# Small World Student Network at the University of Texas at Dallas in Times of
Social Distancing ††thanks: The authors wish to thank Christopher Simmons and
Jaynal Pervez of the Office of Information Technology at UT Dallas for their
help in using the high performance computing cluster for data analysis.

Kailash Subramanian  University of Texas at Dallas, Richardson, Texas 75080,
U.S.A.  Joshua M. Williams  University of Texas at Dallas, Richardson, Texas
75080, U.S.A.  Daniel C. DeAnda  University of Texas at Dallas, Richardson,
Texas 75080, U.S.A.  Aditya A. Agrawal  University of Texas at Dallas,
Richardson, Texas 75080, U.S.A.  Andrei Racila  University of Texas at Dallas,
Richardson, Texas 75080, U.S.A.  Aditi R. Prabhu  University of Texas at
Dallas, Richardson, Texas 75080, U.S.A.  Lawrence Redlinger  University of
Texas at Dallas, Richardson, Texas 75080, U.S.A.  Christopher Wendt
University of Texas at Dallas, Richardson, Texas 75080, U.S.A.  Ravi Prakash
University of Texas at Dallas, Richardson, Texas 75080, U.S.A.

###### Abstract

To limit the spread of the novel coronavirus on college campuses, a common
strategy for the Fall 2020 and Spring 2021 terms has been to offer instruction
weighted toward hybrid or fully online modalities. Colleges are now
considering whether and how to expand hybrid or fully in-person instruction
for future terms, and learn lessons from this experience for future use. Our
paper uses Fall 2019 enrollment data for a medium-sized public American
university to analyze whether some student groupings by class standing or
course level are more susceptible to the spread of infectious disease through
academic enrollment networks. Replicating Weeden and Cornwell [8], we find
that enrollment networks at the institution are “small worlds” characterized
by high clustering, short average path lengths, and multiple independent
connections. We also find that connectivity decreases as class standing
(graduate vs. undergraduate; senior vs. freshman) and course level increase;
as students move from generalized to specialized course loads, networks
cluster by major. Holding other factors constant, our analysis indicates that
policies focusing on in-person instruction for lower division students to
capture the freshman experience conflict with the greater risk of infectious
spread through a lower division network in the absence of additional steps to
minimize academic connectivity. There are academic and financial incentives
for emphasizing the freshman experience, including concerns about student
attrition from the first to second academic year and recouping costs of
infrastructure investments in dormitories that are disproportionately used by
freshmen. Possible solutions could include (i) restricting face-to-face or
hybrid instruction to courses in students’ academic majors, which would
disrupt larger networks into smaller ones and thus restrict the spread of
infection across majors, and (ii) take a “scalpel” approach to instruction
modes by moving online courses most likely to facilitate epidemic spread.

##  1 Introduction

The COVID-19 epidemic resulted in the suspension of face-to-face teaching on
university campuses in the United States in the spring of 2020. The subsequent
debate on when and how to return to face-to-face instruction has been informed
by several recent studies. Notably, Weeden and Cornwell [8] studied course
enrollment at Cornell University and concluded that a graph with edges between
students registered for the same course section forms a small world network,
with high clustering, short average path length, and multiple independent
paths between pairs of students. This appears to indicate that infections
could spread quickly among the student population unless several courses were
converted to online instruction, with face-to-face instruction limited to
small sections only.

Given that a vaccine is unlikely to produce community immunity before the
start of the fall semester of 2021 and fully online instruction is considered
suboptimal for student learning and institutional finances, the key question
for the remainder of Academic Year 2021 is to what extent institutions can
return to face-to-face or hybrid instruction. How many classes can resume
face-to-face or hybrid instruction, how large can they be, and which students
can populate them? Also, what are the lessons to be learned from this
experience so that we are better prepared for course planning and scheduling
if such epidemics are in the our future?

This paper describes an effort to replicate Weeden and Cornwell’s study in the
context of the University of Texas at Dallas (henceforth, “UT Dallas”), a
medium-sized public university. Cornell and UT Dallas have a similar
enrollment, though UT Dallas has a higher share of undergraduates ($71\%$ vs.
$63\%$ per the 2019-2020 Common Data Set), a higher ratio of incoming transfer
students to incoming freshmen ($1:2$ vs. $1:6$), and an undergraduate
population that is older on average ($13\%$ of UT Dallas undergraduates are
over $25$, vs. $1\%$ at Cornell University). UT Dallas has a smaller
fraternity/sorority population ($\sim 5\%$, vs. $\sim 25\%$) and a smaller
share of freshmen ($52\%$, vs. $99\%$) and undergraduates ($24\%$, vs. $52\%$)
living on campus. In addition, UT Dallas has a higher student/instructor ratio
($24:1$ vs. $9:1$). Whereas $56\%$ of Cornell’s undergraduate sections have
fewer than $20$ students, at UT Dallas the share is only $20\%$. Most
critically for our study, UT Dallas has a core curriculum requirement for all
undergraduate students, under which students must take some courses outside
their discipline (usually as freshmen or sophomores). Subsequent sections of
this paper discuss whether these differences result in any significant impact
on clustering, average path length, and bi-connectivity of students.

The Weeden and Cornwell study analyzed enrollment data for the entire
university, for undergraduate students only, and for the Liberal Arts College.
For the UT Dallas study, we perform a similar analysis for the entire
university, as well as separately for graduate and undergraduate students.
Additionally, due to the general core requirement (which for the most part
correspond to lower division courses, i.e., those at the freshmen and
sophomore levels), we separately analyze the data for lower division and upper
division undergraduate courses. We also investigate the relationships at
inter-school level.

While a $1000$-level course at UT Dallas is usually interpreted as a freshman-
level course, the reality is a little different. Even though most $1000$-level
courses primarily have freshmen enrolled in them, some also include
sophomores, juniors and seniors trying to fulfill general core requirements.
Likewise, freshmen with ample AP and other college-level credits may be taking
higher level courses. Therefore, we also analyze networks formed among
undergraduate students by considering the graphs formed using only (i) edges
between students taking courses at the same ($1000,~{}2000,3000,or~{}4000$)
level, and (ii) edges between students at the same class rank, namely
freshmen, sophomores, juniors, or seniors. Several interesting trends emerge
when juxtaposing these networks together.

As described in Section 5, we notice a pattern that calls into question the
efficacy of the strategy followed by some universities as they started classes
in Fall 2020. We observed that all students taking freshmen-level classes form
a highly connected and low-diameter network, with students across all majors
reachable from each other along short paths. By comparison, graduate students
- and to a slightly lesser extent those taking senior-level classes - tend to
exhibit a higher level of clustering by major or school. Students taking
graduate and senior level courses from the same school are densely clustered
together with several short paths among them, and there are significantly
fewer links connecting these intra-school or intra-major clusters. This leads
us to conclude that, rather than inviting freshmen to take in-person classes,
it would have been more prudent to limit in-person classes to graduate
students and upper-division students while offering only online/remote
instruction for lower-division courses.

##  2 Related Work

Milgram and Travers introduced the notion of small-world networks among people
[5, 6]. In [6], they showed that participants in Nebraska and Boston managed
to get letters delivered to targets in Massachusetts using relatively short
chains of acquaintances. They also refer to the presence of a small number of
“sociometric stars” through which a disproportionate number of acquaintance
chains passed. Their work spurred investigation of small-world phenomenon in a
number of disciplines, including sociology, medicine, and computer science.
Graph theory has been applied to this phenomenon by all of these disciplines.
People are modeled as nodes in a graph. An edge is drawn between two nodes if
a relation relevant to the problem being studied exists between the people
corresponding to those nodes.

Watts and Strogatz [7] proposed a framework for modeling networks ranging from
truly random networks with low diameters to uniform networks with high
diameters. They considered a uniform lattice with edges connecting neighboring
nodes and only a small number of edges connecting neighbors replaced by edges
between random pairs of nodes. As the fraction of these random edges
increases, the graph transitions from uniform to random. Watts and Strogatz
showed that, even with a small fraction of edges between distant nodes, the
diameter of the graph reduces significantly and the graph starts exhibiting
the small-world phenomenon. One effect of this phenomenon is that the diameter
of the graph becomes a logarithmic function of the number of vertices. This is
relevant to our study of student enrollment in courses. A two-mode graph of
course enrollment would have students and courses as nodes, with an edge
between student $x$ and course $y$ if student $x$ is enrolled in course $y$.
From such a graph, it is easy to obtain a single-mode graph with only students
as nodes and edges between pairs of students taking the same course. While
students of the same rank (freshmen, sophomore, etc.) and the same major may
take most of their courses with other students of the same rank and major,
there are always a small number of edges connecting random students. For
example, a computer science freshman may be in the same statistics class as a
junior majoring in sociology.

Amaral et al. [1] studied three classes of small-world networks, namely scale-
free networks, broad-scale networks, and single-scale networks. Scale-free
networks in particular are small-world networks in which new vertices will
preferentially connect to the more highly connected vertices within the
network, wherein the diameter will increase logarithmically with the number of
vertices. As such, the growth of a scale-free network closely mimics the
transmission of an infectious disease through a population, where a person is
more likely to be infected through a connection with a superspreader who has
contact with many people.

The “sociometric stars” observed by Travers and Milgram in their experiments
[6] point to an important concept in graphs: point centrality. Freeman [3, 4]
described this as the betweenness property of a vertex. A vertex has high
centrality if it falls on the shortest path between a large number of pairs of
vertices. In effect, this vertex can connect many otherwise distant pairs or
clusters of vertices. From a course enrollment perspective, consider a
mathematics major at UT Dallas. Though the number of students majoring in
mathematics is small, this particular mathematics major may be in an
algorithms class with computer science majors, a statistics course with
business administration majors, and a calculus course with physics and
mechanical engineering majors. This student would connect otherwise disparate
clusters of students, lie on the shortest path between several pairs of
students, and exhibit high centrality or betweenness. From an epidemiological
perspective, if such a student were to contract an infectious disease like
COVID-19, they could be a superspreader. Hence, in our analysis of enrollment
data, we attempt to identify the characteristics of students exhibiting high
centrality.

Caley et al. [2] quantified the effect of social distancing on the spread of
influenza in Sydney during the 1918-19 epidemic, as well as how successive
waves of infection could be explained by varying levels of social distancing
as people’s perception of risk fluctuated. One way to minimize the risk of
spreading an infectious disease through classroom attendance would be to move
courses and students with high centrality to online instruction. This would
have the effect of increasing network diameter and, therefore, enrollment-
driven social distance between distinct groups of students.

As stated earlier, the work that is most relevant to the present study is by
Weeden and Cornwell [8]. They examine the potential for epidemic spread of an
aerosolized virus (such as SARS-CoV-2) through in-person instruction at a
medium-sized private American university (Cornell). Modeling connectivity
between students co-enrolled in course sections, they find that the potential
for spread is high due to college campuses meeting the criteria for a “small
world” network [7], namely high clustering (nearly all students are connected)
and short average path lengths (the average student-pair has a low degree of
separation). Despite academic concentration creating disciplinary silos,
students are connected through hub courses such as general education
requirements or gateway courses shared across disciplines. Furthermore,
students are connected through multiple paths (shared sections or third
parties), so removing even the most centralized sections or students will not
significantly impact the diameter of the network. Weeden and Cornwell’s
analysis indicates that about half of all students were still connected in
four or fewer steps even when all sections of 30 or more were eliminated from
the network. Given full-time course loads, and that individuals can transmit
SARS-CoV-2 for days before recognizing the need to quarantine, four steps is a
good approximation of a week’s worth of potential exposure on a college
campus. Whereas Cornell’s small class sizes means that a 30-student threshold
forced the removal of approximately 16 percent of sections with an in-person
component, at UT Dallas (a medium-sized, public university), a 30-student
threshold affected $34\%$ of sections with an in-person component in Fall
2019, as well as $73\%$ of in-person student-section enrollments.

##  3 UT Dallas Enrollment Data

For our primary analysis we consider student enrollment data for the
University of Texas at Dallas, a medium-sized public university that is part
of the University of Texas System. Total enrollment in the Fall 2019 semester
was $29,449$ students registered in $5,235$ courses. As in the Weeden-Cornwell
work, we removed all online courses from consideration and those students who
registered only for such courses. This left us with $28,852$ students
registered in $4,994$ courses. Of these, $20,953$ were undergraduate students
and $7,899$ were graduate students. Freshmen, sophomore, junior and senior
level courses are numbered $1xxx$, $2xxx$, $3xxx$ and $4xxx$, respectively,
while graduate level courses have higher numbers: $5xxx$ through $8xxx$. The
second digit in each course number indicates the number of student contact
hours per week for that course. Each course number is preceded by an
alphabetic string representing the academic program that offers the course.
Based on this prefix it is possible to uniquely associate a course with the
school offering it.

The university is organized into eight schools, namely (i) Arts and Humanities
(AH), (ii) Arts, Technology and Emerging Communication (ATEC), (iii)
Behavioral and Brain Sciences (BBS), (iv) Economic, Political and Policy
Sciences (EPPS), (v) Engineering and Computer Science (ECS), (vi)
Interdisciplinary Studies (IS), (vii) Management (SOM), and (viii) Natural
Sciences and Mathematics (NSM). Of these schools, Engineering and Computer
Science and Management have the largest enrollment at both the undergraduate
and graduate levels.

###  3.1 Common Core

As our study is based on the one described in [8], it is important to state
some of the important differences between Cornell and UT Dallas curriculum.
While the exact number varies from one major to another, all undergraduate
students need to complete just over 120 semester credit hours of coursework to
graduate. Unlike Cornell, all undergraduate students at UT Dallas are required
to complete a general education core curriculum of forty-two semester credit
hours, in addition to the required courses in their major, and any elective
courses.

The credit requirements for the general education core curriculum include six
credit hours in Communication, three credit hours in Mathematics, six credit
hours in Life and Physical Sciences, three credit hours in Language,
Philosophy and Culture, three credit hours in Creative Arts, six credit hours
in American History, six credit hours in Government/Political Science, three
credit hours in Social and Behavioral Sciences, and six credit hours
classified as Component Area Options. In each of these nine categories there
are a large number of options. So, within the framework of core curriculum
there is still a fair amount of flexibility. However, some courses like
calculus, general chemistry, modern biology tend to have very high enrollment
due to a large number of students interested in majoring in engineering,
physical and life sciences related fields. These courses, for the most part,
tend to have lectures with very large number of students, and laboratory
sessions with comparatively smaller groups of students.

UT Dallas admits approximately $2,000$ transfer students every year who come
in primarily as sophomores and juniors, having completed most of their general
education core curriculum and some major-specific introductory courses at
regional community colleges. Also, some UT Dallas students choose to take
these general education core curriculum courses at community colleges and use
the credits towards their graduation requirement. Courses taken at community
colleges are not considered in our analysis. So, it is possible that based
solely on UT Dallas courses we may infer that two students do not have any
course in common when they may be taking a common course in a community
college.

###  3.2 Adjacency Matrix of Student Interaction

We represent student enrollment information as a two-mode $n\times m$ matrix,
$D$, where $n$ is the number of students, and $m$ is the number of sections
offered across all courses (high enrollment courses do have multiple sections,
with a subset of students taking the course getting enrolled in a given
section). Students enrolled in the same section share the same physical space,
while those in different sections do not. Each student is assigned a unique
index in the range $[1,n]$. Similarly, each section is assigned a unique index
in the range $[1,m]$. If student $i$ is enrolled in section $j$ then
$D[i,j]=1$, otherwise $0$. From this matrix, we compute another matrix
$A=D\times D^{T}$, where $D^{T}$ represents the transpose of matrix $D$.
Matrix $A$ is an $n\times n$ matrix where:

$\forall~{}1\leq i,j\leq n:A[i,j]=\sum_{k=1}^{m}D[i,k]*D^{T}[k,j]$ (1)

If students $i$ and $j$ are both enrolled in section $k$, then
$D[i,k]*D^{T}[k,j]$ is equal to $1$. $A[i,j]$, where $i\neq j$ represents the
number of sections in which students $i$ and $j$ are both enrolled. Hence, $A$
is a one-mode matrix that represents the strength of interaction between
students: higher the value of $A[i,j]$, the greater the number of sections in
which they are both enrolled and, therefore, the stronger the interaction
between them. If two students, $x$ and $y$ are enrolled in no common section
then $A[x,y]=0$.

We also employ two variants of the $A$ matrix. We refer to them as
$A_{binary}$ and $A_{weighted}$. Matrix $A_{binary}$ is a binary matrix such
that if $A[i,j]>0$ then $A_{binary}[i,j]=1$, and if $A[i,j]=0$ then
$A_{binary}[i,j]=0$. This is the matrix and its corresponding graph, described
later in Section 4, that we use to compute the distance between students,
average student network diameter, etc.

The matrix $A_{weighted}$ attempts to capture the strength of interaction
between pairs of students by not only considering the number of sections they
have in common, but the duration they could, in theory, spend in common
classrooms during a week. While a common one-credit-hour course section
between students $i$ and $j$ would contribute the same as a common three-
credit-hour course section in matrix $A$, the element $A_{weighted}[i,j]$
represents the total amount of time students $i$ and $j$ would spend in common
classrooms during a week, assuming there are no class absences.

##  4 Formal Definition of Terms and Metrics

Consider $A_{binary}$, the matrix of student interactions in courses. As
stated in Section 3.2, if $A_{binary}[i,j]=1$, then students $i$ and $j$ have
at least one course section in common. We also set $A_{binary}[i,i]=1$ because
doing so helps in some calculations described later. We use this matrix to
represent an undirected graph $G=(V,~{}E)$ with the following properties:

  * •

$V$ is the set of vertices of the graph and contains one vertex for each
student. The total number of vertices, $\mid V\mid$, is equal to $n$. Vertex
$v_{i}:1\leq i\leq n$ represents the $i^{th}$ student.

  * •

$E$ is the set of edges of the graph. There is an edge $(v_{i},v_{j})\in E$ if
an only if $A_{binary}[i,j]=1$.

Let $s(i,j)$ denote the length of the shortest path between vertices $v_{i}$
and $v_{j}$ in graph $G$. Note that there are $\binom{n}{2}=\frac{n(n-1)}{2}$
pairs of vertices in a graph of $n$ vertices. Therefore, the average path
length, $\ell_{G}$, of a connected, undirected graph $G$ of $n$ vertices
(which refers to the mean of the shortest distances between all pairs of
nodes) can be calculated as follows:

$\ell_{G}=\frac{2}{n(n-1)}\sum_{1\leq i<j\leq n}s(i,j)$ (2)

The diameter of a connected graph $G$ is:

$diameter_{G}=\max_{v_{i},~{}v_{j}\in V}s(i,j)$ (3)

Low values of $diameter_{G}$ and $\ell_{G}$ for the student enrollment graph
would be indicative of a small world network.

A vertex $v_{i}$ is said to be part of a triangle if it has two neighbors
$v_{j}$ and $v_{k}$ that are also neighbors of each other. In other words, the
edges $(v_{i},v_{j})$, $(v_{j},v_{k})$ and $(v_{k},v_{i})$ form a triangle. If
vertex $v_{i}$ has degree $d$, then $v_{i}$ can be part of up to
$\binom{d}{2}=\frac{d(d-1)}{2}$ triangles. Let $T(v_{i})$ represent the actual
number of triangles that $v_{i}$ belongs to. The unweighted clustering
coefficient, $c(v_{i})$, which indicates the cliquishness between vertex
$v_{i}$ and its neighbors ($i.e.$, the fraction of all possible triangles
involving $v_{i}$ that are actually present in the graph), can be calculated
as follows:

$c(v_{i})=\frac{2T(v_{i})}{d(d-1)}$ (4)

The mean value of $c(v_{i})$ over all vertices $v_{i}\in V$ is the average
local clustering coefficient, $C$, for the network:

$C_{G}=\frac{1}{n}\sum_{v_{i}\in V}c(v_{i})$ (5)

A set of three vertices that are reachable from each other is called a triad.
A triad could either be a triangle or a pair of edges between three vertices.
The global clustering coefficient, also referred to as transitivity, of a
graph $G$ is expressed as:

$T_{G}=\frac{3\times\textnormal{number of triangles}}{\textnormal{number of
triads}}$ (6)

The second power of $A_{binary}$ is represented as
$A_{binary}^{2}=A_{binary}\times A_{binary}$. If $A_{binary}^{2}[i,j]$ is non-
zero, the shortest path between vertices $v_{i}$ and $v_{j}$ is at most two.
Generalizing this notation, $A_{binary}^{k}=A_{binary}^{k-1}\times
A_{binary}$, and if $A_{binary}^{k}[i,j]$ is non-zero, the shortest path
between $v_{i}$ and $v_{j}$ is no more than $k$.111Note that setting
$A_{binary}[i,i]$ to $1$ for all $i$ ensures that once $A^{k}_{binary}[i,j]$
becomes non-zero, it stays so for higher powers of $A_{binary}$. The density
of an $n\times n$ adjacency matrix $A_{binary}$ raised to the power $k$ is the
fraction of non-zero elements in $A_{binary}^{k}$:

$\rho_{A_{binary}^{k}}=\frac{\textnormal{number of non-zero elements in
}A_{binary}^{k}}{n^{2}}$ (7)

where $0\leq\rho\leq 1$. This property measures the reachability of students
within path length $k$ and may not reach $1$ for finite values of $k$ if the
graph represented by $A$ is disconnected.

The network density $r$ is the ratio between the number of observed edges in
an undirected graph $G$ and the total number of possible edges. For a graph
with $m$ edges and $n$ vertices, it is expressed as:

$r_{G}=\frac{2m}{n(n-1)}$ (8)

As $r$ approaches 1, $G$ gets closer to a complete graph.

The betweenness centrality of a node $v$ in a weighted, connected graph $G$ is

$b_{v}=\sum_{s,t\in V}\frac{\sigma_{st}(v)}{\sigma_{st}}$ (9)

where $\sigma_{st}$ is the number of shortest paths between $s$ and $t$ and
$\sigma_{st}(v)$ is the number of those paths that also pass through $v$. By
this definition, if $v=s$ or $v=t$, then $\sigma_{st}(v)$ = 0, and if $s=t$,
then $\sigma_{st}=0$.

The betweenness centrality described above is a function of the number of
vertices in the graph. So, to compare the betweenness centrality of vertices
in graphs of different sizes, it is necessary to normalize the value as a
function of graph size. For a connected graph of $n$ vertices, the maximum
value of betweenness centrality of a vertex is
$\binom{n-1}{2}=\frac{(n-1)(n-2)}{2}$, which arises when that vertex is the
hub of a star network with $n-1$ neighbors. Hence, we report the normalized
betweenness centrality of a vertex as follows:

$b_{v,norm}=\frac{2\times b_{v}}{(n-1)(n-2)}$ (10)

A student who is represented by a vertex with high betweenness centrality
connects a large number of pairs of students. Such a student, if infected with
a contagious disease, could be a proverbial superspreader. Hence, in our
analysis, we not only determine the betweenness centrality of vertices, but
also try to identify the kinds of vertices that exhibit high betweenness
centrality. Course delivery-based strategies to minimize rapid spread of
infectious diseases should try to move the corresponding students to online
instruction to minimize their physical interaction with other students.

Thus far, we have considered an unweighted graph of students where all edges
have unit weight. Such a graph does not differentiate between a pair of
students who are enrolled in a single common course that meets for an hour
each week and another pair of students who are enrolled in three common
courses, each of which meet for three hours per week. Hence, we define a new
metric for weekly contact duration between students. Let $enrollment(x)$ be
the set of students enrolled in course $x$, and let $contact(x)$ be the number
of weekly class contact hours for course $x$. Then, the amount of time two
students spend together in classrooms each week (assuming no absences) can be
represented as:

$contact\\_duration(i,j)=\sum\\{contact(x):i,j\in enrollment(x)\\}$ (11)

Using this metric, we consider a weighted graph $G_{weighted}$ with the
corresponding adjacency matrix $A_{weighted}$ such that
$A_{weighted}[i,j]=\frac{1}{contact\\_duration(i,j)}$. The more time students
$i$ and $j$ spend together in class, the lower the weight of the edge between
them. We determine minimal-weight paths between pairs of vertices and the
betweenness centrality of vertices in this weighted graph. Our assumption is
that the betweenness centrality metric for this weighted graph will be a
better indicator of the potential for an infected student to spread a
contagion through classroom interaction than the corresponding metric for an
unweighted graph.

##  5 Analysis of Data for Fall 2019

The results of this study pertaining to the entire university are summarized
in Table 1. A total of $28,849$ students registered in Fall 2019 courses, but
the largest connected component of student-to-student network had only
$27,080$ students. In this largest connected component of all students at the
university, the average path length, $\ell_{G}$ (also referred to as the
average geodesic distance), is equal to $2.97$. This implies that, on an
average, two students are within a distance of three hops of each other. This
short distance between students is more pronounced if we limit our analysis to
undergraduate students for whom the average geodesic distance is a low $2.44$.
Such a short distance in a large graph is indicative of a small-world network.

###  5.1 Graduate versus undergraduate level courses

Graduate students constitute $27\%$ of all students. Like the university-wide
network, a majority ($79.1\%$) of the nodes in the graduate network belong to
the main component, where defining small world characteristics emerged. The
local clustering coefficient was much higher between graduate students
($0.63$) than the university wide network ($0.46$). In particular, we notice
that unlike the interconnectivity between nodes of several majors in the
university-wide network, the arrangement of nodes in the graduate student
subnetwork involves students having more connections with their intra-school
cohorts because they register in major-specific coursework. Figure 1c also
reveals the distinct clustering of students in ECS, EPPS, AH, and SOM.
Furthermore, NSM nodes cluster near the center, and many of these nodes act as
bridges between major clusters, such as SOM and ECS.

The average geodesic distance in the graduate network is approximately 4.1.
So, on average, connecting any two students in the graduate student network
takes about 2 more steps than doing so requires in the undergraduate network.
Although students may be more interconnected with other students of the same
major, the formation of separate clusters may lead to an increased number of
steps to connect students that may be in different majors, causing the average
shortest path length to increase.

Nodes in the graduate network have a significantly lower average degree
($94.5$) than those in the undergraduate network ($326.6$), indicating that,
on average, graduate students have fewer direct connections to others. We also
find that the average edge weight in the graduate network is greater ($11.03$
credit hours) than that in the undergraduate network ($9.50$ credit hours).
Taken together, these two findings explain that, although graduate students
may be connected to fewer people on average, they spend more time in classes
with each other compared to undergraduates.

The diameter of the largest connected component among graduate students, which
corresponds to the distance between two farthest nodes, is only $9$. When
limited to undergraduate students only, this number drops to $5$. So, more
than $99\%$ of undergraduate students can reach each other within five steps.

As can be seen in the table, the largest connected component of students
exhibits a high level of local ($0.46$) and global ($0.38$) clustering. Yet
another indicator of the small world network is that such a high level of
clustering is achieved with a relatively low level of network density, close
to $0.01$.

Metric | University | Graduates | Undergraduates
---|---|---|---
Nodes, full graph | 28,849 | 7,899 | 20,950
Edges, full graph | 3,714,254 | 298,316 | 3,394,312
Nodes, $n$ | 27,080 | 6,252 | 20,784
Edges, $m$ | 3,711,518 | 295,445 | 3,394,310
Average degree | 274.12 | 94.512 | 326.63
Percent nodes in largest comp. | 93.861 | 79.149 | 99.208
Average edge weight | 9.6231 | 11.025 | 9.4966
Average geodesic distance, $\ell_{G}$ | 2.9694 | 4.1021 | 2.4425
Diameter of network | 9 | 10 | 5
Unweighted local $C_{G}$ | 0.46487 | 0.62702 | 0.41971
Unweighted global $T_{G}$ | 0.38386 | 0.45109 | 0.38247
Network density, $r_{G}$ | 0.010122 | 0.015120 | 0.015716
Table 1: Metrics for the largest connected component of student-to-student
networks at the university-wide, graduate, and undergraduate levels.

###  5.2 Diving deeper: comparison between course levels

Next, let us consider the relationships between students enrolled in all
courses at specific levels (i.e., $1xxx$ through $7xxx$) as shown in Table 2.
As there are very few $8xxx$ level courses, and they typically correspond to
individual research credits taken by doctoral students, we do not report the
corresponding numbers. Among undergraduate students, a low average geodesic
distance between students is evident at all levels of courses ($1xxx$ through
$4xxx$). These are markedly lower than the average geodesic distances for
$5xxx$ and $6xxx$ level courses, which constitute the bulk of graduate level
courses with classroom instruction. Also, the diameter (the longest path
between any two nodes) for the $1xxx$ level courses is significantly lower
($5$) than it is at the $4xxx$ level courses ($8$), and the diameters at all
four undergraduate course levels are lower than the diameter for $5xxx$ and
$6xxx$ level courses ($11$).

| Course Catalog Prefix
---|---
Metric | 1xxx | 2xxx | 3xxx | 4xxx | 5xxx, 6xxx | 7xxx
Nodes, full graph | 9,149 | 12,454 | 13,285 | 8,454 | 7,346 | 738
Edges, full graph | 879,833 | 1,346,377 | 866,593 | 435,992 | 305,919 | 10,167
Nodes, $n$ | 9,145 | 12,413 | 13,183 | 7,999 | 6,354 | 178
Edges, $m$ | 879,823 | 1,346,377 | 866,410 | 435,956 | 303,180 | 5,586
Percent nodes in largest comp. | 99.956 | 99.671 | 99.232 | 94.618 | 86.496 | 24.119
Average edge weight | 7.5897 | 9.1007 | 10.175 | 10.436 | 10.983 | 6.3960
Average geodesic distance, $\ell_{G}$ | 2.5027 | 2.50434 | 2.98580 | 3.3771 | 4.1291 | 2.1117
Diameter of network | 5 | 5 | 6 | 8 | 11 | 6
Unweighted local $C_{G}$ | 0.72779 | 0.71452 | 0.66008 | 0.758890 | 0.64202 | 0.88821
Unweighted global $T_{G}$ | 0.54725 | 0.57823 | 0.52284 | 0.63947 | 0.45031 | 0.85617
Network density, $r_{G}$ | 0.021043 | 0.017477 | 0.0099714 | 0.013629 | 0.015021 | 0.354600
Table 2: Metrics for the largest connected component of student-to-student
networks at the various course levels.

Freshmen students are not limited to taking $1xxx$ level courses, sophomores
are not limited to taking $2xxx$ level courses, and so on. For example, a
senior may be enrolled in a $1xxx$ or $2xxx$ level course that satisfies a
general core requirement. So, we analyze course registration data for students
classified as freshmen, sophomores, juniors and seniors separately. Similar
trends, including a high level of clustering, low diameter, and low average
geodesic distance, are evident, as shown in Table 3.

Metric | Freshman | Sophomore | Junior | Senior
---|---|---|---|---
Nodes, full graph | 4,330 | 3,220 | 6,119 | 7,088
Edges, full graph | 663,636 | 203,254 | 350,073 | 426,513
Nodes, $n$ | 4,277 | 3,211 | 6,092 | 6,985
Edges, $m$ | 663,608 | 203,249 | 350,071 | 426,509
Average degree | 310.315 | 126.595 | 114.9281 | 122.1214
Percent nodes in largest comp. | 98.776 | 99.720 | 99.559 | 98.547
Average edge weight | 8.2339 | 10.373 | 10.226 | 11.414
Average geodesic distance, $\ell_{G}$ | 1.9951 | 2.3124 | 2.6136 | 2.7737
Diameter of network | 6 | 5 | 5 | 6
Unweighted local $C_{G}$ | 0.42119 | 0.44724 | 0.4261 | 0.4914
Unweighted global $T_{G}$ | 0.39850 | 0.46357 | 0.4252 | 0.4656
Network density, $r_{G}$ | 0.072571 | 0.039444 | 0.018868 | 0.017486
Table 3: Metrics for the largest connected component of student-to-student
networks for undergraduate freshmen, sophomores, juniors, and seniors.

Figure 1 shows the cumulative distribution of student pairs that are reachable
from each other within a given distance. Let us first consider the graph on
the left. As seen in the plot for all students (blue curve), more than $80\%$
of student pairs are within a distance of $4$ of each other. Freshmen students
are even more strongly interconnected (orange curve), with almost all students
pairs within a distance of $4$ of each other (more than $90\%$ of freshmen
students pairs are within a distance of $2$ of each other). Graduate students
are not as closely related (green curve), with less than $40\%$ of them within
a distance of $4$ of each other. Even when we consider a greater distance,
interconnectivity remains somewhat low; only $60\%$ of graduate student pairs
are within a distance of $6$ of each other. Similar trends are visible when we
analyze the data for different course levels (i.e., $1xxx$ through $7xxx$; see
graph on the right). The plots for the $1xxx$ and $2xxx$ courses are almost
identical and appear as one (orange curve). Especially noteworthy, as an
outlier, is the plot for $7xxx$-level courses, which indicates a high level of
fragmentation for advanced graduate-level courses.

Figure 1: Cumulative density of matrices for consecutive powers of
$A_{binary}$.

Weeden and Cornwell [8] investigate the impact of moving large enrollment
courses online on interconnection between students taking in-person classes.
They studied the impact of removing courses with enrollment over 30, 40, 50,
75 and 100 from the network on path lengths between students. We performed a
similar simulation. The results are shown in Figure 2. Instead of choosing
arbitrary thresholds, we relied on the quintile thresholds for course
enrollment. Sections with enrollment less than $25$ account for $20\%$ of
student enrollment in classes (first quintile). Sections with enrollment less
than $40$ account for $40\%$ of class enrollments (second quintile).
Similarly, the third and fourth quintile thresholds are sections with sizes of
$55$ and $85$, respectively.

Removing all sections with enrollment of $40$ or more, over $65\%$ of student
pairs are within a distance of $4$ and more than $80\%$ are within a distance
of $8$. When all sections with enrollment of $25$ or more are removed from the
network, almost $40\%$ of student pairs are still within a distance of $4$ of
each other and over $60\%$ are within a distance of $8$.

Figure 2: Cumulative density of matrices for consecutive powers of
$A_{binary}$, keeping courses by size.

###  5.3 Academic silos or strong interconnections

Let us now dive deeper and consider the connectivity between individual
students. Whereas Weeden and Cornwell [8] considered a dual-mode graph of
courses and students, we chose to focus on a single mode graph among students.
The student-to-student network is drawn using the force-directed Fruchterman-
Reingold algorithm. The more edge connections (shared courses) that two nodes
(students) have in common, the closer together they are drawn.

Nodes in Figures 3, 4, and 5 were colored according to the school or
concentration that their corresponding students were primarily registered
with: Management (SOM, green), Engineering & Computer Science (ECS, pink),
Natural Science & Mathematics (NSM, maroon), Behavior & Brain Sciences (BBS,
yellow), Arts/Technology/Emerging Communications (ATEC, dark yellow),
Economic/Political and Policy Sciences (EPPS, blue), Interdisciplinary Studies
(IS, dark green), Arts & Humanities (AH, cyan), Executive Management (EMGT,
teal), and unspecified (grey). Though Executive Management courses are offered
by the School of Management, students in these courses tend to form their own
tight knit community for the most part and interact very little with other
Management students. A very small number of students fall into the unspecified
category.

The size of nodes scales linearly relative to the value of the highest
betweenness centrality and students that share at least one course are
connected by a black edge. Edge thickness represents the number of course
hours shared by the student-pair. Although the positions of individual nodes
might not provide meaningful information, the relative positions of distinct
regions in the network convey the interdisciplinary relationship of node
clusters.

(a) University-wide network

(b) Undergraduate network

(c) Graduate network

Figure 3: University wide, undergraduate, and graduate networks. Larger nodes
have higher weighted betweenness centralities.

In Figure 3(a) we see that all students are closely connected, even though
students belonging to the two largest schools by enrollment (ECS and SOM) tend
to be clustered together. Looking at Figures 3(b) and 3(c), it can be inferred
that graduate students have a higher level of clustering than undergraduate
students, likely caused by increased registration in major-specific
coursework. We also find that interdisciplinary relationships are visible
through the relative distances of the clusters, especially between the AH,
NSM, and ECS clusters, and that the NSM and ECS nodes are influential bridges
between several clusters.

####  5.3.1 Lower division courses

Figure 4 shows the interconnections among students enrolled in lower-division
undergraduate courses, $i.e.$, $1xxx$ and $2xxx$ level courses. As in the
previous figure, larger sized nodes have higher weighted betweenness
centralities. The graphs are clumped edges often connect students from
different majors since many individuals share lower-level introductory
classes. SOM and ECS dominate the graphs because they are the two most popular
schools of study.

(a) 1000’s level courses

(b) 2000’s level courses

Figure 4: Lower division student-to-student networks partitioned by course
level.

####  5.3.2 Upper division courses

Figure 5 shows the connections between students taking upper division
undergraduate courses ($3xxx$ and $4xxx$), classroom-based graduate level
courses ($5xxx$ and $6xxx$), and graduate level seminar and independent study
courses ($7xxx$). Once again, larger nodes have higher weighted betweenness
centralities. Clusters become increasingly visible in higher-level courses and
they begin to rely more on bridge nodes to connect clusters. At the $7xxx$
level, clusters break down into components that mostly specialize in a
specific field of study; only about 24% of nodes are in the largest component.
As the $7xxx$ level courses tend to be populated primarily by doctoral and
master’s thesis students, the significant level of intra-program clustering
and disconnections between programs indicate that these courses could be the
first ones to be conducted in-person due to the low risk of spreading an
infectious disease, compared to undergraduate courses.

(a) 3000’s level courses

(b) 4000’s level courses

(c) 5000’s and 6000’s level courses

(d) 7000’s level courses (full network is shown)

Figure 5: Upper division student-to-student networks partitioned by course
level.

(a) Freshman level network

(b) Sophomore level network

(c) Junior level network

(d) Senior level network

Figure 6: Student-to-student networks partitioned by student rank (freshmen,
sophomores, juniors, and seniors). Larger nodes have higher weighted
betweenness centralities.

###  5.4 Courses contributing to high betweenness centrality

As shown in Figure 2, even if all sections with enrollment of $25$ or more are
removed from the network, a significant number of students are still able to
reach each other through short paths. Hence, we investigated another
possibility of disrupting the small world network between students:
eliminating courses that contribute to high betweenness centrality.

As betweenness centrality in a student-to-student network is a property of
individual nodes, we investigated this idea as follows. For a given set of
students, first we identified the one hundred students with the highest
betweenness centrality. We refer to them as the pivotal students. Then, we
identified all sections in which these pivotal students were enrolled. This
simulation was conducted on the unweighted network of students as well as the
weighted network described in Section 4. Our conjecture is that removing
classes with high pivotal student enrollment from the student-to-student
network ($i.e.$, moving these courses online) could disrupt the small world
network, increase average path length between student pairs, and also increase
the diameter of the network.

(a) University-wide network

(b) Undergraduate students only

(c) Graduate students only

Figure 7: The classes most frequently taken by pivotal students (nodes with
high betweenness centralities) in student-to-student networks.

Figure 7 lists sections in which pivotal students are enrolled. Figure 7(a)
shows such courses for the entire student population, while Figure 7(b) and
7(c) are limited to undergraduate and graduate students, respectively. Courses
shown in Figure 7 are not surprising. They either correspond to upper division
undergradaute or graduate courses (e.g., ENTP4340, CS 4485, CS 6375) from ECS
and SOM, the two schools with largest enrollment; or they correspond to lower
division courses that satisfy the general core requirement (e.g., GOVT 2305,
GOVT 2306). Eight out of the one hundred students with the highest betweenness
centralities are enrolled in ENTP 4340.

An interesting situation emerges when we limit the analysis to undergraduate
(Figure 7(b)) or graduate (Figure 7(c)) students. While eight students taking
ENTP 4340 were among the one hundred nodes with highest betweenness centrality
for the university-wide network, only four of them are among the top one
hundred when the analysis was limited to undergraduate students. The other
four most likely had high betweenness centralities because they were also
taking some graduate level courses and were on the shortest paths between
several graduate and undergraduate students. Those links disappeared when the
ananlysis was limited to undergraduate students. Similarly, only three of the
one hundred students with the highest betweenness centralities were in GOVT
2305. However, when the analysis was limited to undergraduate students, that
number jumped to thirteen.

(a) 1000’s level courses

(b) Freshmen level students only

(c) 2000’s level courses

(d) Sophomore level students only

Figure 8: The classes most frequently taken by pivotal students (nodes with
high betweenness centralities) in lower division student-to-student networks.

Figure 8 shows the courses taken by the one hundred pivotal students among
freshmen-rank students, sophomore-rank students, $1xxx$-level courses, and
$2xxx$-level courses. As expected, students with high betweenness centralities
for $1xxx$ level courses (Figure 8(a)) were taking courses corresponding to
the core curriculum such as HUMA 1301, CHEM 1312, PHYS 1301, etc. When we turn
our focus from $1xxx$-level courses to freshmen rank students, we find that
some freshmen with high betweenness centralities were enrolled in $2xxx$
courses that most likely satisfy their general core requirements, such as GOVT
2305, GOVT 2306, ECON 2301.

(a) 3000’s level courses

(b) Junior level students only

(c) 4000’s level courses

(d) Senior level students only

(e) 5000’s and 6000’s level courses

(f) Graduate students only

Figure 9: The classes most frequently taken by pivotal students (nodes with
high betweenness centralities) in upper division student-to-student networks.

Figure 9 shows the courses taken by the one hundred pivotal students when we
limit the analysis to juniors, seniors, graduate students, students taking
$3xxx$-level courses, students taking $4xxx$-level courses, and students
taking $5xxx$ and $6xxx$ level courses. Following a course like CS 4485 across
the university wide network, undergraduate student network, and senior-rank
student network, an interesting trend emerges. Only six of the one hundred
pivotal students in the university wide network were enrolled in CS 4485. This
number increased to seven when the network was limited to undergraduate
students only, and rose to eight for the network of senior-rank students. Some
of these students with the highest betweenness centrality in smaller
populations cease to occupy positions of similar distinction when the scope of
the network is enlarged because these students are closely linked only to
students within their own small group. Persuading these students to move to
online learning may not have much of an impact on achieving the desired
outcomes of reducing clustering and increasing the diameter of the network at
the university level. On the other hand, doing the same for students who
continue to be part of the pivotal student group as the scope of the network
is enlarged may achieve the desired result.

###  5.5 Scalpel approach: selective removal of minimal subset of courses

As stated above, only when a large number of courses with high enrollment are
moved online does the graph based on in-person classes show a significant
increase in network diameter and average path length. If such a solution is
not available to decision-makers, we explore an alternative approach that
requires moving a much smaller subset of courses online. We refer to this less
intrusive approach as the scalpel approach. In this approach, we exploit the
betweenness centrality information about pivotal students, as described in
Section 5.4.

Pivotal students lie on a disproportionately large number of short paths
between pairs of students. We studied the impact of reducing the number of
face-to-face classes in which such students participate, which would reduce
the number of shortest paths passing through these students. Specifically, we
identified all sections corresponding to the twenty-five courses with the
highest enrollment of pivotal students for the unweighted network in Figure
7(a). We simulated moving all these sections to online instruction.
Consequently, edges between students contributed by these classes were removed
from the university-wide enrollment graph. The number of sections declined
from $4,956$ to $4,888$, a very modest reduction of $68$ sections (just under
$1.4\%$ of all sections). Yet, the change in graph connectivity is noteworthy,
as shown in Table 4.

Metric | University | Scalpel Approach
---|---|---
Nodes | 28,849 | 28,793
Edges, full graph | 3,714,254 | 3,302,030
Nodes, $n$ | 27,080 | 26,954
Edges, $m$ | 3,711,518 | 3,297,916
Average degree | 274.12 | 244.70698
Percent nodes in largest comp. | 93.861 | 93.613
Average edge weight | 9.6231 | 9.5522
Average geodesic distance, $\ell_{G}$ | 2.9694 | 3.0547
Diameter of network | 9 | 9
Node pairs within distance $4$ | 84.219% | 82.238%
Unweighted local $C_{G}$ | 0.46487 | 0.47189
Unweighted global $T_{G}$ | 0.38386 | 0.39536
Network density, $r_{G}$ | 0.010122 | 0.009079
Table 4: Metrics for the largest connected component of student-to-student
networks with and without the scalpel approach of removing courses.

The scalpel approach increases average path length by approximately $0.1$,
reduces the average degree of vertices significantly (by about $30$), and
increases the local and global clustering coefficients while reducing network
density, making it more of a small-world network. These changes in
connectivity and path length are significant considering the small number of
sections that are moved online.

By far the most remarkable impact of the scalpel approach is the reduction in
the percentage of student pairs that are within a distance of $4$ of each
other. As shown in Figure 2, removing the top $20\%$ of courses by enrollment
decreased this from $84.219\%$ to $81.477\%$, a reduction of $2.74\%$. In
comparison, removing barely $1.4\%$ of classes using the scalpel approach
reduced this metric by $1.98\%$ (from $84.219\%$ to $82.238\%$)

Hence, the scalpel approach appears to be promising. A two-pass scalpel
approach could be worth considering. In the first pass, courses would be moved
online based on the approach described above. In the next pass, the pivotal
students for the graduate-students-only and undergraduate-students-only graphs
could be identified and a limited number of courses with high number of
pivotal students from corresponding populations could be moved online. This
would further disrupt the small-world networks among these sub-populations.

##  6 Discussion and Recommendations

A vaccine is unlikely to produce community immunity before the start of the
fall semester of 2021. Fully online instruction is considered suboptimal for
student learning and institutional finances. Hence, the key question for the
remainder of Academic Year 2021, and for future occurrences of such a pandemic
is this: to what extent can institutions return to face-to-face or hybrid
instruction? How many classes can resume face-to-face or hybrid instruction,
how large can they be, and which students can populate them?

We replicate Weeden and Cornwell’s study [8] by finding that students at our
medium-sized public American university represent a “small-world” network
susceptible to infections spread. Given the average student course load and
the window for individuals with COVID-19 to move from infectious to
symptomatic, a four-step threshold representing a week’s worth of courses is a
good benchmark for identifying the potential for epidemic spread.

By moving as much as $60\%$ of enrollment by course section (courses with $40$
or more students) online, we were able to reduce the largest network below
$75\%$ connectivity (to $64.1\%$) within four steps. In addition, by moving
$80\%$ of enrollment by course section (courses with $25$ or more students)
online, we were able to reduce the largest network below $50\%$ connectivity
(to $33.2\%$) within four steps, and it took more than eighteen steps for
$71\%$ of students to reach each other.

Building on Weeden and Cornwell, we find an inverse relationship between
network connectivity and class standing (freshmen to senior), as well as a
similar relationship between network connectivity and course level. Seniors
are significantly less connected than freshmen are and doctoral students are
significantly less connected than master’s students. Without removing any
class sections from the sample, $97.5\%$ of freshmen were within four steps of
one another, while $97\%$ of seniors were within four steps of one another.
However, freshmen students are much more closely linked to each other than
seniors: only about $24\%$ of senior pairs are within two steps of each other,
whereas $91\%$ of freshmen pairs are within the same distance. In contrast,
only $35.7\%$ of graduate student pairs were within four steps of one another.

This pattern is also present in the graphs formed by considering enrollment at
the various course levels. For example, $99.9\%$ of student-pairs in the
$1xxx$-level graph were within four steps of each other, a number that drops
to $81.9\%$ in the 4xxx-level graph. Further, it is safe to assume that this
difference is not the result of low enrollment, as the $1xxx$-level graph
features $9,149$ students, which is not prohibitively dissimilar to the
$4xxx$-level graph’s $8,454$ students.

An even lower percentage (only $41.9\%$) of student pairs in a graph using
$5xxx$ or $6xxx$ level courses (master’s level) were within four steps of one
another. The percentage further declined to $10.2\%$ of student pairs enrolled
in $7xxx$ level courses (doctoral level) being within four steps of one
another.

A key mechanism underpinning our findings is specialization. As students at
American universities increase in class standing, they move from generalized
(“core curriculum”) to specialized (“major”) course loads. This increases
clustering within academic divisions but decreases clustering between
divisions. As a result, networks of students with higher class standing tend
to be split into smaller groups by major and overall connectivity decreases.
Similarly, due to core curriculum and course requirements, students are more
likely to take lower-level courses outside their division.

In addition, we found that certain courses have a greater potential for
epidemic spread due to their centrality to the core curriculum or to majors in
our largest divisions (ECS and SOM). These courses are more likely to enroll
highly connected students, so taking them online may be sufficient to reduce
the potential for epidemic spread without broadly restricting courses by size
or division. For example, we found that $25$ courses have more than three of
the $100$ most connected students in the university-wide network. Removing
them from the network reduces connectivity within four steps from $84.219\%$
to $82.238\%$.

Given that this analysis is concerned with a topic of immediate relevance to
thousands of higher education institutions, there are a number of questions
that must be addressed when considering whether and how to convert the trends
that we have observed into policy decisions. First, how generalizable are our
findings to other institutions? As a medium-sized, public, four-year
institution that transitions students from general to specialized courses, we
expect our analysis to be most relevant to similar institutions. One potential
difference is our mix of majors; in Fall 2019, $30\%$ of our undergraduate
students were majoring in the School of Engineering and Computer Science while
a further $22\%$ were majoring in the School of Management. Beyond the core
curriculum, programs from both schools (e.g., Computer Science, Finance)
require courses in mathematics, natural sciences (e.g., physics), and/or
social sciences (e.g., economics, political science). Relative to an
institution with a larger share of arts and humanities or social sciences
students, students at our institution may be more connected due to our
programmatic distribution.

Furthermore, as Weeden and Cornwell note, our analysis is necessarily limited
to academic networks. It does not consider residential patterns, social
activities, contact in research spaces (libraries or laboratories), transit
between courses, shared instructors or class spaces, transit modes to and from
the institution (mass vs. personal transit), or exposure to the broader
community through activities such as grocery shopping. Additionally, our
analysis does not consider how the presence or absence of mitigating policies,
such as testing and contact tracing, mandated masking, or restrictions on
high-transmission activities (such as indoor dining), affect spread at an
institution. The impact of each factor will vary by student demographics; the
characteristics of the surrounding community; and institutional, local, and
state policies.

In general, we can identify factors that are likely to reinforce the
difference in the susceptibility to infectious spread and other factors likely
to reduce this difference. As lower-division students are more likely to live
in dorms, they are in closer contact with other students through shared
bedrooms, bathrooms, and common areas. Similarly, undergraduates are more
likely than graduate students to engage in extracurricular activities that
increase the risk of transmission, notably fraternities/sororities and campus
athletics. Conversely, upper division students and graduate students are more
likely to have legal access to bars, which are associated with a high risk of
transmission, or to live with non-college students (e.g., roommates or a
spouse), which increases the risk of spread from outside the student
community. In addition, we should not discount that older students are
generally less susceptible to risk-taking behaviors [1]. Given the relatively
low number of freshmen who live on campus and the relatively high share of
transfer and older ($25+$) undergraduates, UT Dallas undergraduates are likely
less connected outside the classroom than students at the archetypical
flagship or major private university.

A common strategy for Fall 2020 was to focus on bringing lower-division
students back to campus in order to maintain the “freshman experience”. There
are academic and financial incentives for pursuing this strategy. Student
attrition is highest from the first to the second academic year, making that
early experience crucial for habituating students to the rigors of higher
education and strengthening connections to their institution (and each other).
Financially, first year students are the most likely to utilize on campus,
university-owned dormitories, which represent a significant, fixed cost for
institutions.

Our analysis indicates that a strategy returning lower-division students to
campus runs contrary to the goal of minimizing infectious spread through an
academic network in the absence additional steps to reduce connectivity. One
potential mitigating step would be to recreate the specialization of upper-
division students through academic pods, restricting face-to-face or hybrid
instruction to courses in their academic division (Engineering, Management,
etc.) while limiting student enrollment in courses outside their division to
online instruction. This method would maximize student habituation to their
discipline, where a student will spend most of their academic careers, while
minimizing the risk of infectious spread across divisions. Furthermore, though
housing arrangements for Academic Year 2021 are effectively locked in,
institutions facing similar challenges in the future could combine academic
and habitation choices, assigning students to dormitories by academic
division. This strategy of creating bubbles, combined with a rigorous testing
regime, proved successful for major American sports leagues in 2020. However,
our analysis also indicates that institutions with sufficient analytic
capacity might also take a scalpel approach to instruction modes by
identifying the specific courses most likely to facilitate epidemic spread.

##  7 Conclusion

In this work we have performed a graph-theoretic analysis of interconnections
among students in a medium-sized public university. This analysis is similar
to that performed by Weeden and Cornwell [8] for a medium-sized private
university. Students, through enrollment in various courses, form a small
world network.

Due to the comparatively larger section sizes in the public university
considered in this study, the solution proposed by Weeden and Cornwell to
prevent the spread of contagion through classroom contact may not be as
effective. This work proposes a complementary solution, the scalpel approach,
that disrupts paths for the spread of contagion by moving courses that
contribute to high betweenness centrality to online instruction. Another
recommendation is to maintain classroom teaching of highly specialized
graduate and some upper-division courses as students in these courses tend to
be tightly knit with little to know connection with other students, while
moving all other courses online.

While the development of vaccines may result in some restoration of normalcy
by Fall 2021, recommendations provided in this work could be useful for policy
makers if faced with similar pandemics in the future.

## References

  * [1] Mariam Arain, Maliha Haque, Lina Johal, Puja Mathur, Wynand Nel, Afsha Rais, Ranbir Sandhu, and Sushil Sharma.  Maturation of the adolescent brain.  Neuropsychiatric disease and treatment, 9:449, 2013.
  * [2] Peter Caley, David J Philp, and Kevin McCracken.  Quantifying social distancing arising from pandemic influenza.  Journal of the Royal Society Interface, 5(23):631–639, 2008.
  * [3] Linton C Freeman.  A set of measures of centrality based on betweenness.  Sociometry, pages 35–41, 1977.
  * [4] Linton C Freeman.  Centrality in social networks conceptual clarification.  Social networks, 1(3):215–239, 1978.
  * [5] Stanley Milgram.  The small world problem.  Psychology today, 2(1):60–67, 1967.
  * [6] Jeffrey Travers and Stanley Milgram.  An experimental study of the small world problem.  In Social Networks, pages 179–197. Elsevier, 1977.
  * [7] Duncan J Watts and Steven H Strogatz.  Collective dynamics of ‘small-world’networks.  Nature, 393(6684):440, 1998.
  * [8] Kim A Weeden and Ben Cornwell.  The small-world network of college classes: Implications for epidemic spread on a university campus.  Sociological Science, 7:222–241, 2020.