Course-Prerequisite Networks for Analyzing and Understanding Academic Curricula Pavlos Stavrinides1and Konstantin Zuev1 1Department of Computing and Mathematical Sciences

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Course-Prerequisite Networks for Analyzing and Understanding Academic Curricula

Pavlos Stavrinides1and Konstantin Zuev1, ∗

1Department of Computing and Mathematical Sciences,

California Institute of Technology, Pasadena, CA 91125, USA

Understanding a complex system of relationships between courses is of great importance for the

university’s educational mission. This paper is dedicated to the study of course-prerequisite net-

works (CPNs), where nodes represent courses and directed links represent the formal prerequisite

relationships between them. The main goal of CPNs is to model interactions between courses, rep-

resent the ﬂow of knowledge in academic curricula, and serve as a key tool for visualizing, analyzing,

and optimizing complex curricula. First, we consider several classical centrality measures, discuss

their meaning in the context of CPNs, and use them for the identiﬁcation of important courses.

Next, we describe the hierarchical structure of a CPN using the topological stratiﬁcation of the

network. Finally, we perform the interdependence analysis, which allows to quantify the strength

of knowledge ﬂow between university divisions and helps to identify the most intradependent, in-

ﬂuential, and interdisciplinary areas of study. We discuss how course-prerequisite networks can be

used by students, faculty, and administrators for detecting important courses, improving existing

and creating new courses, navigating complex curricula, allocating teaching resources, increasing

interdisciplinary interactions between departments, revamping curricula, and enhancing the overall

students’ learning experience. The proposed methodology can be used for the analysis of any CPN,

and it is illustrated with a network of courses taught at the California Institute of Technology. The

network data analyzed in this paper is publicly available in the GitHub repository.

I. INTRODUCTION

An academic curriculum is a complex system of courses

and interactions between them that lies at the heart of

a university and underlies its educational mission. Un-

derstanding a university curriculum as a whole is an im-

portant prerequisite for providing students with a high

quality education and meaningful learning experiences.

Moreover, designing an appropriate curriculum is of great

importance not only from an academic point of view, but

also for organizational and ﬁnancial management.

A full list of courses together with their descriptions

given in the university catalog allows, at least in princi-

ple, to know everything about the curriculum and answer

any question about it. However, it is hard to comprehend

this raw data, extract actionable knowledge, and make

data-driven decisions.

A network, where nodes represent courses and links

represent certain relationships between them, is a nat-

ural model for conceptualizing, representing, and ana-

lyzing a curriculum. For example, links can represent

the temporal relationships between courses based on how

many students move from one course to another through

out their studies [1]. These temporal networks can be

used for forecasting course enrollments, predicting stu-

dent performance [2], and estimating the relative contri-

bution of courses [3]. Alternatively, links between courses

can reﬂect the inﬂuence that some subjects have on oth-

ers based on the expert knowledge of the professors. Such

inﬂuence network models can be used for the curriculum

design and recommendations [4].

∗Corresponding author, email: kostia@caltech.edu

This paper focuses of the study of course-prerequisite

networks (CPNs), where nodes represent courses and di-

rected links represent the formal prerequisite require-

ments between them listed in the university catalog. Un-

like temporal and inﬂuence networks, CPNs are objec-

tively deﬁned and, when in a steady state, don’t change

substantially from year to year. Over the last decade,

CPNs have attracted a lot of interest from researchers

due to their key role in the understanding of the complex

structure of academic curricula. For example, Slim et al.

used CPNs for detecting crucial courses that have a high

impact on students progress and graduation rates [5],

Aldrich discussed applications of CPNs to advising and

curriculum reform [6], and Molontay et al. introduced a

data-driven probabilistic approach for studying the dis-

tribution of graduation time based on the CPN topol-

ogy [7].

In this paper, we propose a general network-science-

based framework for analysis of CPNs and illustrate it

with a CPN based on the courses oﬀered at the Califor-

nia Institute of Technology in the 2021-2022 academic

year. We show that a CPN is an indispensable tool for

visualizing, understanding, and optimizing an academic

curriculum. It can be used not only for identiﬁcation of

important courses, but also for improving existing and

creating new courses. We discuss how students can use

a CPN to navigate their complex curriculum, and how a

CPN can help faculty and administrators to meaningfully

allocate teaching resources, increase interactions between

divisions and departments, revamp the curriculum, and

enhance the overall students’ learning experience.

The proposed framework is based on network sci-

ence [8–11], which is an interdisciplinary ﬁeld that

emerged at the intersection of graph theory, compu-

tational statistics, computer science, and statistical

arXiv:2210.01269v3 [physics.soc-ph] 28 Apr 2023

physics. The basic idea of network science is to use a

network as a simpliﬁed representation of a complex sys-

tem that captures the pattern of connection between sys-

tem’s components and represents its structural skeleton.

Networks have been used to represent a variety of so-

cial, technological, information, and biological systems

consisting of many interconnected, interacting compo-

nents. Modeling complex systems with networks has

proved to be useful for understanding systems as intri-

cate and diverse as the Internet, the world wide web, food

webs, power grids, protein interactions, interwoven social

groups, and even the human brain.

The rest of the paper is organized as follows. In Sec-

tion II, we deﬁne an abstract CPN and basic related no-

tions and describe the Caltech CPN and its giant con-

nected component. Section III is dedicated to the identi-

ﬁcation of important courses and diﬀerent measures for

importance quantiﬁcation. In Section IV, we construct

the topological stratiﬁcation of a CPN and discuss how

the emergent hierarchical structure on the CPN can be

used for ﬁnding hidden prerequisites and creating com-

prehensive schedules for diﬀerent areas of study. In Sec-

tion V, we perform an interdependence analysis of a CPN

that provides a bird’s eye view of the whole curriculum

and allows to quantify the strength of ﬂow of knowledge

from one university division or area of study to another

and identify the most intradependent, inﬂuential, and in-

terdisciplinary divisions and areas of study. Finally, Sec-

tion VI concludes with a brief summary and speciﬁc rec-

ommendations on how students, faculty, and administra-

tors can use the results of the CPN analysis for eﬃcient

navigation and optimal enhancement of the curriculum.

II. NETWORK REPRESENTATION OF

UNIVERSITY COURSES

The main object of study in this paper is a course-

prerequisite network (CPN), which is a directed network

that describes interactions between university courses. In

a CPN, nodes represent diﬀerent courses and directed

links between nodes represent the course-prerequisite re-

lationships between the corresponding courses. A course

Xis called a prerequisite for course Yif taking Xis re-

quired before taking Y. Usually prerequisites cover ma-

terial that is necessary for understanding more advanced

courses. For example, a calculus course is often listed as

a prerequisite for a course on diﬀerential equations. If X

is a prerequisite for Y, then, in the CPN, this is repre-

sented by a directed link from node Xto node Y. In this

case, Yis called a postrequisite of X.

It is convenient to mathematically represent a CPN by

its adjacency matrix. If a CPN has nnodes labeled by

1, . . . , n, then its adjacency matrix is the n×nmatrix

with elements Aij ,i, j = 1, . . . , n, deﬁned as follows:

Aij =(1,if there is a link from ito j,

0,otherwise. (1)

The adjacency matrices of CPNs are sparse (most of Aij

equal to zero), since a typical course has a small number

of prerequisites and serves as a prerequisite to a small

number of courses.

As an example, consider a toy curriculum consisting of

six courses: A, B, C, X, Y, and Z. The course-prerequisite

relationships are summarized in Table I. Course X has

Course Prerequisites

A —

B —

C —

X A, B

Y B, C

Z —

TABLE I. Example of a curriculum consisting of six courses.

two prerequisites, A and B, course Y has two prerequi-

sites, B and C, and all other courses have no prerequi-

sites. If a course does not have any prerequisites, like

courses A, B, C, and Z, then it can be taken any time. If

a course does not have any prerequisites and postrequi-

sites, like course Z, then it is represented by an isolated

node, i.e. a node without incoming and outgoing links.

Figure 1shows the CPN induced by the toy curriculum.

AB C

X Y Z

FIG. 1. The course-prerequisite network (CPN) for the toy

curriculum with six courses deﬁned in Table I.

The adjacency matrix of the this toy CPN is

ABCXYZ













A 0 0 0 1 0 0

B 0 0 0 1 1 0

C 0 0 0 0 1 0

X 0 0 0 0 0 0

Y 0 0 0 0 0 0

Z 0 0 0 0 0 0

(2)

A course-prerequisite network represents the ﬂow of

knowledge between diﬀerent courses in a university cur-

riculum. The main goal of this paper is to show how

CPNs can be used for visualization of complex curric-

ula, drawing important observations and insights about

the courses, and helping students to navigate and faculty

and administrators to optimize their curricula.

The methods described here can be applied to the anal-

ysis of any CPN. We will illustrate them with a real CPN

based on the courses that were oﬀered at the California

Institute of Technology (Caltech) in the 2021-2022 aca-

demic year. The Caltech CPN, consisting of both under-

graduate and graduate courses, is shown in Fig. 2. All

network visualizations in this paper are done in Gephi,

an open-source and free network visualization software

package [12]. For visual clarity, the network visualization

in Fig. 2omits the node labels (course names). A larger

and more detailed visualization of the Caltech CPN is

shown in Fig. 15 in the Appendix. The network data is

publicly available in the GitHub repository [13].

FIG. 2. The 2021-2022 Caltech CPN. Nodes in gray are iso-

lated, i.e. have no prerequisites and do not serve as prereq-

uisites. Nodes in color other than gray represent connected

components. The network has 771 nodes and 772 links.

Any university curriculum contains courses that are

completely independent of each other: they are not pre-

requisites for each other, not prerequisites for another

course, there is not a course which is a common pre-

requisite for them, etc. This independence between two

courses manifests itself in the CPN by the absence of a

path between the nodes representing the courses. Inde-

pendent courses belong to diﬀerent connected components

of the CPN. Technically, a (weakly) connected compo-

nent of a CPN is a subset of nodes such that for any two

nodes in the subset there exists at least one path through

the network connecting the nodes, where paths are al-

lowed to go in both ways along any link (the directions

of the links are ignored). Each isolated node constitutes

a trivial connected component. Isolated nodes usually

represent seminars, projects, outreach, and special top-

ics courses. For example, the toy CPN on Fig. 1has two

connected components: consisting of nodes {A, B, C, X,

Y}and one isolated node Z.

Real-world CPNs have several small connected compo-

nents and one “giant” connected component, called the

largest connected component (LCC), which contains the

largest fraction of nodes, almost all links, and consti-

tutes the most interesting and nontrivial part of a CPN.

The LCC of a CPN, denoted by G, is the main part of

the network, which represents its complex structure and

function.

In the Caltech CPN, in addition to isolated nodes,

there are 10 connected components represented by dif-

ferent colors in Fig. 2and, in more detail, in Fig. 15. All

but one are very small, with sizes not exceeding six nodes.

The largest connected component G(shown in pink) con-

tains n= 436 nodes (57% of all nodes) and m= 747 links

(97% of all links). In what follows, we focus our analysis

on the largest connect component Gof the Caltech CPN,

which is shown in Fig. 16 in the Appendix.

III. CENTRALITY MEASURES

One of the most interesting and intriguing questions

about a university curriculum is the following: “Which

are the most important courses in the curriculum?”. In

other words, “Which are the most important nodes in

the CPN?”. Knowing the most important courses, the

courses that form the “backbone” of the curriculum,

could help to a) better allocate university resources to

provide students with better experiences in these courses

and b) inform students about these courses, so that they

can pay special attention to them.

There are diﬀerent ways to deﬁne the “importance” of

a node in a CPN. Here, we will consider three widely used

in network science centrality measures, which quantify

the node importance: degree, PageRank centrality, and

betweenness centrality.

A. Degree Distributions

The total degree of a node is the total number of links

connected to it. In directed networks like CPNs, nodes

have two kinds of degree: an in-degree, the number of in-

coming links, and an out-degree, the number of outgoing

links. The in-degree kin(i) of a node iis the number of

prerequisites course ihas, and the out-degree kout(i) of i

is the number of courses for which iis a prerequisite. In

terms of the adjacency matrix A, the in- and out-degrees

of node iare given by

kin(i) =

j=1

Aji and kout(i) =

j=1

Aij .(3)

The total degree of node iis then

k(i) = kin(i) + kout(i) =

j=1

(Aij +Aji).(4)

The in-degree of a node measures how specialized the

corresponding course is: the larger kin(i) is, the more

prerequisites course ihas, the more specialized it is. The

out-degree, on the other hand, measures how fundamen-

tal a course is: the larger kout(i) is, the more courses

have ias a prerequisite, the more fundamental course i

is. We expect that in real-wold CPNs, the in- and out-

degrees of nodes are negatively correlated. The absolute

value of the Pearson correlation coeﬃcient between in-

and out-degrees of nodes,

ρin,out =

i=1 kin(i)−¯

kinkout(i)−¯

kout

i=1 kin(i)−¯

kin2sn

i=1 kout(i)−¯

kout2

(5)

can be used to measure the structural diﬀerence between

fundamental and specialized courses.

To elaborate more on this, consider two extreme cases

shown in Fig. 3.

…

nodes

…

/2 nodes

(a)

(b)

FIG. 3. Two extreme cases. Curriculum (a) (blue): all courses

are equivalent. Curriculum (b) (green): there is a substantial

structural diﬀerence between fundamental courses (bottom

row) and specialized courses (top row).

In curriculum (a), essentially all courses are struc-

turally equivalent (except for the ﬁrst and the last one);

there are no fundamental and specialized courses. The

sequences of in- and out-degrees are kin = (0,1,...,1)

and kout = (1,...,1,0), and the correlation coeﬃcient

between them is

ρ(a)

in,out =−1

n−1≈0,for large n. (6)

In curriculum (b), however, there are fundamental

courses (bottom row) and specialized courses (top row).

These two types of courses are structurally very dif-

ferent: all fundamental courses are prerequisites for

all specialized courses. The sequences of in- and out-

degrees are kin = (0,...,0, n/2, . . . , n/2) and kout =

(n/2, . . . , n/2,0,...,0), and the correlation coeﬃcient

between them is

ρ(b)

in,out =−1,for any n. (7)

In real-world CPNs, the correlation coeﬃcient ρin,out ∈

(−1,0), and its magnitude |ρin,out|quantiﬁes the

structural “gap” between fundamental and specialized

courses: the larger |ρin,out|is, the more signiﬁcant the

split of the curriculum into fundamental and specialized

courses.

Figure 4shows the histograms of in-, out-, and total

degrees in the LCC Gof the Caltech CPN. All degree

distributions are right-skewed and have long right tails.

0 1 2 3 4 5 6 7

In-degree, kin

100

120

140

160

180

200

Number of Courses

0 5 10 15 20 25 30

Out-degree, kout

100

150

200

250

300

Number of Courses

0 5 10 15 20 25 30 35

Total degree, k

100

120

140

Number of Courses

FIG. 4. The in-, out-, and total degree distributions in the

largest connected component Gof the Caltech CPN.

A network is called scale-free, a term coined in the

seminal paper [14], if its degree distribution follows a

power-law,

P(k)∝k−γ,for k≥kmin,(8)

where P(k) is the probability that a node chosen uni-

formly at random has degree k,kmin is the lower cut-

oﬀ for the scaling region, and γis the power-law expo-

nent. Many real-world networks are approximately scale-

free, with the power-law exponent typically in the range

2< γ < 3 [8]. For example, both the in- and out-degrees

of the World Wide Web approximately follow power-law

distributions with γin = 2.1 and γout = 2.7 [15]. For

the LCC Gof the Caltech CPN, the hypothesis test for

discerning and quantifying power-law behavior in empiri-

cal data developed in [16] accepts the hypothesis that the

total degree distribution (bottom panel in Fig. 4) approx-

imately follows a power-law, but rejects that hypothesis

for the in- and out-degrees. The maximum-likelihood ﬁt-

ting method developed in [16] estimates the power-law

exponent γand the lower cut-oﬀ kmin as follows:

γ= 2.48 and kmin = 3.(9)

Most courses in G(74%) have one or two prerequisites

(top panel in Fig. 4) and many (63%) are “dead ends,”

i.e. they do not serve as prerequisites for other courses

(middle panel in Fig. 4). The average in-, out-, and total

degrees are (note that always ¯

kin =¯

kout =¯

k/2):

kin =1

i=1

kin(i)=1.70,

kout =1

i=1

kout(i) = 1.70,

k=1

i=1

k(i) = ¯

kin +¯

kout = 3.40.

(10)

The Pearson correlation coeﬃcient between in- and

out-degrees of nodes is

ρin,out =−0.13.(11)

It is instructive to see how exactly the in- and out-

degrees contribute to the total degrees of nodes. Let

Gd⊂ G be the subset of nodes with total degree dand

ndbe the number of nodes in Gd,

Gd={i:k(i) = d}, nd=|Gd|.(12)

Then for any node in Gdthe sum of its in- and out-degrees

is exactly d,

kin(i) + kout(i) = d, ∀i∈ Gd.(13)

Let ¯

k(d)

in and ¯

k(d)

out be the average in- and out-degrees of

nodes in Gd,

k(d)

in =1

ndX

i∈Gd

kin(i) and ¯

k(d)

out =1

ndX

i∈Gd

kout(i).(14)

Then, by averaging (13) over all nodes i∈ Gd, we have

k(d)

in +¯

k(d)

out =d, (15)

for any total degree d. Figure 5shows the decomposition

of the total degree dinto the in- and out- components

k(d)

in and ¯

k(d)

out.

0 5 10 15 20 25 30

Total degree, d

k(d)

out

k(d)

FIG. 5. Decomposition d=¯

k(d)

in +¯

k(d)

out of the total degree d

into the in- and out- components ¯

k(d)

in and ¯

k(d)

out.

In all subsets Gd⊂ G for d > 5, ¯

k(d)

out strongly dominates

k(d)

in and contributes the most to the total degree d. The

in-degree component can be viewed as a “noise” added

to the out-degree component.

Tables II,III, and IV list the top courses with respect

to the in-, out-, and total degrees.

In-Degree Top 9

Course Title In-Deg

1 CMS 139 Algorithm Analysis 7

2 Ge 270 Continental Tectonics 7

3 ACM 106 Computational Math 5

4 ChE 111 Sustainability 5

5 Ch 21 Physical Chem 5

6 Ch 25 Biophysical Chem 5

7 CMS 144 Networks 5

8 ME 50 Modelling in Mech. Eng. 5

9 Ph 6 Physics Lab 5

TABLE II. The top 9 courses with the largest in-degree. Cut-

oﬀ was set to in-degree 5.

As expected, the top in-degree nodes in Table II in-

clude some of the most specialized courses oﬀered at Cal-

tech. These courses are often at the graduate level and

require a considerable amount of previous coursework in

order to be taken. The top out-degree nodes in Table III

correspond to some of the most fundamental courses in

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Course-PrerequisiteNetworksforAnalyzingandUnderstandingAcademicCurriculaPavlosStavrinides1andKonstantinZuev1,1DepartmentofComputingandMathematicalSciences,CaliforniaInstituteofTechnology,Pasadena,CA91125,USAUnderstandingacomplexsystemofrelationshipsbetweencoursesisofgreatimportancefortheuniversity'...

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Course-Prerequisite Networks for Analyzing and Understanding Academic Curricula Pavlos Stavrinides1and Konstantin Zuev1 1Department of Computing and Mathematical Sciences

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