# MAGNETO-ACOUSTIC WAVE OSCILLATIONS IN SOLAR SPICULES

A. Ajabshirizadeh1,2,3, E. Tavabi1,4, S. Koutchmy4
1 Department of Theoretical Physics and Astrophysics, Tabriz University, 51664
Tabriz,
2 Research Institute for Astronomy and Astrophysics of Maragha,(RIAAM),
3 Research Institute for Applied Physics and Astronomy of Khadjeh Nassiraldin,
Iran,
4 Institut d’Astrophysique de Paris and UPMC, 98 Bis Boulevard Arago, F-75014
Paris, France.
(E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Some observations suggest that solar spicules show small amplitude and high
frequency oscillations of magneto-acoustic waves, which arise from
photospheric granular forcing. We apply the method of MHD seismology to
determine the period of kink waves. For this purposes, the oscillations of a
magnetic cylinder embedded in a field-free environment is investigated.
Finally, diagnostic diagrams displaying the oscillatory period in terms of
some equilibrium parameters are provided to allow a comparison between
theoretical results and those coming from observations.

PACS: 96.60.-j

Keywords: Spicules; Oscillations; Kink waves; Flux tube.

##  1 Introduction

Spicules are jet-like chromospheric structures and are usually seen all around
the limb of the Sun arising in different directions. The mechanism of spicule
formation and evolution is not well understood (for the propulsive mechanisms,
see reviews of Sterling 2000; Lorrain and Koutchmy 1996; Filippov et al.
2006). Spicules are relatively homogeneous in height along their life time of
approximately 5-15 min., i.e. they are short-lived and comparable to the
photospheric granules lifetime. They have typical up flow speeds of 20-50km/s,
spicules diameter at chromospheric layers are of the order of 200-500km. The
mean number of spicules per supergranule cells at height of km is
approximately 40 (Pataraya et al. 1990), which is covered about 1 percent of
the Sun’s surface and they are usually concentrated between supergranule
cells. Their temperatures and density are higher than those of the surrounding
environment,Parenti et al. (1999) estimated $n_{e}=10^{10}cm^{-3}$ and
$T_{e}\approx 2\times 10^{5}$ ${}^{\circ}K$ in giant spicules see Koutchmy and
Loucif, also named macrospicules, are observed over 20,000 km off-limb and
live 40 min. (Xia et al. 2005) in case of macro-spicules. This means that a
magnetic field of 10G or more is needed for the low-$\beta$ (ratio of the
thermal pressure to the magnetic pressure) conditions (Wilhelm 2000) in case
of macro-spicules. Spicule usually have oscillation behavior, the existence of
5 minutes oscillations in spicules have been firstly reported by
Kulizhanishvili and Nikolsky (1978) and others including spectroscopically
resolved observations. Recently image sequences were studied by De Pontieu et
al. 2003, 2004; Xia et al. (2005); Ajabshirizadeh et al. (2007). These
oscillations seem to be related to p-modes, but it is evident that if spicules
are driven by p-modes, crucial details about their formation are still
missing. Clearly, not all spicular flows are periodic, whereas most
photospheric oscillations are. In addition, the horizontal scale for amplitude
coherence of p-modes ($\approx 8000km$) is well beyond the width of fibrils
(De Pontieu et al. 2003). On the-other-hand, oscillations in spicule with
shorter period have been reported by Nikolsky and Platova (1971). They found
that spicules oscillate along the limb with a characteristic period of about 1
min. If spicules are formed in thin magnetic flux tubes, then the periodic
displacement of the axis observed by them was probably due to the propagation
of kink waves. More recently, Kukhianidze et al. 2006 have reported the
observational signature of propagating kink waves in spicules. The period of
waves was estimated to be 35-70s for a spicule with height 3500 km which may
carry photospheric mechanical energy into the corona. The cutoff period of
kink waves due to stratification in the hydrostatic photosphere is $\approx
660s$ so the expected period of kink waves is well below the cutoff value
(Singh and Dwivedi, 2007). The wavelength was found to be $\approx 1000km$ at
the photosphere level which indicated a granular origin of the waves. Magnetic
flux tubes support transverse kink waves that can be generated in photospheric
magnetic flux tubes through buffeting action of granular motions (Roberts,
1979; Hasan and Kalkofen, 1999) although the extrapolation of the network
magnetic field toward the corona is still a matter of discussion. In this
study, the effect of gravitational stratification has been ignored, Singh and
Dwivedi 2007 have considered kink mode periods with this effect and confirms
with our results.

##  2 Basic MHD Equations

###  2.1 The Dispersion Relations

Magneto-hydrodynamics (MHD) is one of the key tools to understand the
hydrodynamics of magnetized plasmas. It concerns virtually all phenomena
observed in the solar atmosphere: coronal loops, filaments, spicules, etc.
Thanks to high spatial resolution, image processing, and time cadence
capabilities of the SoHO and TRACE spacecraft, oscillating loops (and
spicules) and propagating waves have been identified and localized in the
transition region (TR) and chromosphere, and studied in detail since 1996.
Using seeing free observations, they evidently complement what has been
studied for a long time at ground-based, including spectroscopic analysis.
These discoveries established a new discipline has become know as solar
atmosphere seismology. Many astrophysical plasmas are characterized by a set
of equations that is called ideal MHD equations and includes the continuity,
the momentum, Maxwell’s equations and Ohm’s law. To understand the various
oscillations and waves we observe in the spicule plasma we have to find wave
solutions of the MHD equations (Roberts, 1981), the existence of wave
solutions is generally derived by introducing a small perturbation of physical
parameters (such as: density, velocity and magnetic field) of the plasma, and
to derive dispersion relations $\omega(k)$, which tell us either the group
velocity or phase speed of wave. The cylindrical flux tube appearance of many
magnetic structures in the low-$\beta$ plasmas of the magnetosphere and more
specifically the solar chromosphere and corona encourages an investigation of
propagation in cylindrical geometries. Edwin and Roberts (1983) consider a
uniform cylinder of magnetic field $B_{0}\widehat{z}$ confined to a region of
radius 2b, surrounded by a uniform magnetic field $B_{e}\widehat{z}$, the gas
pressure and density within the cylinder are $P_{0},\rho_{0}$ , outside
$P_{e},\rho_{e}$ respectively (see figure 1). The Fourier form of the velocity
disturbance $v_{1}$ in cylindrical coordinate is:

$\begin{array}[]{c}v_{1}=v_{1}(r)\exp[i(wt+n\theta-k_{z}z)],\end{array}$ (2.1)

where n is integer (n=0, 1, 2 ) which describes the azimuthally behavior of
the oscillating tube i.e. the cylindrically symmetric (sausage or pulsation
mode given by n=0, the asymmetric (kink or taut-wire) mode given by n=1 and
higher mode with n=2, 3 are called the fluting or interchange modes. The
general governing equation is (Edwin and Roberts 1983):

$\begin{array}[]{c}\frac{d}{dr}[\frac{\rho_{\alpha}(r)(k^{2}_{z}V^{2}_{A\alpha}-\omega^{2})}{(m^{2}_{\alpha}+\frac{n^{2}}{r^{2}})}\frac{1}{r}\frac{d}{dr}(rv_{1})]-{\rho_{\alpha}(r)(k^{2}_{z}V^{2}_{A\alpha}-\omega^{2})}v_{1}=0,\end{array}$
(2.2)

where $m_{0}$ and $m_{e}$ are defined by $m_{\alpha}$ (with $\alpha=0$ or
$\alpha=e$ inside or outside of the tube respectively):

$\begin{array}[]{c}m_{\alpha}=\frac{(k^{2}_{z}C^{2}_{\alpha}-\omega^{2})(k^{2}_{z}V^{2}_{A\alpha}-\omega^{2})}{(C^{2}_{\alpha}+V^{2}_{A\alpha})(k^{2}_{z}C^{2}_{T\alpha}-\omega^{2})}\end{array}$
(2.3)

where $C_{\alpha}=(\frac{\gamma P_{\alpha}}{\rho_{\alpha}})^{\frac{1}{2}}$ and
$V_{A\alpha}=\frac{B_{\alpha}}{(\mu\rho_{\alpha})^{\frac{1}{2}}}$ are the
sound and Alfv n speed inside (or outside) the cylinder, and $C_{T\alpha}$ is
defined as:

$\begin{array}[]{c}C_{T\alpha}=\frac{C_{\alpha}V_{A\alpha}}{\sqrt{(C^{2}_{\alpha}+v^{2}_{A\alpha})}}\end{array}$
(2.4)

( $\gamma$ is the ratio of specific heats.) The external and internal
solutions of MHD equations need to be matched at the boundary by the
continuity of pressure and the perpendicular component of velocity. After some
algebra one gets the dispersion relation for magneto-acoustic waves in a
cylindrical magnetic flux tube is found to be (Edwin and Roberts 1983; D az et
al. 2002):

$\begin{array}[]{c}\rho_{e}(\omega^{2}-k^{2}_{z}v^{2}_{Ae})m_{0}\frac{I^{\prime}_{n}(m_{0}b)}{I_{n}(m_{0}b)}+\rho_{0}(\omega^{2}-k^{2}_{z}v^{2}_{A0})m_{e}\frac{K^{\prime}_{n}(m_{e}b)}{K_{n}(m_{e}b)}=0,\end{array}$
(2.5)

where $I_{n}$ and $K_{n}$ are modified Bessel functions of order n, with
$I^{\prime}_{n}$ and $K^{\prime}_{n}$ being the derivatives with respect to
the argument x. This dispersion relation describes both surface (for
$m^{2}_{0}>0$ ) and body waves (for $m^{2}_{0}<0$ ).

###  2.2 Kink-mode period

Magneto-acoustic oscillations of kink mode have now been directly observed in
$H\alpha$ line using 53-cm coronagraph the Abastumani Astrophysical
Observatory at different heights above the photosphere (Kukhianidze et al.
2006). The ratio of the spicule width 2b to the spicule full length 2L is
$\frac{b}{L}=0.01-0.4$ , which is correspond to the dimensionless wave number
(kL).
Therefore, the observed kink-mode oscillations are in the long-wavelength
regime of $KL<<1$, where the phase speed of kink-mode is practically equal to
the kink speed (Spruit, 1981; Roberts et al. 1984) given by:

$\begin{array}[]{c}C_{k}=(\frac{\rho_{0}V^{2}_{A0}+\rho_{e}V^{2}_{Ae}}{\rho_{0}+\rho\rho_{e}})^{\frac{1}{2}},\end{array}$
(2.6)

In the low-$\beta$ plasma limit and for the field free environment, the
expression for the kink speed $C_{k}$ reduces to:

$\begin{array}[]{c}C_{k}\approx(\frac{2}{1+\frac{\rho_{e}}{\rho_{0}}})^{\frac{1}{2}}V_{A0},\end{array}$
(2.7)

If we denote the full spicule length l=2L, the wavelength of the fundamental
standing wave is the double spicule length (due to the forward and backward
propagation) i.e., $\lambda=2l$ and thus the wave number of the fundamental
mode (N=1) is $k_{z}=\frac{2\pi}{\lambda}=\frac{\pi}{l}$, while higher
harmonics (N=2, ) would have wave numbers $k_{z}=N(\frac{\pi}{l})$, then the
time period P of a kink-mode oscillation at the fundamental harmonic is:

$\begin{array}[]{c}P\approx\frac{2l}{C_{k}}=\frac{2l}{V_{A0}}(\frac{1+\frac{\rho_{e}}{\rho_{0}}}{2})^{\frac{1}{2}},\end{array}$
(2.8)

and for higher harmonics,

$\begin{array}[]{c}P\approx\frac{2l\sqrt{\mu}}{NB_{0}}(\frac{\rho_{0}+\rho_{e}}{2})^{\frac{1}{2}}.\end{array}$
(2.9)

##  3 Results

Such as has been pointed out before, the determinant coming from dispersion
equation (2-5) must be truncated by taking into account a finite number of
basis functions and we will use surface wave ($m^{2}_{0}>0$ in Eq. 2-5). As we
know, the kink wave is essentially non-dispersive and has a phase speed equal
to the kink speed, $v_{ph}=\frac{\omega}{k_{z}}\approx C_{k}\approx V_{A0}$ so
we introduce dimensionless frequency which is given by $\frac{\omega
L}{V_{A0}}$ for odd modes and the fundamental modes and their harmonics have a
cutoff frequency for odd modes equal $\pi$ (see Diaz et al. 2002).
In figure 2 the eigenfrequencies of the kink (n=1) odd modes
($\omega_{cutoff}=\pi$ ) have been plotted in term of the spicule half-
thickness. In this plot we can see that for small value of $\frac{b}{L}$ only
the fundamental and lower harmonics could appear and for large ratio of
$\frac{b}{L}$ the frequency of the fundamental mode is insensitive to the
spicule thickness. One of the essential results of these plots is that the
oscillatory frequency is quite insensitive to the exact value of the ratio
$\frac{b}{L}$ (the ratio of the spicule thickness to the half-length of
magnetic field lines inside the spicule), i.e., for a given length 2L,
spicules with different thickness oscillate with the same frequency, this is
perhaps an interesting result, and this subject have been firstly reported by
D az et al. 2002, for kink waves in the prominence fibrils. We continue by
choosing two typical values of the spicule full length, namely, l=3500 and
8500km, and then plot the frequency of odd modes below the cutoff as a
function of $(\frac{\rho_{e}}{\rho_{0}})$ (see figure 3-a, b).
In these plots, we keep $\frac{b}{L}=0.05,0.03$ constant for two lengths of
spicule respectively. For all harmonic modes the frequency is seen to slowly
decrease with increasing $(\frac{\rho_{e}}{\rho_{0}})$ and more oscillatory
modes are present for large values of this quantity. To compute the
corresponding periods for each harmonic modes, we used l=3500 and 8500km,
$B_{0}=30G$ and $\rho_{0}=3\times 10^{-10}kgm^{-3}$. The periods obtained from
these quantities and using Eq. (2-9) is labeled P and correspond to the right
vertical axes in figure 3 (see also Table 1).
The period, P, (figure 3- a, b right vertical axes and table 1) for different
values of $\frac{b}{L}$ , L calculated from the expression of Eq. (2-5) and
(2-9), show that the fundamental and first harmonic have typical periods of
$\approx 30-80s$. These periods which are obtained from a simple MHD approach
are found in the more recently reported observational results (Kukhianidze et
al. 2006). where oscillation periods in the range of kink wave of $\approx
35-70s$ are found for spicules with height 3500-8700 km. From the theoretical
point of view, we therefore expect kink wave periods in the range of $\approx
80-120s$ for length of $\approx 10,000-14,000km$ which could be found from
future observations of kink wave inside spicules (Hinode observations).
_Acknowledgments._ The authors are most grateful to A. J. D az for useful
discussions and critical reading of the manuscript.

## References

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Figure Captions

Figure-1: Fig.1- Sketch of the equilibrium configuration used in this study.
The density and magnetic field inside the spicule are $B_{e},\rho_{e}$, and in
chromosphere environment are $B_{0},\rho_{0}$. The magnetic field is uniform
and along z-axis..

Figure-2: Fig. 2. Variation of the frequency with the spicule half-thickness
for Kink-Modes for the set of value 2L=8500km, and for the density ratio
$\frac{\rho_{e}}{\rho_{0}}$=0.03.

Figure-3: Fig. 3- a, b. Frequency of the kink odd modes
vs.$\frac{\rho_{e}}{\rho_{0}}$ for two full length of sipules and
$\frac{b}{L}=0.05,0.03$ for the parameters . The right-hand axis provides the
period P after estimation that magnetic field strength, the spicule density,
and full-length of spicules are $\rho_{0}=3\times 10^{-10}kgm^{-3}$
(corresponding to a $2\times 10^{11}$ number density), $B_{0}=30G$, and
2L=3500, 8500km, respectively.

Figure-4: TABLE 1


# A Comparative Analysis of Student Performance Predictions in Online Courses
using Heterogeneous Knowledge Graphs

Thomas Trask

Georgia Institute of Technology<EMAIL_ADDRESS>Dr. Nicholas Lytle

Georgia Institute of Technology<EMAIL_ADDRESS>Michael Boyle

Georgia Institute of Technology<EMAIL_ADDRESS>Dr. David Joyner

Georgia Institute of Technology<EMAIL_ADDRESS>Dr. Ahmed Mubarak

IBB University, Yemen<EMAIL_ADDRESS>
###### Abstract

As online courses become the norm in the higher-education landscape,
investigations into student performance between students who take online vs
on-campus versions of classes become necessary. While attention has been given
to looking at differences in learning outcomes through comparisons of
students’ end performance, less attention has been given in comparing
students’ engagement patterns between different modalities. In this study, we
analyze a heterogeneous knowledge graph consisting of students, course videos,
formative assessments and their interactions to predict student performance
via a Graph Convolutional Network (GCN). Using students’ performance on the
assessments, we attempt to determine a useful model for identifying at-risk
students. We then compare the models generated between 5 on-campus and 2
fully-online MOOC-style instances of the same course. The model developed
achieved a 70-90% accuracy of predicting whether a student would pass a
particular problem set based on content consumed, course instance, and
modality.

###### keywords:

Clickstream, knowledge graphs, online courses

##  1 Introduction

The COVID-19 pandemic accelerated the adoption of online education. In
addition to MOOCs and other online-only courses, traditional "on campus"
classes made greater use of online course delivery, live or asynchronous
discussion, and assessment. However, unresolved concerns about the
effectiveness of online courses remain, in particular issues with MOOC-style
courses such as low completion rate or low rates of learner engagement [19,
10]. In response, there is an increased use of Learning Analytics to analyze
the large volume of data generated by online learning platforms to develop
metrics and models to improve learning outcomes. Among other applications,
Learning Analytics has been used to predict learner performance [7], to
identify at-risk students in need of early intervention[4, 5], and to drive
improvements in the design of online courses [12, 15].

In this paper we apply Learning Analytics to Georgia Tech’s GTX1301:
Introduction to Python course, which offers both "on campus" and "online"
versions of the course. In both variants, the same lectures and assessments
and other course content are delivered on the EdX platform. However, the
learner populations and contexts between the two variants differ, most
importantly, the on-campus course is taken by Georgia Tech’s undergraduates
for a letter grade on a traditional semester calendar, while the online
variant is open to any learner without a fixed start or end date. This novel
structure allows for opportunities to understand the strengths and limitations
of online course delivery and contrast its outcomes with those of a more
traditional on-campus learning experience. Our investigation uses a
clickstream dataset provided by the EdX platform, covering five on-campus and
two online MOOC-style instances of the course, offered in 2021 and 2022. This
interaction data, along with information about the overall course design, was
used to construct a Heterogeneous Knowledge Graph (HKG), which serves as input
to a Graph Convolutional Network (GCN). This GCN model was then used to
predict a learner’s likelihood of achieving a passing grade on a given problem
set based on the content the learner has previously consumed. A secondary
focus of this work was to understand how the models differ between the two
course modalities.

Table 1: Course Structure Subcourse | Videos |  Ungraded exercises |  Coding exercises |  Graded problems
---|---|---|---|---
Fundamentals | 160 | 56 | 54 | 67
Control structures | 122 | 85 | 77 | 85
Data structures | 117 | 58 | 44 | 111
Objects & Algorithms | 43 | 17 | 60 | 32
Total | 442 | 216 | 235 | 295

##  2 Literature Review

###  2.1 In-person vs. Online Classes

While there had been dedicated work in evaluating and designing online
educational opportunities in higher education for decades [6], the Covid-19
pandemic and the rapid switch to remote and then hybrid learning environments
accelerated research interest in this space. It became imperative to
understand the differences in educational outcomes for students who
participate in different modalities of the same course to identify if efforts
were needed to close ‘educational gaps’ caused by the pandemic. Fortunately,
the switch to remote settings offered instructors an opportune natural
experiment to conduct research on these new environments. Research comparing
course structures of in-person and online has come to several conclusions.
First, each modality brings with it a set of affordances that students on the
whole respond positively to. Students in online courses enjoy the freedom of
having material available on the onset and can better self-regulate their
learning by choosing topics, activities, and assessments at their own pace.
Affordances of in-person classes tend to center around the social dimensions
of courses, especially the face to face and spontaneous interactions between
students and instructors as well as among students. Recent research in the
structure of online courses has focused on how to replicate these types of
interaction for online courses in order to create a better sense of learning
community [18, 11] . Traditionally, online courses referred to massively
online open courses (MOOCs) where anyone could participate and completion was
historically low. However, new online course programs that structure their
offerings around the semester system and have a steeper ‘buy-in’ (i.e.
enrollment in a degree program, more rigid structure for course activities)
has allowed course completion and student achievement to more closely match
those found in a traditional degree program [8, 14]. These successes point to
a trend of blending the affordances of both modalities into offerings that
give students the best of all worlds.

###  2.2 Clickstream Data and Neural Networks

Educational clickstream data has a number of uses in the analysis of learning
behavior, from seeing how and when students access or complete resources to
redesigning courses to better account for how students engage with material
[3]. Past studies have applied a variety of models to clickstream data to
predict at-risk learners in self-paced online courses. Al-azazi and Ghurab[2]
used a "day-wise" model with an Artificial Neural Network (ANN) with Long
Short Term Memory and predicted student performance with 70% accuracy in the
third month of the course. Rohani[16] used an ANN to predict student
performance in a MOOC of 3,000 students, achieving an AUC of 81% with data
from the first week of the course. The input to this model was a transition
probability matrix rather than an HKG.

Figure 1: the relationship of users, pages, videos, and assessments.

Representing the learning environments of online courses as graphs has a
number of affordances. Graphs are useful in modeling a student’s progression
in engaging in learning material at the level of both an individual problem or
an entire course [17, 13]. Graph structures as a useful model for online
interaction is not unique to educational settings and this usefulness of
modeling online structures as a set of nodes and edges has led to the
proliferation of graph neural networks. Graph Convolutional Neural Networks
have recently emerged as a means of classifying in semi-supervised
environments and have been shown to scale better in a wide variety of
structured graphs than similar RNN or CNN models alone[9]. In prior work [13]
a Graph Convolutional Network was developed as a semi-supervised
classification task to classify students’ engagement in various behavioral
patterns. This work first developed a label function to label datasets instead
of manual labeling, in which input and output data are labeled for
classification to provide a learning foundation for future data processing.
Accordingly, four behavioral patterns were hypothesized, namely (“High-
engagement”, “Normal-engagement”, “At-risk”, and “Potential-At-risk”) based on
students’ engagement with course videos and their performance on the
assessments/quizzes conducted after. A heterogeneous knowledge graph
representing learners was then built using course videos as entities, and
capturing semantic relationships among students according to shared knowledge
concepts in videos. The model intrinsically works for heterogeneous knowledge
graphs as a semi-supervised node classification task. It was evaluated on a
real-world dataset across multiple settings to achieve a better predictive
classification model. While this work was successful in classifying students
in a real-world setting, this current context extends prior work in two ways.
First, our course context with both on-campus and on-line learners allows us
to compare student engagement behavior depending on modality of learning.
Second, this work includes assessment data, providing ground truth values in
student scores on various assessments.

##  3 Methodology

###  3.1 Course Context

Our data comes from several semesters of Georgia Tech’s GTX1301: Introduction
to Python course, which offers both "on campus" and "online" versions of the
course. The course matches a standard introduction to computer science course
covering topics such as basic program construction. The course is intended for
students with zero or near-zero programming experience. In both variants, the
same lectures and assessments are delivered on the EdX platform, which has
typically been used for online-only MOOC courses. Table 1 shows the number of
students who engaged and completed all course requirements for each semester
and modality. The course content and instructional staff was the same for all
offerings. Table 1 gives an overview of the course structure for the online
course instances. The online instances are divided into 4 submodules and users
could choose to take any of the four submodules. Each submodules is composed
of 3-7 chapters, delivering course content primarily through short videos and
texts. Formative assessments inclduded multiple-choice exercises, short answer
exercises, and interactive coding exercises which were checked but did not
contribute to the learner’s final grade. There were graded problem sets at the
end of each unit, as well as a graded final test (and ungraded practice tests)
at the end of each module. Online learners had to pay a fee to have access to
the graded problem sets and the graded final test.

The original dataset consists of 16M clickstream events from the 5 on-campus
course instances and 55M events from the 2 online course instances, each
representing a year’s worth of user interaction. This data was then filtered
to focus on user-initiated clickstream events, namely video interactions and
assessment submissions as these categories of event were more directly
relevant to understanding how users interacted with the course content.
Clickstream events originating from the edX mobile app were also removed, as
there was not a comparably straightforward way to link page content together
for those records. The data was then collated into the following datasets:

  1. 1.

Student feature matrix which includes the following student metadata: session
count, total counts for various browser interactions, first, last access date,
and student interaction level based on the calculation found in appendix B.
Each student was labeled with one of the following interaction levels: high
engagement, normal engagement, potential at-risk, or at-risk.

  2. 2.

Video/assessment feature matrices consisting of video duration.

  3. 3.

Course Content Edge Matrix: Each video and assessment link to a page. Videos
and assessments were grouped together by whether they shared the same [page]
property.

  4. 4.

Student->Content edge matrix: Each student-video edge weight was calculated
based on the final % of the video the student watched, normalized. Each
student-assessment edge represents a feature set containing the users first
grade, final grade, and number of attempts.

  5. 5.

Student->Page Edge Matrix: A student->page edge is a binary representation of
whether a student met the page success criteria . For pages containing
assessments, this represents whether the student achieved an average score of
>70% on all page assessments. For pages containing only videos, this
represents whether the user had watched >70% of the page’s videos.

Table 2: Course User/Interaction Totals Modality | Term | Users |  Browser Events | Sessions
---|---|---|---|---
On campus | Spring 2021 | 341 | 113,016 | 2,120
Summer 2021 | 94 | 27,849 | 276
Fall 2021 | 513 | 1,607,457 | 4,011
Spring 2022 | 371 | 1,179,669 | 1,386
Summer 2022 | 288 | 849,683 | 1,942
Total | 1,607 | 3,777,675 | 9,735
Online | 2021 | 25,670 | 11,833,984 | 79,664
2022 | 21,346 | 3,916,560 | 24,386
Total | 47,016 | 15,750,544 | 104,050
Figure 2: GCN Model Loss/AUC Results, On-campus vs Online

###  3.2 Data Limitations

As seen in Figure 1, course videos and assessments are linked together via
course “pages”. Although there is a ‘page’ field in each EdX log record, only
30% of course videos and assessments could be directly linked to course pages
via that property. Only assessments records had associated display name within
the logs; course videos did not. More research must be done to determine a
more robust method of linking course content.

###  3.3 Data Governance

As noted, the clicktream data used in this experiment was gathered from
multiple instances of the GTX1301 on the edX platform. As part of edx’s
platform, users agree to automated collection of the tracking data in the
experiment. Data was gathered automatically while users interacted with the
edx course site in accordance with the GDPR and openEdx data governance
guidelines.[1] Each data record included a user’s edx user ID, along with the
datetime of the record’s event. Other than two polls indicating a user’s
previous experience with course material, no demographic information was
included in the dataset. Data governance for the experience was managed via GT
IRB. During the experiment, the dataset was accessed by GT researchers through
GT private cloud and a secured, offline dataset instance.

###  3.4 Graph Convolution Model

This heterogeneous knowledge graph structure above was converted into a
Pytorch Geometric (PyG) graph convolutional network (GCN) for analysis. The
GCN uses a 3 stage classifier with 2 Sage Convolution layers and a rectified
linear unit (ReLU) activation function between them. Input node data was
processed using a standard embedding and linear transform combination.. The
dot product between the user and page nodes was then taken to derive edge-
level predictions of whether a student ’passed’ a particular page. The GCN was
trained using 64 hidden layers and 4 output layers. For training the model,
the GCN was split into an 80/10/10 subgraph training split based on users’
interaction with course content using the PyG Random Link Splitter, similar to
how standard subsampling splits a dataset along rows. Across 30 epochs, the
training data was split into subgraphs using the PyG link neighbor loader, and
trained using the PyTorch binary cross entropy with logits loss function with
the Adam optimizer to calculate total loss, in-situ AUC, and final AUC
validation. See Appendix A for more information regarding the GCN.

##  4 Results and Discussion

The GCN model achieved an average AUC score of 58-90% predicting whether a
student will pass a particular problem check. Fig 2 shows the model’s loss and
AUC for each individual on-campus course instances, both online course
instances, and an aggregate of each year’s on-campus courses, averaged across
5 iterations.

For on-campus course instances, the AUC the GCN model varied widely, both
comparatively between courses and between individual runs for courses. Fig. 2
shows average AUC scores of individual courses ranging from 50-80%. It shows
that the GCN predictor performs better when looking at each year’s data in
aggregate; the AUC score rises to 90% for 2021 and 82% for 2022. The lower AUC
for 2022 is likely a result of missing data for the Fall 2022 semester. This
is also evident when comparing the model results for the aggregate on-campus
and online programs. While the 2021 on-campus CGN model performs in lock-step
with the 2021 and 2022 online GCN models, the 2022 on-campus model lags. More
research is needed to unpack the variance in results for the on-campus course
instances as there is no direct correlation between a courses user count,
interaction count, session count, variance, or combinations thereof.

Figure 3: GCN Model AUC Variance, On-Campus vs Online

###  4.1 Repeatability and Transferability

It is important to note that there is significant variance within and between
GCN training runs when looking at any individual course instance, as see in
fig. 3. This is likely due to a floor effect when splitting the data for the
individual on-campus courses. As noted, graph training was done on an 80/10/10
split along individual user interactions. It is theorized that, as the course
user count lowers, course data may be more susceptible to
successful/unsuccessful students being grouped together, impacting results.
This is indicated in the relatively higher per-run variance in AUC scores for
the individual on-campus course instances when compared to the aggregated on-
campus courses and online course instances, where the user and interaction
counts are much higher. While the online classes generate an order of
magnitude more data, additional analysis needs to be conducted to determine
the cause and impact of this difference in variance. Even though the GCN model
successfully generated prediction models for each course instance, the
difference in the data’s shape caused the model for one class to be non-
transferable to other course instances. A translation layer must be developed
to normalize available course content between courses to solve this
transferability issue.

##  5 Conclusion and Limitations

This work extends previous clickstream GCN models to include student
assessment interaction but focused on one course offering. Further
investigations into different courses, topics, and degree paths are needed.
Our understanding of the demographics of the students who engaged in these
course offerings was quite limited. As the project continues, we plan on
developing more robust implementations of the Graph network to include these
demographics.

In future work, we plan to include more information from student engagement
into the model in order to extend predictions. This can include information
from participation in class forums like Piazza, better content-mapping between
course concepts and content, and more fine-grained video interaction patterns.
Going forward, a scalable method will need to be developed to infer
connections between different content for each course. A transposition layer
should then be created to link course content across courses and transfer one
course instance’s GCN model to other course instances. Finally, exploring user
interaction via temporal graphs should allow us to examine user interactions
on a per-session and per-interaction-group basis to determine whether key
interaction patterns for target user groups exist.

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##  Appendix A GCN Platform Metrics

The below system was used to aggregate the log data and train the GCN used in
this exercise:

  * •

CPU: Intel Core i5 12600 10-core 16-thread CPU

  * •

GPU: Nvidia Geforce RTX 3060ti GPU

  * •

Memory: 64GB DDR4 3200mts

  * •

Disk: 512GB NVME PCIE 3.0 SSD

  * •

OS: Windows 10 2023H2

  * •

Language: Python 3.11

  * •

MongoDB 7.0, hosted locally

  * •

Software Environment: Spyder IDE 5.4.3

  * •

GCN Training Hyperparameters:

    1. 1.

epoch count: 30

    2. 2.

learning rate: 0.001

    3. 3.

batch size: 64

    4. 4.

Link Neighbor Loader neighborhood: [4,2]*4

On the above system, data generation and GCN training times scaled semi-
linearly based on the size of the course’s content and its’ user count. Data
generation and training for [on-campus Spring 2021, 94 users] took roughly 20
seconds, whereas the Online, 20211, 21k users took 22 minutes. GCN
hyperparameters were chosen based on a combination of computational time
efficiency and accuracy. Modifying the learning rate and Link neighbor loader
neighborhood shape had the largest impact on efficiency and accuracy. The
system used the CUDA framework for GCN training, resulting in a 0.5-3x
performance gain over GCN training using the CPU. System performance will vary
and can be improved by leveraging the Python multithreading libraries. Please
feel free to reach out to the researchers for copies of the GCN processor and
Data aggregator.

##  Appendix B Student Interaction Level Calculation

Figure 4: The calculation used to determine a student’s engagement level.[13]