7B.
Despite their effectiveness, there is a significant gap in our understanding
of the underlying features and feature patterns encapsulated by these
embeddings, hindering our ability to comprehend their effectiveness. The
generalizability of our findings is limited by the study’s execution within a
single context and the lack of publicly available datasets comparable in
richness to Lernnavi. However, our study aligns with and contributes to the
growing body of research in answer forecasting by incorporating student
history into predictive models. We aim to introduce a novel approach that can
be replicated by the EDM community in different ITS and contexts, enabling a
better understanding of the generalizability of our findings and fostering
advancements in personalized instruction.

In conclusion, we introduce MCQStudentBert, a model for student answer
forecasting that leverages LLMs to integrate the contextual understanding of
question and answer texts with students’ historical interactions. Our work
contributes to the field of EDM in the German language context, where such
studies are scarce, and promotes the use of open-source models, facilitating
wider adoption and adaptation within the EDM community. Furthermore, our
model’s utility extends to ITS, where it can be employed to tailor potential
answers for individual learners and give hints dynamically. From the
educator’s and developers’ perspective, it is possible to modify or augment
the answer choices without necessitating a complete retraining of the model.
This feature could allow seamless updates and expansions to the answer sets in
response to evolving pedagogical requirements or teacher/student feedback.

Acknowledgements. This project was substantially funded by the Swiss State
Secretariat for Education, Research and Innovation (SERI) and the Swiss Canton
of St. Gallen.

## References

  * [1] Neil T Heffernan and Cristina Lindquist Heffernan.  The assistments ecosystem: Building a platform that brings scientists and teachers together for minimally invasive research on human learning and teaching.  International Journal of Artificial Intelligence in Education, 24:470–497, 2014.
  * [2] Ghodai Abdelrahman, Qing Wang, and Bernardo Nunes.  Knowledge tracing: A survey.  ACM Computing Surveys, 55(11):1–37, 2023.
  * [3] Tanja Käser, Gian-Marco Baschera, Juliane Kohn, Karin Kucian, Verena Richtmann, Ursina Grond, Markus Gross, and Michael von Aster.  Design and evaluation of the computer-based training program calcularis for enhancing numerical cognition.  Frontiers in psychology, 4:489, 2013.
  * [4] John R Anderson, Albert T Corbett, Kenneth R Koedinger, and Ray Pelletier.  Cognitive tutors: Lessons learned.  The journal of the learning sciences, 4(2):167–207, 1995.
  * [5] Tirth Shah, Lukas Olson, Aditya Sharma, and Nirmal Patel.  Explainable knowledge tracing models for big data: Is ensembling an answer?  arXiv preprint arXiv:2011.05285, 2020.
  * [6] Albert T Corbett and John R Anderson.  Knowledge tracing: Modeling the acquisition of procedural knowledge.  User modeling and user-adapted interaction, 4:253–278, 1994.
  * [7] Chris Piech, Jonathan Bassen, Jonathan Huang, Surya Ganguli, Mehran Sahami, Leonidas J Guibas, and Jascha Sohl-Dickstein.  Deep knowledge tracing.  Advances in neural information processing systems, 28, 2015.
  * [8] Hiromi Nakagawa, Yusuke Iwasawa, and Yutaka Matsuo.  Graph-based knowledge tracing: modeling student proficiency using graph neural network.  In IEEE/WIC/ACM International Conference on Web Intelligence, pages 156–163, 2019.
  * [9] Sami Sarsa, Juho Leinonen, and Arto Hellas.  Empirical Evaluation of Deep Learning Models for Knowledge Tracing: Of Hyperparameters and Metrics on Performance and Replicability.  Journal of Educational Data Mining, 14(2), 2022.
  * [10] Hao Cen, Kenneth Koedinger, and Brian Junker.  Learning factors analysis–a general method for cognitive model evaluation and improvement.  In International conference on intelligent tutoring systems, pages 164–175. Springer, 2006.
  * [11] Philip I Pavlik Jr, Hao Cen, and Kenneth R Koedinger.  Performance factors analysis–a new alternative to knowledge tracing.  Online Submission, 2009.
  * [12] Alisa Lincke, Marc Jansen, Marcelo Milrad, and Elias Berge.  The performance of some machine learning approaches and a rich context model in student answer prediction.  Research and Practice in Technology Enhanced Learning, 16(1):1–16, 2021.
  * [13] Zichao Wang, Angus Lamb, Evgeny Saveliev, Pashmina Cameron, Yordan Zaykov, José Miguel Hernández-Lobato, Richard E Turner, Richard G Baraniuk, Craig Barton, Simon Peyton Jones, Simon Woodhead, and Cheng Zhang.  Diagnostic questions: The neurips 2020 education challenge.  arXiv preprint arXiv:2007.12061, 2020.
  * [14] Shen Shuanghong, Liu Qi, Chen Enhong, Tong Shiwei, Huang Zhengya, Tong Wei, Su Yu, and Wang Shijin.  Which to choose? an order-aware cognitive diagnosis model for predicting the multiple-choice answer of students.
  * [15] Si Chenglei, Wang Shuohang, Kan Min-Yen, and Jiang Jing.  What does bert learn from multiple-choice reading comprehension datasets?  2019\.
  * [16] Wang Zichao, Lamb Angus, Saveliev Evgeny, Cameron Pashmina, Zaykov Yordan, Hernández-Lobato José Miguel, Turner Richard E., Baraniuk Richard G., Barton Craig, Peyton Jones Simon, Woodhead Simon, and Zhang Cheng.  Results and insights from diagnostic questions: The neurips 2020 education challenge.  Proceedings of Machine Learning Research, (133):191–205, 2021.
  * [17] Yuto Shinahara and Daichi Takehara.  Quality assessment of diagnostic questions based on multiple features, 2020.
  * [18] Zhang Hongbo, Qin Xiaolei, Zou Wuhe, Zhu Yue, Liu Ying, Liang Nan, and Zhang Weidong.  How to predict students’ interactions with diagnostic questions: from a perspective of recommender system.
  * [19] Aritra Ghosh and Andrew S Lan.  Option tracing: Beyond binary knowledge tracing.
  * [20] Benoît Choffin, Fabrice Popineau, Yolaine Bourda, and Jill-Jênn Vie.  Das3h: modeling student learning and forgetting for optimally scheduling distributed practice of skills.  arXiv preprint arXiv:1905.06873, 2019.
  * [21] Ahatsham Hayat and Mohammad Rashedul Hasan.  Personalization and Contextualization of Large Language Models For Improving Early Forecasting of Student Performance.  In NeurIPS Workshop on Generative AI for Education (GAIED), 2023.
  * [22] Xu Guowei, Chen Jiaohao, Li Hang, Kang Yu, Liu Tianqiao, Hao Yang, Ding Wenbiao, and Liu Zitao.  Solution for neurips education challenge 2020 from tal ml team.
  * [23] Manh Hung Nguyen, Sebastian Tschiatschek, and Adish Singla.  Large Language Models for In-Context Student Modeling: Synthesizing Student’s Behavior in Visual Programming from One-Shot Observation.  ArXiv, abs/2310.10690, 2023.
  * [24] Shishir Roy, Nayeem Ehtesham, Md. Saiful Islam, and Marium-E-Jannat.  Augmenting bert with cnn for multiple choice question answering.  In 2021 24th International Conference on Computer and Information Technology (ICCIT), pages 1–5, 2021.
  * [25] Joshua Robinson, Christopher Michael Rytting, and David Wingate.  Leveraging large language models for multiple choice question answering.  arXiv preprint arXiv:2210.12353, 2022.
  * [26] Thiemo Wambsganss, Vinitra Swamy, Roman Rietsche, and Tanja Käser.  Bias at a second glance: A deep dive into bias for german educational peer-review data modeling.  COLING, 2022.
  * [27] Xinlong Li, Xingyu Fu, Guangluan Xu, Yang Yang, Jiuniu Wang, Li Jin, Qing Liu, and Tianyuan Xiang.  Enhancing bert representation with context-aware embedding for aspect-based sentiment analysis.  IEEE Access, 8:46868–46876, 2020.
  * [28] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is all you need, 2023.
  * [29] Jiacheng Chen, Hexiang Hu, Hao Wu, Yuning Jiang, and Changhu Wang.  Learning the best pooling strategy for visual semantic embedding.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 15789–15798, 2021.
  * [30] Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, et al.  Mistral 7b.  arXiv preprint arXiv:2310.06825, 2023.

##  5 Appendix

This appendix includes additional figures and material to complement the main
section of the paper.

Figure 4: Example of dataset to illustrate the dataset split into a training
and a validation/test dataset. For clarity, each individual MCQ is
characterized by the same colour.

A dataset example showing multiple-choice questions in four colors: blue,
green, red, and orange. Each question includes a computation problem (e.g.,
Compute 5 + 7) with four answer choices labeled A, B, C, and D, an ID number,
and the selected answer.

Figure 5: Training and validation/test datasets obtained by splitting the
dataset using the second method presented in the report. Different instances
of the same MCQ can be in only one of the two datasets.

Dataset example split into training and validation datasets. The training
dataset includes questions in blue, green, and orange, asking to compute sums
(e.g., Compute 5 + 7, Compute 7 + 7, Compute 6 + 6), each with four answer
choices and an ID number. The validation dataset, in red, contains different
instances of a question asking to compute 6 + 7, also with four answer choices
and an ID number.

Table 2: MCQBert performance in comparison with a baseline model (Dummy Classifier) that predicts the majority class (0) for each answer choice. | MCC | F1 Score | Accuracy
---|---|---|---
Dummy Classifier | 0 | 0.292 | 0.605
MCQBert | 0.472 | 0.702 | 0.740

Figure 6: MCQBERT Architecture. The MCQBERT model for correct answer
prediction involves a binary classifier and a finetuned BERT architecture
predicting answers in sequence. The inset showing the Transformer encoder is
taken from [28].

At the top, a binary classifier receives input from the [CLS] token of a BERT
model. The BERT model processes input sequences including a classification
token [CLS], question tokens, a separator token [SEP], and answer tokens. The
BERT model consists of a transformer encoder, depicted as layers of multi-head
attention and feed-forward neural networks with positional encoding. An inset
on the right illustrates the internal structure of the transformer encoder.

Figure 7: t-SNE visualization of the Mistral 7B embedding space for difference
sequence lengths. The evolution of the same student’s embedding through time
is highlighted.

Four side-by-side t-SNE visualizations showing the Mistral 7B embedding space
at different sequence lengths (10, 20, 30, 40). Each plot features a light
blue circular cluster of points, with the evolution of a single student’s
embedding through time highlighted in colors ranging from orange (old) to
purple (recent).

###  5.1 MCQBert: Correct Answer Prediction

This section introduces MCQBert, a model developed to predict students’
responses to MCQs using only the question text444MCQBert is available at
https://huggingface.co/collections/epfl-ml4ed/student-answer-forecasting-
edm-2024-663b7c20bb2aa3273dda4de2. The downstream task and primary objective
of MCQBert is to accurately identify the correct answer(s) from the given
options in each MCQ in the dataset. The architectural design of MCQBert is
illustrated in Figure 6. It represents the LernnaviBERT model’s application in
processing both the question and a potential answer as input sequences. The
transformer encoder component of LernnaviBERT processes the sequences before
it is passed to a binary classifier, which outputs ‘1’ for a correct answer
and ‘0’ for an incorrect one.

Data Split. To formulate the MCQ prediction challenge as a binary
classification task, each MCQ datapoint is decomposed into separate instances,
each pairing the question with a possible answer option. The model thus aims
to assign a ‘1’ to a correct or student-selected answer and a ‘0’ to an
incorrect or unselected answer.

We implement a partitioning ratio of 80/10/10 for training, validation, and
testing, respectively. Notably, each MCQ occurs multiple times within the
dataset, corresponding to different answers. To rigorously assess the model’s
ability to generalize and accurately answer new MCQs, we ensure that
individual MCQs are exclusively allocated to either the training or the
testing set. In other words, all occurrences of a particular MCQ are confined
to a single subset.

Experiments. To evaluate the performance of our models in predicting correct
answers to MCQs, two distinct experiments were conducted. In the first one,
the model was fine-tuned using the designated training set, and its
generalization capacity was assessed on a separate test set comprising unseen
questions, validating its ability to respond to MCQs beyond the training data.
The second experiment involved training the model on the entire MCQ dataset,
confirming its effective learning of correct answers across all MCQs, ensuring
that any failure in predicting a student’s response did not arise from a lack
of knowledge of the correct answer. In the final phase of our pipeline for
predicting student responses to MCQs, the model trained on the complete
dataset was fine-tuned, allowing it to utilize its comprehensive knowledge of
the correct answers when making predictions.

This section describes the evaluation of MCQBert’s performance in the specific
task of predicting correct answers to MCQs. The evaluation consisted of two
distinct experiments, each employing a different training procedure to assess
the model’s predictive capabilities.

Experiment 1: Model Evaluation Against Unseen MCQs. The first experiment aimed
to evaluate the ability of the model to predict the correct answers of
previously unseen MCQs accurately. The models were fine-tuned for one epoch on
a designated training set and subsequently evaluated on a separate test set.
The evaluation metrics included MCC, F1 score, and accuracy.

The results, as summarized in Table 2, contrast the performance of MCQBert
with that of a Dummy Classifier, a baseline always predicting the majority
class (i.e. 0). This comparison is useful for evaluating the effectiveness of
MCQBert beyond simple chance or biased class distribution.

The performance metrics indicate that MCQBert outperforms the baseline Dummy
Classifier, evidencing its capability to discern correct answers in the
context of MCQs.

Experiment 2: Model MCQs Retention Evaluation. The second experiment was
designed to evaluate MCQBert’s capacity for retaining correct answers after
being fine-tuned on the entire Lernnavi MCQ dataset. The model was then tested
on the same dataset to assess its ability to recall the correct answers,
effectively evaluating its memorization capability. Not surprisingly, MCQBert
achieved an MCC of 0.983, indicating nearly perfect recall of the correct
answers within the dataset. This high level of performance is further
corroborated by F1 scores of 0.993 for class 0 and 0.989 for class 1. The
accuracy score of 0.992 reinforces the model’s strong predictive capability
and suggests that MCQBert model has learnt the correct answers to the vast
majority of the MCQs present in the dataset, and we can therefore exploit this
knowledge in the next step.

###  5.2 Reproducibility for MCQStudentBERT

Similar to the MCQ prediction task without student data, we split the dataset
into training, validation, and test datasets using an 80/10/10 split. In
contrast to the base model (MCQBert), where each question-answer pair was
exclusively assigned to one subset, in this task, individual MCQs can be
present in both training and testing phases due to multiple representations
within the dataset. This decision allows the model to leverage prior history
related to specific questions, enhancing predictive accuracy by considering
responses from students who share similar characteristics with the target
student.

Evaluation Metrics. We assess our models using three different metrics: the
Matthews Correlation Coefficient (MCC) for binary classification, the F1 score
for balancing precision and recall, and the accuracy score for overall
predictions. MCC and F1-score are effective even if the classes are strongly
imbalanced, motivating our evaluation choices. While accuracy and F1-score
range between 0 to 1, MCC is a correlation coefficient value between -1 and +1
(+1: perfect prediction, 0: average random prediction, -1: inverse
prediction).

###  5.3 Impact of sequence length on latent space representations

We examined the sequence length impact on the Mistral 7B embeddings. Similar
to Figure 3, Figure 7 shows a single student’s embeddings across varying
sequence lengths. Echoing the behavior of the LernnaviBERT embedding, we note
a discernible diagonal progression in the embeddings when the sequence length
is greater than 10. The trend suggests that as the sequence length increases,
the student’s representation in the embedding space demonstrates a more
pronounced diagonal trajectory, transitioning methodically from older to more
recent embeddings with a progressively smoother evolution.


# Power Domination and Resolving Power Domination of Fractal Cubic Network

S. Prabhu${}^{\text{a},}$, A.K. Arulmozhi${}^{\text{b}}$, Michael A.
Henning${}^{\text{c}}$, M. Arulperumjothi${}^{\text{d}}$
${}^{\text{a}}$Department of Mathematics, Rajalakshmi Engineering College,
Chennai 602105, India ${}^{\text{b}}$Department of Mathematics, RMK College of
Engineering and Technology, Puduvoyal 601206, India ${}^{\text{c}}$Department
of Mathematics and Applied Mathematics, University of Johannesburg, Auckland
Park, 2006 South Africa ${}^{\text{d}}$Department of Mathematics, St. Joseph’s
College of Engineering, Chennai 600119, India Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

In network theory, the domination parameter is vital in investigating several
structural features of the networks, including connectedness, their tendency
to form clusters, compactness, and symmetry. In this context, various
domination parameters have been created using several properties to determine
where machines should be placed to ensure that all the places are monitored.
To ensure efficient and effective operation, a piece of equipment must monitor
their network (power networks) to answer whenever there is a change in the
demand and availability conditions. Consequently, phasor measurement units
(PMUs) are utilised by numerous electrical companies to monitor their networks
perpetually. Overseeing an electrical system which consists of minimum PMUs is
the same as the vertex covering the problem of graph theory, in which a subset
$D$ of a vertex set $V$ is a power dominating set (PDS) if it monitors
generators, cables, and all other components, in the electrical system using a
few guidelines. Hypercube is one of the versatile, most popular, adaptable,
and convertible interconnection networks. Its appealing qualities led to the
development of other hypercube variants. A fractal cubic network is a new
variant of the hypercube that can be used as a best substitute in case faults
occur in the hypercube, which was wrongly defined in [Eng. Sci. Technol. 18(1)
(2015) 32-41]. Arulperumjothi et al. have recently corrected this definition
and redefined this variant with the exact definition in [Appl. Math. Comput.
452 (2023) 128037]. This article determines the PDS of the fractal cubic
network. Further, we investigate the resolving power dominating set (RPDS),
which contrasts starkly with hypercubes, where resolving power domination is
inherently challenging.

Keywords: fractal cubic network; hypercube variant; power domination; phase
measuring units; resolving power domination
Mathematics Subject Classification (2020): 05C69 $\cdot$ 05C12

##  1 Introduction

The electrical nodes and connection wires constitute an electrical power
network. Electric power corporation must continuously monitor their systems’
conditions. It is necessary to regulate the deployment of PMUs at precise
position within the device.

Owing to the rising cost of PMUs, it is essential to engage as few as feasible
while still tracking the entire system. This problem is introduced as a
theoretical problem in [1], and coined it as a power domination problem after
[2] evince its existence.

Let $G$ be a simple connected graph whose vertex set (electrical hubs) and
edge set (connecting cables), respectively, is denoted by $V(G)$ and $E(G)$.
If the graph $G$ is clear from the context, we write $V=V(G)$ and $E=E(G)$.
Two vertices $u$ and $v$ of $G$ are _adjacent_ if $uv\in E$. Two adjacent
vertices are called _neighbors_. For $r\geq 1$ and for a vertex $v\in V$, the
_open_ $r$-_neighborhood_ of $v$ is denoted by $N_{r}(v)$ and is defined as
the collection of vertices that are at distance $r$ from $v$. The _closed_
$r$-_neighborhood_ of $v$ is $N_{r}(v)\cup\\{v\\}$, which is formally denoted
by $N_{r}[v]$. When $r=1$, the _open_ $r$-_neighborhood_ of $v$ is called its
_open neighborhood_ and the _closed_ $r$-_neighborhood_ of $v$ is called its
_closed neighborhood_. For a set $S\subseteq V$, its _open neighborhood_ is
the set $N_{G}(S)=\cup_{v\in S}N_{1}(v)$, and its _closed neighborhood_ is the
set $N_{G}[S]=N_{G}(S)\cup S$. The _degree_ of a vertex $v$ in $G$ is the
number of vertices adjacent to $v$ in $G$, and is denoted by $\deg_{G}(v)$,
and so $\deg_{G}(v)=|N_{1}(v)|$. We let $\mathbb{N}_{n}:=\\{1,2,\ldots,n\\}$
and $\mathbb{W}_{n}:=\\{0,1,2,\ldots,n\\}$. Also, we use
$\mathbb{N}_{2n+1}-\mathbb{N}_{n}$ to denote $\\{n+1,n+2,\ldots,2n+1\\}$.

A set $S\subseteq V$ is a _dominating set_ of $G$ if every vertex in
$V\setminus S$ is adjacent to at least one vertex in $S$. A vertex $v$ in $G$
_dominates_ itself and all its neighbors, and a set $X$ _dominates_ a set $Y$
in $G$ if every vertex in $Y$ is dominated by at least one vertex in $X$. Thus
a dominating set $S$ of $G$ dominates every vertex in $V(G)$. The _domination
number_ of $G$, denoted by $\gamma(G)$, is the minimum cardinality of a
dominating set in $G$. We refer [3, 9, 4, 5, 6, 7, 8, 10, 11, 12] for a
detailed study of domination and its variants.

If $V$ is observed or recursively observed by the following two rules with
respect to a set $S$ of vertices in $G$, then $S$ is called _power dominating
set_ , abbreviated PDS of $G$.

  1. 1.

Domination:
$M\leftarrow N_{G}[S]$

  2. 2.

Propagation:
$\exists,\enspace x\in M$ s.t. $N(x)\cap(V-M)=\\{y\\}$
$M\leftarrow M\cup\\{y\\}$

In the domination step of PDS, all vertices in $N_{G}[S]$ are _monitored_. In
the event of the propagation phase of PDS, whenever a vertex $x$ is monitored,
and $N_{G}(x)\smallsetminus\\{y\\}$ is monitored, then the vertex $y$ is added
to the set $M$ and is also monitored. An initial set $S$ in the first step is
a PDS for $G$ if upon completion of the propagation phase the resulting set
$M$ is the entire vertex set $V$. The term $\gamma_{P}(G)$ is used to denote
the minimum cardinality of a PDS in $G$ called the _power domination number_.

Investigation of PDS for arbitrary general graphs is NP-complete. It remains
NP-complete for classes of graphs such as bipartite, chordal, and split graphs
[1]. Numerous algorithms for fetching the PDS for a particular graph family
were reported in [1, 14, 13]. This invariant is investigated for generalized
Petersen family of graphs [15, 18, 16, 17], hypercubes [19], circular-arc
graphs [20], block graphs [21], permutaion graphs [22], grids [23], planar and
maximal planar graphs [24], Hanoi and Knödel graphs [25], Kautz and de Bruijn
graphs [26], claw-free regular graphs [27] and octahedral structures [28].
Also, for graph operations like strong and tensor product [29], Cartesian
product [15, 30], join and corona product [31], the PDS problem is examined.
The lower bounds [33], upper bounds [34], and NG (Nordhaus-Gaddum) type
results on PDS were discussed in [32]. An excellent survey on this topic can
be seen in the book chapter by Dorbec [35].

The concept of a $k$-_power dominating set_ , abbreviated $k$-PDS, was
introduced in [16], where a $0$-PDS is a traditional dominating set and a
$1$-PDS is the original power dominating set PDS. This parameter is
investigated for certain interconnection networks [36], weighted trees [37],
Sierpiński networks [38], block graphs [39], regular graphs [40]. Few more
variation namely power dominating throttling [41] and infectious power
domination [42] are recent and interesting problems.

The concept of metric dimension (MD) was primarily discussed in [43] and
separately in [44]. It is equivalent to the least number of landmark vertices
from which any two vertices can be differentiated using the distance
parameter. The applications of this problem arise in various branches of
science and technology.

The _graph geodesic_ between two vertices $u$ and $v$ is the length (in terms
of the number of edges) of the shortest path between $u$ and $v$. The
_diameter_ ${\rm diam}(G)$ of $G$ is the maximum length of a geodesic in $G$.
The maximum distance over all pairs of $V(G)$ is called diameter.

For a vertex $x\in V(G)$, the code of $x$ with respect to the subset $R$
$=\\{r_{1},r_{2},\ldots,r_{k}\\}$ of $V(G)$ is termed as a $k$-vector

$C_{R}(x)=(d_{G}(x,r_{1}),d_{G}(x,r_{2}),\ldots,d_{G}(x,r_{k}))$

where $d_{G}(x,r_{i})$ is the geodesic from $x$ to $r_{i}$ for
$i\in\mathbb{N}_{k}$. The proper subset $R$ is a basis or resolving set for
$G$ if any two distinct vertices of $G$ have nonidentical codes w.r.t $R$.
Equivalently, for any two distinct vertices $x,y\in V(G)$, there exist a
vertex $r\in R$ such that $d_{G}(r,x)\neq d_{G}(r,y)$. See Figure 1. The basis
of $G$ with fewer cardinality is called the _resolving number_ (also called
the _basis dimension_) of $G$ and is denoted by $\dim(G)$. More on this topic
and some of its recent works can be found in [45, 47, 46, 48].

Figure 1: Basis

##  2 Fractal Cubic Network: A New Hypercube Variant

Frequently, an interconnection networks with multiprocessors are essential to
link an eloquent portion of dependably imitated processors (vertices). In lieu
of shared memory, message transient is mostly used to afford complete
transmission and synchronisation between processors for programmed execution.
Owing to the availability of cost-effective, potent memory circuits and
microprocessors, there has been a recent surge in fascination in implementing
and designing multistage networks.

In parallel computing and supercomputers, interconnection networks, the
development of CPUs, and routing algorithms are three primary research areas.
The interconnection network, which connects millions of processors, is
essential for the development of a supercomputer.

Multiple processors, each with their own cognitive connections (edges) and
local sensing that enable data transfer between processors (vertices), compose
an interconnection network. It can be represented as the previously defined
graph $G$ in which two vertices $x_{i}$ and $x_{j}$ are directly connected by
a communication connection. The attribute used to evaluate the productiveness
of the networks are the bisection width, broadcasting time, fault-tolerance,
degree, and diameter [50].

The hypercube is a common interconnection network design distinguished by its
regularity, ease of transit, recursive structure, symmetry, and high
connectedness. In recent years, hypercubes have been the subject of extensive
research into their various properties [51].

There are many hypercube variations in the literature, including exchanged
hypercube [52], folded hypercubes [53, 54], crossed cubes [55, 56], exchanged
crossed cube [57], twisted cubes [58, 59], locally twisted cubes [60], shuffle
cubes [61], spined cubes [62], Möbius cubes [63], and augmented cubes [64].
The hierarchical cubic network (HCN) [65, 66] and its folded version in [67]
have also been ideas put forth based on a hierarchical framework employing the
base hypercube as an introductory module.

Despite the fact that numerous studies have been conducted on the variants of
hypercube enumerated above, the problem resolving number has not been
investigated for any of these variants except fractal cubic network. With this
motivation, we investigate power domination and resolving power domination for
the newly proposed hypercube variant fractal cubic network (FCN) [68]. Though
the definition of this architecture is not clear in [68], we corrected this
definition in [45]. We define $FCN(0)$ as a cycle with four vertices $00$,
$01$, $11$, and $10$. For $d\geq 1$, we define $FCN(d)$ as follows:

An $d$-dimensional FCN is defined as $FCN(d)=(V_{1}(d),E_{1}(d))$, $d>0$, and
can be constructed as follows

$FCN(d)=11\mathbin{\|}FCN(d-1)\cup 01\mathbin{\|}FCN(d-1)\cup
10\mathbin{\|}FCN(d-1)\cup 00\mathbin{\|}FCN(d-1),$

where

$V_{1}(d)=11\mathbin{\|}V_{1}(d-1)\cup 01\mathbin{\|}V_{1}(d-1)\cup
10\mathbin{\|}V_{1}(d-1)\cup 00\mathbin{\|}V_{1}(d-1)$

and

$\begin{array}[]{lcl}E_{1}(d)&=&11\mathbin{\|}E_{1}(d-1)\cup
01\mathbin{\|}E_{1}(d-1)\cup 10\mathbin{\|}E_{1}(d-1)\cup
00\mathbin{\|}E_{1}(d-1)\\\ &&\,\cup\,\\{(001100\ldots 0,101100\ldots
0),(101100\ldots 0,111100\ldots 0)\\}\\\ &&\,\cup\,\\{(111100\ldots
0,011100\ldots 0),(011100\ldots 0,001100\ldots 0)\\}.\end{array}$

Figure 2(a)-(d), respectively, denotes the $FCN(0)$, $FCN(1)$, $FCN(2)$, and
$FCN(3)$.

We denote $11\mathbin{\|}FCN(d-1)$ as the collection of strings obtained by
concatenating $11$ and each of the strings in $V_{1}(d-1)$. For $d\geq 1$,
$FCN(d)$ is constructed from four copies of $FCN(d-1)$ with four additional
edges connecting them.

##  3 Main Results

Before proceeding to the main arguments, we first present some preliminary
lemma and results that are crucial for our investigation.

###  3.1 Twin Nodes

Two vertices $u,v\in V$ are said to be _non_ -_adjacent twins_ (also called
_open twins_ in the literature) if $N_{1}(u)=N_{1}(v)$ and are _adjacent
twins_ (also called _closed twins_ in the literature) if $N_{1}[u]=N_{1}[v]$
(see [69, 70]). The vertices $u$ and $v$ shown in Figure 3(a) are open twins,
while the vertices $u$ and $v$ shown in Figure 3(b) are closed twins. Two
vertices in $G$ are _twins_ if they are open or closed twins in $G$. A set
$T\subseteq V(G)$ is a _open twin set_ or _open twin class_ of $G$, if every
pair of vertices in $T$ are open twins in $G$, while a set $T\subseteq V(G)$
is a _closed twin set_ or _closed twin class_ of $G$, if every pair of
vertices in $T$ are closed twins in $G$. The sets $T_{1}$ and $T_{2}$ shown in
Figure 3(c) are examples of two open twin sets.

Arulperumjothi, Klavžar, and Prabhu [45] determined $\dim(G)$, where
$G=FCN(d)$ and $d\geq 1$.

######  Theorem 1.

[45] For $d>0$, $\dim(FCN(d))=4^{d}$.

(a)  (b)  (c)  (d)

Figure 2: (a) $FCN(0)$; (b) $FCN(1)$; (c) $FCN(2)$; (d) $FCN(3)$

(a)  (b)  (c)

Figure 3: (a) Open twins; (b) Closed twins; (c) Open twin sets $T_{1}$ and
$T_{2}$

We proceed further with the following two key lemmas.

######  Lemma 2.

Let $G$ be a connected graph. If $S$ is a PDS of $G$ and $T$ is an open twin
class of $G$, then $S\cap N_{1}(T)\neq\emptyset$ or $|S\cap T|\geq|T|-1$.

###### Proof.

Let $G$ be a connected graph, and let $S$ be a PDS of $G$ and $T$ an open twin
class of $G$. We note that $T$ is an independent set and $|T|\geq 2$. Moreover
every $x\in N_{1}(T)$ is adjacent to every $y\in T$. Thus, every path
connecting a vertex in $T$ and a vertex in $V(G)-T$ must contain a vertex in
$N_{1}(T)$. We show that $S\cap N_{1}(T)\neq\emptyset$ or $|S\cap
T|\geq|T|-1$. Suppose, to the contrary, that $S\cap N_{1}(T)=\emptyset$ and
$|S\cap T|\leq|T|-2$. Let $R=T\setminus S$ (possibly, $R=T$ which occurs is
$S\cap T=\emptyset$), and so $|R|=|T|-|S|\geq 2$. Since $T$ is an independent
set and since $S\cap N_{1}(T)=\emptyset$, no vertex in $S$ is adjacent to a
vertex in $R$. Thus, $R$ contains no vertex in $N_{G}[S]$, and so no vertex of
$R$ is dominated in the initial domination step of PDS. As observed earlier,
every path connecting a vertex in $T$ and a vertex in $V(G)-T$ must contain a
vertex in $N_{1}(T)$. Hence in order to observe the vertices in $R$ we must
first observe a vertex in $N_{1}(T)$ either by domination or by propagation
from the vertices that do not belong to the set $R$. However, every vertex in
$N_{1}(T)$ is adjacent to all vertices of $R$. Thus the vertices of $R$ cannot
be propagated as every vertex of $N_{1}(T)$ has $|R|\geq 2$ vertices as
neighbors. This contradicts the supposition that $S$ is a PDS of $G$. ∎

######  Lemma 3.

If $G$ is a connected graph with $k$ vertex disjoint twin classes
$T_{1},\ldots,T_{k}$ such that $N_{1}(T_{i})\cap N_{1}(T_{j})=\emptyset$ for
every $1\leq i<j\leq k$, then $\gamma_{P}(G)\geq k$.

###### Proof.

Let $S$ be a PDS of $G$. By Lemma 2, $S\cap N_{1}(T_{i})\neq\emptyset$ or
$|S\cap T_{i}|\geq|T_{i}|-1$ for all $i\in\mathbb{N}_{k}$. Thus since
$|T_{i}|\geq 2$, we note that if $S\cap N_{1}(T_{i})=\emptyset$, then $|S\cap
T_{i}|\geq 1$. Hence at least one of $|S\cap N_{1}(T_{i})|\geq 1$ or $|S\cap
T_{i}|\geq 1$ holds for all $i\in\mathbb{N}_{k}$, implying that
$|S\cap(T_{i}\cup N_{1}(T_{i}))|\geq 1$ for all $i\in\mathbb{N}_{k}$. By
supposition, the twin classes $T_{1},\ldots,T_{k}$ are vertex disjoint.
Further, the sets $T_{i}\cup N_{1}(T_{i})$ and $T_{j}\cup N_{1}(T_{j})$ are
vertex disjoint for every $1\leq i<j\leq k$. From this we infer that

$|S|\geq\sum_{i=1}^{k}|S\cap(T_{i}\cup N_{1}(T_{i}))|\geq k.$

Since $S$ in arbitrary PDS of $G$, we infer that $\gamma_{P}(G)\geq k$. ∎

We are now in a position to prove the following result.

######  Theorem 4.

For $d>0$, $\gamma_{P}(FCN(d))=4^{d}$.

###### Proof.

Let $G=FCN(d)$. From the definition of $G$, we note that
$N_{1}(u_{2d+2}u_{2d+1}\ldots u_{3}01)=N_{1}(u_{2d+2}u_{2d+1}\ldots u_{3}10)$,
where $u_{2d+2}u_{2d+1}\ldots u_{3}\in\\{0,1\\}^{2d}$. That is,
$\\{u_{2d+2}u_{2d+1}\ldots u_{3}01,u_{2d+2}u_{2d+1}\ldots u_{3}10\\}$ is an
open twin set in $G$. The $FCN(2)$ and its $2^{4}$ twin sets are marked in
Figure 4. Hence, $G$ contains $2^{2d}$ twin sets, each containing exactly two
vertices. Therefore by Lemma 3, $\gamma_{P}(G)\geq 4^{d}$.

Figure 4: The $2^{4}$ open twin sets in $FCN(2)$

To show that $\gamma_{P}(G)\leq 4^{d}$, let

$A=\bigcup\\{u_{2d+2}u_{2d+1}\ldots u_{3}01\\}\hskip
14.22636pt\mbox{and}\hskip 14.22636ptB=\bigcup\\{u_{2d+2}u_{2d+1}\ldots
u_{3}10\\}$

where the union is taken over all $u_{2d+2}u_{2d+1}\ldots
u_{3}\in\\{0,1\\}^{2d}$. We note that $|A|=|B|=4^{d}$. Let
$C=V(G)\setminus(A\cup B)$. We claim that the set $A$ is a power dominating
set of $G$. For this, we have to prove that every vertex of $V$ is monitored
by the vertices of $A$ either by domination or by propagation. We note that
$N_{1}(u_{2d+2}u_{2d+1}\ldots u_{3}01)=\\{u_{2d+2}u_{2d+1}\ldots
u_{3}11,u_{2d+2}u_{2d+1}\ldots u_{3}00\\}$, where as before
$u_{2d+2}u_{2d+1}\ldots u_{3}\in\\{0,1\\}^{2d}$. Hence, $N_{G}[A]=A\cup C$.
Thus, all vertices in $A\cup C$ are monitored by the vertices of $A$ by
domination. We note further that $A\cup B$ is an independent set and that
$N_{G}(B)=C$. Also, each vertex $x\in\\{u_{2d+2}u_{2d+1}\ldots u_{3}10\colon
u_{i}\in\\{0,1\\}\\}$ is of degree $2$ in $FCN(d)$ and is adjacent to vertex
$y$, where $y\in N_{1}(u_{2d+2}u_{2d+1}\ldots u_{3}01)$. Moreover, the
vertices in $B$ are pairwise at distance at least $3$ apart, and so no two
vertices in $B$ have a common neighbor that belongs to $C$. Thus every
dominated vertex in $C$ has all its neighbors dominated, except for exactly
one vertex in $B$. Thus, each dominated vertex in $C$ propagates to its unique
neighbor in $B$. Thus, each vertex $y$ is propagated by either
$u_{2d+2}u_{2d+1}\ldots u_{3}11$ or $u_{2d+2}u_{2d+1}\ldots u_{3}00$. Hence,
$V=M(A)$, that is, $A$ is a power dominating set of $G$, implying that
$\gamma_{P}(G)\leq|A|=4^{d}$. As observed earlier, $\gamma_{P}(G)\geq 4^{d}$.
Consequently, $\gamma_{P}(G)=4^{d}$. ∎

##  4 Resolving Power Domination of FCN

This section recalls the definition of a resolving-power dominating set and
its minimum cardinality. A subset $S$ is both resolving and power-dominating.
The subset $S$ of $V$ is called a resolving-power dominating set, and its
minimum cardinality is the resolving power domination number and is notated by
$\eta_{P}(G)$. The first paper on this notion was introduced in [71].

######  Theorem 5.

[71] For any simple connected graph $G$,
$\max\\{\dim(G),\gamma_{P}(G)\\}\leq\eta_{P}(G)\leq\dim(G)+\gamma_{P}(G)$.

Since then, we are not aware of further research on this problem. Prabhu et
al. recently investigated this parameter for probabilistic neural networks
(PNN) in [48]. As a consequence of our main results in this paper, we
determine the resolving power domination number of a fractal cubic network.

######  Theorem 6.

For $d\geq 1$, $\eta_{P}(FCN(d))=4^{d}$.

###### Proof.

The proof follows immediately from Theorems 1, 4 and 5. ∎

##  5 Conclusion

Multistage interconnection networks are essential to parallel computing
because their performance on a large scale is determined by their
connectivity. Communication efficacy is a key prosecution indicator in
parallel computing. The diameter of a network’s interconnections is an
essential metric of transmission efficiency. Hypercube is prevalent in all
architecture owing to its advantageous properties. There are numerous
prospective variants of this network that can be created by modifying certain
links. This variant of hypercube is one such variant. In the present
investigation, we concentrate on power domination and resolving-power
domination for this new invariant in the most optimal manner. In this paper,
we completely determine these parameters for fractal cubic networks.

Compliance with ethical standards
Conflict of interest: The authors declare that there is no conflict of
interest regarding the publication of this paper.
Data Availability Statement: No Data associated with the manuscript.

## References

  * [1] T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Domination in graphs applied to electric power networks, SIAM Journal on Discrete Mathematics 15(4) 519–529.
  * [2] T.L. Baldwin, L. Mili, M.B. Boisen, R. Adapa, Power system observability with minimal phasor measurement placement, IEEE Transactions on Power systems 8(2) (1993) 707–715.
  * [3] D. Chalupa, An order-based algorithm for minimum dominating set with application in graph mining, Information Sciences 426 (2018) 101–116.
  * [4] W. Chen, Z. Lu, W. Wu, Dominating problems in swapped networks, Information Sciences 269 (2014) 286–299.
  * [5] C. Pang, R. Zhang, Q. Zhang, J. Wang, Dominating sets in directed graphs, Information Sciences 180(19) (2010) 3647–3652.
  * [6] W.C.K. Yen, The bottleneck independent domination on the classes of bipartite graphs and block graphs, Information Sciences 157 (2003) 199–215.
  * [7] Y.T. Tsai, Y.L. Lin, F.R. Hsu, Efficient algorithms for the minimum connected domination on trapezoid graphs, Information Sciences 177(12) (2007) 2405–2417.
  * [8] L. Wu, E. Shan, Z. Liu, On the $k$-tuple domination of generalized de Brujin and Kautz digraphs, Information Sciences 180(22) (2010) 4430–4435.
  * [9] Z. Shao, M. Liang, C. Yin, X. Xu, P. Pavlič, J. Žerovnik, On rainbow domination numbers of graphs, Information Sciences 254 (2014) 225–234.
  * [10] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), _Topics in Domination in Graphs_. Series: Developments in Mathematics, Vol. 64, Springer, Cham, 2020. viii + 545 pp.
  * [11] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), _Structures of Domination in Graphs_. Series: Developments in Mathematics, Vol. 66, Springer, Cham, 2021. viii + 536 pp.
  * [12] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, _Domination in Graphs: Core Concepts_. Series: Springer Monographs in Mathematics, Springer, Cham, 2023. xx + 644 pp.
  * [13] A. Aazami, K. Stilp, Approximation algorithms and hardness for domination with propagation, SIAM Journal on Discrete Mathematics 23(3) (2009) 1382–1399.
  * [14] J. Guo, R. Niedermeier, D. Raible, Improved algorithms and complexity results for power domination in graphs, Algorithmica 52 (2008) 177–202.
  * [15] R. Barrera, D. Ferrero, Power domination in cylinders, tori, and generalized Petersen graphs, Networks 58(1) (2011) 43–49.
  * [16] G.J. Chang, P. Dorbec, M. Montassier, A. Raspaud, Generalized power domination of graphs, Discrete Applied Mathematics 160(12) (2012) 1691–1698.
  * [17] M. Zhao, E. Shan, L. Kang, Power domination in the generalized Petersen graphs, Discussiones Mathematicae Graph Theory 40(3) (2020) 695–712.
  * [18] G. Xu, L. Kang, On the power domination number of the generalized Petersen graphs, Journal of Combinatorial Optimization 22 (2011) 282–291.
  * [19] N. Dean, A. Ilic, I. Ramirez, J. Shen, K. Tian, On the power dominating sets of hypercubes, In 2011 14th IEEE International Conference on Computational Science and Engineering (2011) pp. 488–491.
  * [20] C.S. Liao, D.T. Lee, Power domination in circular-arc graphs, Algorithmica 65(2) (2013) 443–466.
  * [21] G. Xu, L. Kang, E. Shan, M. Zhao, Power domination in block graphs, Theoretical Computer Science, 359(1-3) (2006) 299–305.
  * [22] S. Wilson, Power domination on permutation graphs, Discrete Applied Mathematics 262 (2019) 169–178.
  * [23] M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154(6) (2006) 1023–1027.
  * [24] P. Dorbec, A. González, C. Pennarun, Power domination in maximal planar graphs, Discrete Mathematics & Theoretical Computer Science 21(4) (2019) 1–24.
  * [25] S. Varghese, A. Vijayakumar, A.M. Hinz, Power domination in Knödel graphs and Hanoi graphs, Discussiones Mathematicae Graph Theory 38(1) (2018) 63–74.
  * [26] J. Kuo, W.L. Wu, Power domination in generalized undirected de Bruijn graphs and Kautz graphs, Discrete Mathematics, Algorithms and Applications 7(1) (2015) 1550003.
  * [27] C. Lu, R. Mao, B. Wang, Power domination in regular claw-free graphs, Discrete Applied Mathematics 284 (2020) 401–415.
  * [28] S. Prabhu, S. Deepa, R.M. Elavarasan, E. Hossain, Optimal PMU placement problem in octahedral networks, RAIRO-Operations Research 56(5) (2022) 3449–3459.
  * [29] P. Dorbec, M. Mollard, S. Klavžar, S. Špacapan, Power domination in product graphs, SIAM Journal on Discrete Mathematics 22(2) (2008) 554–567.
  * [30] K.M. Koh, K.W. Soh, On the power domination number of the Cartesian product of graphs, AKCE International Journal of Graphs and Combinatorics 16(3) (2019) 253–257.
  * [31] I. Yuliana, I.H, Agustin, D.A.R. Wardani, On the power domination number of corona product and join graphs, Journal of Physics: Conference Series 1211(1) (2019) 012020.
  * [32] K.F. Benson, D. Ferrero, M. Flagg, V. Furst, L. Hogben, V. Vasilevska, Nordhaus–Gaddum problems for power domination, Discrete Applied Mathematics 251 (2018) 103–113.
  * [33] D. Ferrero, L. Hogben, F.H. Kenter, M. Young, Note on power propagation time and lower bounds for the power domination number, Journal of Combinatorial Optimization 34 (2017) 736–741.
  * [34] M. Zhao, L. Kang, G.J. Chang, Power domination in graphs, Discrete Mathematics 306(15) (2006) 1812–1816.
  * [35] P. Dorbec, Power domination in graphs. _Topics in Domination in Graphs,_ 521–545. Series: Developments in Mathematics, Vol. 64, Springer, Cham, 2020.
  * [36] R.S. Rajan, S Arulanand, S Prabhu, I Rajasingh, 2-power domination number for Knödel graphs and its application in communication networks, RAIRO-Operations Research 57(6) (2023) 3157–3168.
  * [37] C. Cheng, C. Lu, Y. Zhou, The $k$-power domination problem in weighted trees, Theoretical Computer Science 809(2) (2020) 231–238.
  * [38] P. Dorbec, S. Klavžar, Generalized power domination: propagation radius and Sierpiński graphs, Acta Applicandae Mathematicae 134 (2014) 75–86.
  * [39] C. Wang, L. Chen, C. Lu, $k$-Power domination in block graphs, Journal of Combinatorial Optimization 31(2) (2016) 865–873.
  * [40] P. Dorbec, M.A. Henning, C. Löwenstein, M. Montassier, A. Raspaud, Generalized power domination in regular graphs, SIAM Journal on Discrete Mathematics 27(3) (2013) 1559–1574.
  * [41] B. Brimkov, J. Carlson, I.V. Hicks, R. Patel, L. Smith, Power domination throttling, Theoretical Computer Science 795 (2019) 142–153.
  * [42] B. Bjorkman, Infectious power domination of hypergraphs, Discrete Mathematics 343(3) (2020) 111724.
  * [43] P.J. Slater, Leaves of trees, Congressus Numerantium 14 (1975) 549–559.
  * [44] F. Harary, R.A. Melter, On the metric dimension of a graph, Ars Combinatoria 2 (1976) 191–195.
  * [45] M. Arulperumjothi, S. Klavžar, S. Prabhu, Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension, Applied Mathematics and Computation 452 (2023) 128037.
  * [46] S. Prabhu, V. Manimozhi, M. Arulperumjothi, S. Klavžar, Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks, Applied Mathematics and Computation 420 (2022) 126897.
  * [47] S. Prabhu, D.S.R. Jeba, M. Arulperumjothi, S. Klavžar, Metric dimension of irregular convex triangular networks, AKCE International Journal of Graphs and Combinatorics https://doi.org/10.1080/09728600.2023.2280799 (2023).
  * [48] S. Prabhu, S. Deepa, M. Arulperumjothi, L. Susilowati, J.-B. Liu, Resolving-power domination number of probabilistic neural networks, Journal of Intelligent & Fuzzy Systems 43(5) (2022) 6253–6263.
  * [49] S. Prabhu, T. Flora, M. Arulperumjothi, On independent resolving number of TiO${}_{2}[m,n]$ nanotubes, Journal of Intelligent & Fuzzy Systems 35(6) (2018) 6421–6425.
  * [50] S.B. Akers, B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Transactions on Computers 38(4) (1989) 555–566.
  * [51] F. Harary, J.P. Hayes, H.J. Wu, A survey of the theory of hypercube graphs, Computers and Mathematics with Applications 15(4) (1988) 277–289.
  * [52] P.K. Loh, W.J. Hsu, Y. Pan, The exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16(9) (2005) 866–874.
  * [53] A. El-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Transactions on Parallel and Distributed Systems 2(1) (1991) 31–42.
  * [54] Q. Zhu, S.Y. Liu, M. Xu, On conditional diagnosability of the folded hypercubes, Information Sciences 178(4) (2008) 1069–1077.
  * [55] K. Efe, The crossed cube architecture for parallel computation, IEEE Transactions on Parallel & Distributed Systems 3(5) (1992) 513–524.
  * [56] J. Fan, X. Jia, Embedding meshes into crossed cubes, Information Sciences 177(15) (2007) 3151–3160.
  * [57] K. Li, Y. Mu, K. Li, G. Min, Exchanged crossed cube: a novel interconnection network for parallel computation, IEEE Transactions on Parallel and Distributed Systems 24(11) (2013) 2211–2219.
  * [58] S. Abraham, K. Padmanabhan, The twisted cube topology for multiprocessors: A study in network asymmetry, Journal of Parallel and Distributed Computing 13(1) (1991) 104–110.
  * [59] C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Information Sciences 113(1-2) (1999) 147–167.
  * [60] X. Yang, D.J. Evans, G.M. Megson, The locally twisted cubes, International Journal of Computer Mathematics 82(4) (2005) 401–413.
  * [61] T.K. Li, J.J. Tan, L.H. Hsu, T.Y. Sung, The shuffle-cubes and their generalization, Information Processing Letters 77(1) (2001) 35–41.
  * [62] W. Zhou, J. Fan, X. Jia, S. Zhang, The spined cube: a new hypercube variant with smaller diameter, Information Processing Letters 111(12) (2011) 561–567.
  * [63] P. Cull, S.M. Larson, The Mobius cubes, IEEE Transactions on Computers 44(5) (1995) 647–659.
  * [64] S.A. Choudum, V. Sunitha, Augmented cubes, Networks: An International Journal 40(2) (2002) 71–84.
  * [65] K. Ghose, K.R. Desai, Hierarchical cubic networks, IEEE Transactions on Parallel and Distributed Systems 6(4) (1995) 427–435.
  * [66] S.K. Yun, K.H. Park, Comments on hierarchical cubic networks, IEEE Transactions on Parallel and Distributed Systems 9(4) (1998) 410–414.
  * [67] D.R. Duh, G.H. Chen, J.F. Fang, Algorithms and properties of a new two-level network with folded hypercubes as basic modules, IEEE Transactions on Parallel and Distributed Systems 6(7) (1995) 714–723.
  * [68] A. Karci, B. Selçuk, A new hypercube variant: fractal cubic network graph, Engineering Science and Technology: An International Journal 18(1) (2015) 32–41.
  * [69] G.A. Barragán-Ramírez, A. Estrada-Moreno, Y. Ramírez-Cruz, J.A. Rodríguez-Velázquez, The local metric dimension of the lexicographic product of graphs, Bulletin of the Malaysian Mathematical Sciences Society 42(5) (2019) 2481–2496.
  * [70] M.S. Lin, Simple linear-time algorithms for counting independent sets in distance-hereditary graphs, Discrete Applied Mathematics 239 (2018) 144–153.
  * [71] S. Stephen, B. Rajan, C. Grigorious, A. William, Resolving-power dominating sets, Applied Mathematics and Computation 256 (2015) 778–785.