Jeffrey Wigdahl
Candidate
Electrical and Computer Engineering
Department
This thesis is approved, and it is acceptable in quality and form for publication:
Approved by the Thesis Committee:
Dr. Greg Heileman, Chairperson
Dr. Chaouki Abdallah
Dr. Simon Barriga
i
ASSESSMENT OF CURRICULUM GRAPHS WITH RESPECT
TO STUDENT FLOW AND GRADUATION RATES
by
JEFFREY WIGDAHL
B.S. ELECTRICAL ENGINEERING, ST. MARY’S
UNIVERSITY, 2010
THESIS
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Master of Science
In
Electrical Engineering
The University of New Mexico
Albuquerque, New Mexico
December 2013
ii
ASSESSMENT OF CURRICULUM GRAPHS WITH RESPECT TO STUDENT
FLOW AND GRADUATION RATES
by
Jeffrey Wigdahl
B.S. Electrical Engineering, St. Mary’s University, 2010
ABSTRACT
Universities have been predicting graduation rates for incoming student classes
using a multitude of factors. These factors can be broken down into two categories: preinstitutional and institutional. Pre-institutional factors include socioeconomic (race,
gender, economic status) and high school performance data (G.P.A, S.A.T). The
combination of this data can be incorporated into a statistical model for predicting
graduation rates. While this information can give a broad student success estimate to the
university, the refinement comes from the institutional factors. These can range from
semester grade point averages to the amount of time a student spends at the gym. One
particular factor that will be presented and studied in this thesis is curriculum difficulty,
defined through features of a curriculum represented as a graph.
A curriculum can be presented as a directed graph, with each class as an
individual node, and co/prerequisites as the edges between them. Features of these
graphs are defined and studied over several comparable universities’ electrical
engineering programs. Further, student data at the class level is adapted to create a
weighted digraph form which a cumulative curricular difficulty/complexity metric is
obtained. The purpose of this tool is to give administration quick access to curricular
iii
features and the ability to compare with other universities to guide future decisions.
Computer code was developed to support this effort and has been adapted to pull
information directly from the University of New Mexico (UNM) curriculum website to
automatically generate the graphs and features.
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TABLE OF CONTENTS
LIST OF FIGURES ......................................................................................................... vi
LIST OF TABLES .......................................................................................................... vii
CHAPTER 1 INTRODUCTION ......................................................................................1
1.1 Student Success ......................................................................................................1
1.2 Effects of Curriculum on Success ..........................................................................4
1.3 Overview of Thesis ................................................................................................5
CHAPTER 2 BACKGROUND .........................................................................................6
2.1 Graph Theory .........................................................................................................6
2.2 Curriculum Graphs.................................................................................................9
2.3 Features of Curricular Graphs ..............................................................................10
2.4 University Data ....................................................................................................12
CHAPTER 3 CURRICULAR ANALYSIS ...................................................................16
3.1 Graphs and Results ..............................................................................................16
3.2 Graph Difficulty Through Weighting ..................................................................23
CHAPTER 4 DISCUSSION............................................................................................27
4.1 Proposed Tool ......................................................................................................27
4.2 Limitations ...........................................................................................................27
4.3 Curriculum Changes ............................................................................................28
APPENDIX A ...................................................................................................................33
REFERENCES .................................................................................................................39
v
LIST OF FIGURES
Figure 1. HERI Expected Graduation Rate Calculator .......................................................2
Figure 2. Examples of undirected and directed graph notation ..........................................7
Figure 3. Example of a Complete Graph ............................................................................8
Figure 4. Example classes in graph form ............................................................................9
Figure 5. Distributions for UNM graduates ......................................................................13
Figure 6. Breakdown of student population at UNM main campus .................................15
Figure 7. The University of Houston Electrical Engineering graph .................................17
Figure 8. Arizona State University Electrical Engineering graph ....................................19
Figure 9. University of Central Florida Electrical Engineering graph ..............................20
Figure 10. University of New Mexico Electrical Engineering graph ...............................22
Figure 11. UNM EE graph with difficulty metric calculated ...........................................25
Figure 12. UNM Computer Engineering Degree graph....................................................29
Figure 13. UNM proposed 120 hour Computer Engineering Degree graph.....................29
Figure 14. Propose UNM Electrical Engineering 120 hour curriculum graph .................31
vi
LIST OF TABLES
Table 1. HERI predicted graduation rates for UNM.........................................................14
Table 2. Summary of feature results calculated for each university .................................22
Table 3. Difficulty metrics calculated for each EE program ............................................26
Table 4. Features and difficulty metric for UNM’s Computer Engineering curriculum
and new 120 hour proposed curriculum.............................................................................30
Table 5. Features and difficulty metric for UNM’s Electrical Engineering curriculum and
new 120 hour proposed curriculum. ..................................................................................31
vii
Chapter 1- Introduction
1.1 Student Success
Many definitions of student success exist in the literature. While these vary from
grades and persistence to self-improvement, many consider graduation the ultimate
measure of student success (1). For the student, having a bachelor’s degree has become a
necessity with attainment rates topping 30% for adults over the age of 25 from the latest
census numbers (2). From the university perspective, and especially for public
universities, the definition of student success broadens from not only graduation, but
student retention rates and time to degree. These factors are important because many
states, including New Mexico, tie a percentage of university funding directly to success
metrics (3). This performance funding has become a popular way to incentivize
universities to help students graduate in a timely fashion. Also, since 1990, institutions
have been required to release graduation and retention rates to the public. These numbers
are often used in college rankings, and can impact student decisions on where to attend
college. Studies have also a shown a rise in graduation percentage as state appropriations
per student increase (4).
Breaking down student success can be a tricky thing. There are many factors that
have been studied that correlate to a student graduating. These factors can be broken
down into two categories, pre-institutional and institutional factors. Pre-institutional
factors include high school and socioeconomic data. The Higher Educational Research
Institute (HERI) out of UCLA has collected this data for incoming freshmen classes over
the last 15 years and publishes national norms every few years. These norms include
graduation rates by ethnicity, race, sex, SAT/ACT, and high school grade. These factors
1
have been studied at length and their contribution to success is well documented. They
also conduct a freshmen survey that looks a little deeper into pre-institutional factors as
well as early institutional factors that will be discussed later. Using data from 356 fouryear (separated by institutional type: public, private) non-profit institutions and 210,000
first-time full-time students, HERI has created a calculator that can predict graduation for
a single student or an entire incoming class (5).
Figure 1: HERI graduation rate calculator. Institutions can input data for a single student or entire
incoming class to predict a graduation rate. http://www.heri.ucla.edu/GradRateCalculator.php
Figure 1 shows their basic calculator, which can be expanded to include information from
their freshmen survey. All graduation prediction is done using logistic regression and the
2
output can be seen as a baseline graduation rate that can vary based on what happens to
the student due to institutional factors.
Institutional factors attributing to success can be any quantifiable metric that occurs
during the student’s time at an institution. In 1987, Tinto stated that success is clearly
dependent on institutional experiences and that those satisfied with their experience, stay
at a higher rate than those who are not (6). With the advent of web-enhanced learning
tools for classes and smart student ID cards, the amount and scope of data being collected
by an institution has grown tremendously to the point of needing sophisticated software
and full-time employees to manage it. This allows for the tracking of student
engagement by seeing how often a student is going to the library or the length of time
they stay on a class website. It is obvious that the number of factors is only limited by
imagination and covering these is beyond the scope of this work, but the purpose of all
this data is to better track student success and intervene when a student gets off track in a
much timelier manner.
Many important institutional factors exist separate from any student involvement.
Simply providing certain amenities and infrastructure is enough to change student
success. Studies have identified things such as learning centers, freshman year programs,
dorms, study rooms etc. as all attributing to student success (7). For most new students,
college is the first time they will be away from home for an extended period of time and
creating a positive learning environment can help ensure students are engaged and
comfortable.
3
1.2
Effect of Curriculum on Success
One institutional factor that is often overlooked is the curricula. This may be due to a
feeling of lack of control from administrators due to certain classes being required by the
state or certain certification bodies. For instance, students in Texas are required to take 6
hours of history as part of their core requirements regardless of where they are from (8).
A core curriculum is usually set to provide a well-rounded learning experience of which
students, regardless of major must take. These core classes may fit well into some
majors, but can be burdensome to others, adding extra hours to already rigorous programs
such as those in the science, technology, engineering, and mathematics fields (STEM).
At the University of Houston, the electrical engineering program requires 131 hours to
graduate, where a sociology degree can be completed in 120. This adds close to an extra
semester worth of hours that is supposed to fit into the same 4-year workload.
So how can a curriculum affect the success of a student? The purpose of all the data
being collected, programs being offered, and infrastructure being built is to provide the
easiest and most efficient path to graduation. This should also be the goal of any
curriculum in a much more literal sense. The ease of flow through a curriculum should
be as simple as possible which means students should have as many options as possible
to fulfill their graduation requirements. The important classes in a curriculum should be
identified and special attention should be given by administration and faculty to ensure
student flow. This also means going back and examining what is truly needed to take a
certain class. Unnecessary pre-requisites can set students back in their studies and create
bottlenecks to graduation.
4
Due to the nature of a curriculum, it can be studied using graph theory, providing a
mathematical foundation for determining the ease of flow and detecting important
classes. Analyzing the features of a curriculum graph can help administrators make
decisions on when to offer certain classes, who should teach them, and what is truly
necessary for a degree in a certain field.
1.3 Overview of Thesis
In the following chapters, I will expand upon the graph theory used to create
curriculum graphs and extract the important features from them, giving examples from
several electrical engineering programs at a number of different institutions. In chapter 2,
I will introduce graph theory and how it applies to a curriculum. I will also introduce the
features that are used to determine curricular difficulty and how they were calculated, as
well as the curricula and student data used to create the graduation difficulty models. In
chapter 3, I will look at the graphs from several electrical engineering programs,
including UNM’s, and compare the results of the features extracted from them. A
curriculum difficulty metric is also presented based on weights assigned to the nodes of a
graph. Chapter 4 will give a discussion on the proposed tool along with the limitations
of the data and proposed metrics. Future work is also proposed consisting of a
longitudinal study of the computer engineering curriculum, which will be changing in the
near future.
5
Chapter 2 – Background
2.1 Graph Theory
Graph theory and network analysis has been a topic of study since the 1700’s (9),
with the first textbook on the subject being written in 1936 by Denes Konig. Later
publications unified graph theory methods over multiple disciplines, providing a common
language to the broad range of researchers (10). Graphs are a main structure in discrete
mathematics and have applications in a wide variety of subjects. As examples, graph
theory is used in computer science to model networks of communication, or to show the
structure of the web (11). In chemistry, the connection of molecules, showing atoms and
bonds can be considered a graph (12). And in sociology, networks can be created linking
people who are friends, or actors who have worked together (13). The following sections
are meant to give a brief introduction to graphs and some of the terms used to describe
types of graphs and their individual parts.
Graphs consist of nodes called vertices that are connected by lines called edges.
Formally, a graph is a pair of sets (V, E), where V is the set of vertices V = {v1, v2, v3} and
E is the set of edges E = {(v1, v2), (v1, v3), (v2, v3)}. Edges are commonly labeled as e1, e2,
e3 …en instead of using the start and end node. Figure 2 shows examples of this notation.
6
Figure 2: Examples of undirected and directed graph notation along with examples of different types of
nodes and edges (14).
Based on the example above, we can define the following terms. If the edges in a graph
have direction, the graph is called a Directional graph or digraph. This means the order
of vertices in an edge matters and are often called arcs, directed edges, or arrow. This is
opposed to the undirected graph, where the order of vertices in an edge doesn’t matter
and the edge (v1, v2) = (v2, v1). Other types of graphs include Mixed graphs: where some
edges are directed and some are not, Multi-graphs: which can have multiple edges
connecting the same nodes, and Quiver graphs: which are multi-graphs with direction.
The edges in a graph can also be weighted, which can represent a number of factors, such
as distance or cost.
7
Figure 3: Example of a complete graph where every node is connected to every other node in the graph.
The connectedness of a graph also gives us a few common terms. A graph that has no
edges is called empty, while a graph with no vertices is null. A graph with a single vertex
is trivial. When each vertex has the same number of neighbors, it is deemed a regular
graph. A complete graph has all possible edges present. This means that each vertex is
joined to every other vertex by an edge. Figure 3 shows an example of a complete graph.
Graphs can be finite or infinite. An infinite graph has either an infinite number of
vertices or edges.
Other common terms are derived from the connectedness of single nodes or edges. Two
edges are adjacent if they share a common vertex. From figure X, edges e1 and e2 are
adjacent because they share vertex v2. Adjacency is also defined for vertices in the
opposite manner. If two vertices share an edge, they are considered adjacent. If it is a
digraph, and the edge shared by two vertices contain the tail and head of an arrow, the
vertices are called consecutive. The degree of a node, d(v), is the number of edges with v
as an end vertex. Each vertex of a digraph has an in-degree and an out-degree. The indegree is the amount of arrows leading into vertex while the out-degree is the number of
arrows leaving a vertex. Vertices with degree zero, such as v3 from figure 2 are isolated
vertices. A vertex with degree one is a pendent vertex, and any edge that has a pendent
8
vertex in it is considered a pendent edge. A path is a set of alternating vertices and edges
in which all edges and vertices are distinct between vertices u and v. Any path forms a
sub-graph of the original as it contains a subset of its vertices and edges.
2.2 Curriculum Graphs
A curriculum can be represented as a directed graph with each individual class as a
vertex and pre/co-requisites as the edges between them. Figure 4 shows a simple
example of a few classes in graph form.
Figure 4: Classes in graph form. Writing 1 is a prerequisite of writing 2. POLS 1336 has no prerequisites
and is not a prerequisite for any other class.
First year writing 1 has no prerequisites, but is a prerequisite for First year writing 2.
Therefor there is a directed edge connecting the two class nodes. POLS 1336 has no
prerequisites and is not a prerequisite for any class and is thus disjoint or isolated from
the main graph. ENGL 1303 has an in-degree of zero and an out-degree of 1 and this is
reversed for ENGL 1304. This connection is also the only path found in this graph. For
most STEM related curriculums, there is generally one large main graph with many
individual isolated nodes. Most core courses have little to no prerequisites, but once you
get into the major curriculum, classes begin to advance on each other creating longer
paths and higher in and out degrees. This isn’t seen as much in other degree programs
9
where the classes have much fewer prerequisites and can be taken at any time during a
student’s tenure.
At most universities, degree plans are available to students to show them one possible
way of advancing through a program. At the University of New Mexico, we now have
all the curriculums up on a website with the ability to quickly turn these into graph files
using JSON and RUBY to pull the data from the website and convert the information.
Appendix A shows the code necessary to do this.
2.3 Features of Curricular graphs
While there are many features that can be calculated from graphs, we have come up
with a handful we believe best explain curricular difficulty with respect to student
success along with measurements associated with the curriculum that are not necessarily
graph related. These features are listed along with a brief description and explanation of
each:
Degree Hours: Though each node does not have the number of hours built into it and is
thus, not directly a graph related measure from our stand point, the number of hours in a
degree plan significantly impacts the number of hours a student must take per semester to
graduate in 4 years. This trickles down to the number of hours a student must spend per
week on school related activities. In our curricular graphs, it usually is indicative of the
number of nodes present.
Maximum In-Degree and In-Degree over 3: As mentioned previously, the in-degree
of a directed graph represents the number of arrows going into a node. In a curricular
graph, this represents the number of prerequisites a class has. These high in-degree
classes show us possible bottleneck points in the graph and possible important classes.
10
Maximum Out-Degree and Out-Degree over 3: As mentioned previously, the outdegree of a directed graph represents the number of arrows going out of a node. In a
curricular graph, this represents the number of classes that require this class as a
prerequisite. These high out-degree classes show us bottleneck points, important classes,
and opportunities for better performance (i.e., increasing pass rates in these classes).
Important Classes (Bottlenecks): Many universities have different definitions of
important classes in a curriculum. For the purposes of our graph approach, we have
defined important classes as those that have an in-degree or out-degree above 3 or a
combination above 5. These classes represent bottlenecks to graduation where failure
can lead to the inability to progress in a timely manner.
Longest Path and number of Long Paths: Paths were defined earlier as a set of
alternating vertices and edges in which all edges and vertices are distinct between two
vertices. In a curriculum graph, the longest path represents the longest chain of
prerequisites through a graph. An example would be having to take several math classes
in a certain order, such as Calculus 1 followed by, Calculus 2, Differential Equations, and
then some advanced mathematics class. The path length would be 3 and if none of these
classes is a co-requisite, would take 4 semesters to complete. Long paths represent
chains of classes that must be taken in order. Failing a class in a chain often requires
summer school to get back on track or falling behind by a semester or year, depending on
the availability of the class. The logic is that the more long paths, the more likely a
student is to get off track, get frustrated, and leaves a program. Our definition of a long
path is any path length of 5 or above (5 edges, 6 nodes).
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2.4 University Data
Each of the curriculums used were publicly available on each universities website.
Electrical engineering programs from the following schools were chosen for analysis:
The University of Houston, The University of New Mexico, Arizona State University,
and the University of Central Florida (15), (16), (17), (18). These were chosen because
of their peer status as public state universities with similar student acceptance criteria and
rates. Also, engineering graduation rates were available for these universities to help test
the hypothesis of curriculum affecting these rates.
Thanks to the analytics department at the University of New Mexico, we also have a
collection of student data with the same information HERI collects on incoming
freshmen. Cohorts from 2002 to 2011 were made available totaling 31,518 students with
graduation data available in 2006 and earlier. From this data, we can look at the type of
students graduating from UNM. Figure 5 shows distributions of graduates based on their
high school GPA, ACT, UNM undergraduate GPA, and the number of hours taken at
time of graduation. An average graduate from UNM takes 146 hours to graduate, came
in with an ACT score of 22.5 and high school GPA of 3.45, and maintained a 3.35 GPA
through his studies.
12
Figure 5: Distributions for students that graduate from UNM based on ACT, HS GPA, Undergrad GPA,
and hours to graduation.
We also have the expected HERI graduation rate and the actual graduation rate dating
back to 1995. This gives us a chance to see how well the university has done compared
to the HERI expected graduation rate. Table 1 shows these results. For all but one year,
the HERI statistics overestimated the graduation rate. On average, it overestimated the
graduation rate by 3.6%. From HERI documents, the variables in the HERI logistic
regression model explain approximately 65% of the variance, with much of the rest being
explained by institutional factors. From the graduation rates shown, at the University of
New Mexico, it would appear that these institutional factors are affecting student success
in a negative way, at least up until 2006.
13
HERI
Cohort
Actual
Difference
Model
1995
50.0%
45.3%
4.70%
1996
49.8%
46.0%
3.80%
1997
49.6%
42.6%
7.00%
1998
39.5%
40.2%
-0.70%
1999
47.8%
41.1%
6.70%
2000
47.6%
43.2%
4.40%
2001
47.6%
44.2%
3.40%
2002
47.4%
44.3%
3.10%
2003
47.3%
43.1%
4.20%
2004
47.1%
44.5%
2.60%
2005
47.4%
45.1%
2.30%
2006
47.3%
45.8%
1.50%
Table 1: HERI predicted graduation rates versus actual graduation rates for the University of New Mexico
These are six-year graduation rates, which haven’t topped 46% during this 12-year span.
According to HERI, the average six-year graduation rate for public universities was
65.6% (5). This tells us that UNM is underperforming the general public university
population, although it would be unfair to make a head-to-head comparison as many
schools have different acceptance rates and thus different entering student populations.
A report from UNM’s Office for Equity and Inclusion released in 2011 gives the
breakdown of student enrollment by gender and ethnicity for the years of 2005-2009 (19).
14
Figure 6: Breakdown of Student populations at UNM main campus for the years 2005-2009 by gender and
ethnicity.
Figure 6 shows a trend of steadily increasing male population as well as a decreasing
White population. UNM also has a high American Indian enrollment percentage mainly
due to the percentage of American Indians in the State of New Mexico (10%) compared
to the national average (1%) (20). These are important factors because based on HERI
calculations being male, Hispanic, or American Indian, are three of the most negative
factors when creating their logistic regression models. Positive factors include being
female, having a high average high school grade, and being White or Asian/Pacific
Islander (5).
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Chapter 3 – Curricular Analytics
3.1 Graphs and Results
Graphs were created for the electrical engineering curricula from The University of
New Mexico, University of Houston (UH), University of Central Florida (UCF), and
Arizona State University (ASU). The graphs for these curriculums were created using
either flow charts provided by the school or a document containing all the classes
necessary for graduation plus prerequisites or a combination of the two. As a note, other
qualifications to take a class were not considered in these graphs. For example, you may
need to be an upper classmen to take a certain class. The graphs were coded in Ruby
using the ruby graph library (rgl) by creating a directed adjacency graph that is basically
a two-dimensional structure (21). The first dimension of the structure is an individual
vertex while the second dimension of the structure is a set of the adjacent vertices. This
second set implies the edges between the two vertices. Once a curriculum is in this form,
standard graph algorithms are provided (i.e., topological sort, depth-first search).
We have also created the ability to create a graph by pulling information straight from
our UNM curriculum website (http://degrees.unm.edu). The classes and prerequisites are
opened via a JSON file from the website and saved as a .dat file with individual classes in
parenthesis and prerequisite edges as a pair of classes separated by a comma. The first
class in the pair is the prerequisite to the second class. This data is then manipulated in
order to fit the rgl format. This includes removing parenthesis and white spaces from the
.dat file and iterating over all of the vertices and edges. Once in the proper form, features
can then be extracted. The entire code base for the project can be found in Appendix A.
This will make calculating graph statistics simple and fast for any curriculum at UNM.
16
Figure 7 shows the curriculum graph created for The University of Houston. As a
note, the isolated nodes are placed in the graph to shrink the size and do not have
anything to do with when the classes should be taken. The computed features can be
found in Table 2 for the four universities. This curriculum is an example of a very
complex road to graduation.
Figure 7: The University of Houston Electrical Engineering program in graph form.
It required 131 hours to get an electrical engineering degree at the University of Houston
in 2010. This is approximately 16.25 hours per semester for a four-year plan. The
maximum node in-degree is 7, which is for their Circuit Analysis class, and the maximum
node out-degree is 6 for their Calculus 2 class. These two classes are typically
bottlenecks in most electrical engineering programs. If someone fails Calculus 2, you
likely would not want them moving forward in an electrical engineering degree, while
passing opens the door to the more rigorous engineering classes. Circuit Analysis is also
17
an important class in electrical engineering as it marks the first in a chain of engineering
classes based around circuit design/analysis and electronics. The University of Houston
has made reaching this class difficult by requiring 7 pre/co-requisites. This may stunt the
advancement of a student through the program, forcing the student to take classes that do
not fill a requirement for graduation. A report by UNM shows that students at the
University of Houston’s engineering department are taking 138% of the hours that are
needed to graduate (22). For this program, it means students are graduating with
approximately 180 hours on average. Another feature that shows the complexity is the
edge-to-node ratio. There are 1.48 edges per node, meaning each class has an average of
one and a half prerequisites. The longest path is also concerning since its length, 9, is
longer than the number of semesters it should be completed in. The length 9 longest path
means that there is a 10 class chain to be completed in 8 semesters. Obviously there are
several co-requisites in this chain, meaning that two classes can be taken together, but
this length makes a failure very costly in terms of timely graduation and hours to
graduation. Also, the longest path for this program contains 4 courses that would be
considered bottlenecks by our definition and 8 total in the curriculum.
On the opposite end of the spectrum is the curriculum graph from Arizona State
University. Figure 8 shows the graph that looks very different from the University of
Houston graph, shorter class names disregarded. This electrical engineering curriculum
only requires 120 hours to complete. The highest in-degree is 3 and out-degree is 4.
Calculus 2 is once again the highest out-degree and there are many classes that have an
in-degree of 3, one of these being their Circuit Analysis class. However, these are the
only two classes in the curriculum that would be considered bottlenecks. The longest
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path is just 6 and the edge to node ratio is .82, almost half that of the University of
Houston. From the UNM report, engineering students at Arizona State take 123% more
hours than are needed to graduate. This comes out to around 148 total hours on average
for students at the time of graduation. Compare this with 180 hours students at UH are
taking and we see that ASU students are taking a year less of hours when they graduate.
Figure 8: Arizona State University Electrical Engineering curriculum.
How can two electrical engineering programs offer the same degree, but be so different in
terms of student outcomes. Both of these programs are accredited by the Accreditation
Board for Engineering and Technology (ABET), which gives general criteria for
curriculum requirements. It would seem that ASU has leveraged these requirements into
a much more flexible program. This gives students more options and the ability to stay
on track even if failure occurs at an important node.
19
The previous two curriculums showed how widely the same degree plan can differ at
different universities. We now want to look at a few EE programs that are in the middle,
with the UNM program being one of these for which we have graduation data. First,
however, The University of Central Florida was chosen as a peer institute that would
possibly be open to giving us graduation data for their engineering department in the
future. Figure 9 shows their curriculum graph.
Figure 9: University of Central Florida Electrical Engineering Curriculum Graph.
An interesting feature of this graph is that the maximum in-degree is two. This is lower
than ASU and any other STEM curriculum graph that we have studied. Since no class
has more than two prerequisites, the program has few bottlenecks: mainly the two
mentioned before, Calculus 2 and what they call their Networks and Systems class which
is equivalent to the Network Analysis class described earlier. The maximum out-degree
is 6 and the longest path is 8. Their edge to node ratio is .9. It will be seen shortly that
20
UCF shares very similar features with UNM, with the exception being the max in-degree.
Going forward, it will be interesting to see the effect of this on student success. From the
UCF Electrical Engineering website, the administration has highlighted what they believe
to be critical path courses in the program (18). They mention 14 classes that they believe
will indicate student success, which is likely based on previous student data and a
consensus on what the important topics in the program are. It is probably not a
coincidence that 5 of these courses are on the longest path.
The University of New Mexico is presented last because we have much more data to
work into the graph. It is also an interesting graph because we have the ability to make
changes in the future to see the effect on student success. This will be looked at further
in a later section, when the UNM Computer Engineering program curriculum graph is
introduced along with proposed changes to simplify a few of the program features.
Similar changes will also be presented for the UNM Electrical Engineering program.
Figure 10 shows the UNM Electrical Engineering curriculum graph. On average, UNM
engineering students accumulate 168 hours by the time they graduate. This falls in
between ASU and UH, as do most of the curricular features. As mentioned previously,
UNM shares many of the same features with UCF. UNM has a slightly lower edge-tonode ratio of .83, but a higher maximum in-degree of 4. The maximum in-degree is once
again Circuit Analysis and the maximum out-degree is Calculus 2. This becomes a little
more interesting due to the student data that has been collected. On top of the HERI type
student data collected, there is also class level data. We can now look at the pass rates for
these classes to determine if they truly are bottlenecks in the curriculum.
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Figure 10: University of New Mexico Electrical Engineering Curriculum Graph.
University # of
hours
University
of
Houston
University
of New
Mexico
Arizona
State
University
University
of Central
Florida
Max
Outdegree
6
Longest Bottlenecks
Path
131
Edges/node Max
Indegree
1.48
7
9
10
128
.83
4
6
7
2
120
.83
3
4
6
2
128
.9
2
6
8
2
Table 2: Summary of calculated results for the four universities.
From the class level data, approximately 32% of students fail to earn credit for
Calculus 2. This particular data did not come with major information so it is not possible
to tell the percentage of electrical engineering students who failed the course. However,
22
since circuit analysis is an electrical engineering course, we are able to relate the failure
percentage to the curriculum. Approximately 9% of students failed to receive credit for
this class while another 21% received a grade in the C range. This is a low failure rate
meaning that although this class is considered a bottleneck by our graph definition, it is
unlikely that it is affecting student success and graduation rate on its own. It is, however,
part of a prerequisite chain of classes where the failure rates will have a partial
compounding effect on each other. If we had the electrical engineering outcomes for the
math classes early in the program, an educated guess for the graduation rate could be
made by following the failure rates along the longest path.
3.2 Graph Difficulty Through Weighting
Knowing information, such as pass rate, about each node can allow for the creation of
a different kind of graph. Graph theory allows us the ability to assign weights to nodes,
edges, or both. This also affords us the ability to create our own difficulty metric for
weighting the nodes in our graph. We propose a metric based on a combination of a
nodes pass/fail rate and the graph metrics discussed earlier in the chapter. Weighting the
nodes instead of the edges allows us to incorporate isolated nodes, which would not be
considered if only the edges were weighted. In many networks, the weighting affects the
flow through a graph, with the path of least resistance being the choice route. However,
in curriculum graphs, a student has the ability to be at more than one node at a time, and
must reach every node in order to graduate. This means our weighting scheme will have
a cumulative effect and the combination of weights can create a curriculum wide metric
for difficulty/complexity.
23
Each node has a pass/fail rate, representing the difficulty of that node separate from
the other structures in the graph. The in-degree of a node and the path length to a node
represent the complexity of reaching the node in a graph. The in-degree represents the
number of immediate prerequisites needed while the path length represents a prerequisite
chain needed to reach a certain node. The following simple formula calculates the
difficulty per node
Where dm refers to the difficulty metric, fr is the fail rate for the node, in is the in-degree,
and pl is the path length. Path length is calculated as the maximum number of path edges
tracing back from an origin node (node with 0 in-degree). The maximum in-degree and
maximum path length are derived from the UNM graph and are used to normalize the
feature values between 0 and 1 (for UNM). All other schools will be compared to UNM,
and thus, will not necessarily be normalized between 0 and 1. These features are then
multiplied by a scaling factor of .15 to bring these values in to line with typical class
failure rates so as not to dominate the formula. The theoretical maximum value for a
UNM node would be 1.3 if the class was the last class in the longest path, had the
maximum in-degree, and a 100% fail rate. Isolated nodes will only have the contribution
of their fail rate. Figure 11 shows the node metrics for the UNM electrical engineering
curriculum along with the summation of these values to create the overall curriculum
difficulty/complexity metric.
24
Figure 11: UNM Electrical Engineering curriculum with calculated metric for each node and the summed
total for the whole graph.
The highest node value was .39 for calculus 2 due mainly to a high fail rate (33.5%).
One would expect the numbers to increase as you moved lower in the graph; however,
the fail rate for most of the upper level engineering classes is very low, which
compensates for being farther along a path. Summarizing the graph values, early math
and science classes typically have high fail rates (20-35%) while engineering classes had
much lower rates (0-12%). One class had a value of 0 (Senior design 1), although the
sample size for calculating engineering fail rates was much smaller than core, math, and
sciences classes that are shared among many majors. The total value for the curriculum
is 7.68, which comes out to a value of .19 per node. Table 3 shows the values calculated
for the 3 other electrical engineering programs. Since class level data was not available
25
for the other schools, the fail rates were dropped to make a fair comparison. This gives a
value of 0 to isolated nodes and origin nodes.
EE program
Total Curriculum Difficulty
University of New Mexico
2.623
University of Houston
4.602
Arizona State University
2.504
University of Central Florida
3.491
Table 3: Difficulty metrics calculated for each electrical engineering program
Without taking into account node fail rates, we can see that the UNM curriculum is
much closer in difficulty to ASU than it is to UCF. This is interesting because of UCF’s
lower maximum in-degree of 2. This is apparently offset by the total number of edges in
the graph. UCF has an edge-to-node ratio close to .1 higher than UNM. When excluding
maximum out-degree, ASU and UNM are much more similar, making differences in
graduation rate more likely based on acceptance criteria and the student body make-up.
Creating a simpler curriculum could still help student success metrics at UNM going
forward.
26
Chapter 4 – Discussion
4.1 Proposed Tool
The curriculum analysis metrics proposed provide a tool for university faculty and
administrators, giving them the ability to quickly analyze the complexity of a curriculum
and have information on hand to help guide decisions going forward. From the graphs
created from other institutions, comparisons can be made to see what classes they are
providing, what prerequisites they have, and possible areas for change at UNM. Since
other Universities do not have their curriculum information in a set format, each graph
must be made individually. While this is not difficult, it does require a working
knowledge of Ruby and the ruby graph library. For curriculum graphs from UNM, we
have created the ability to pull information straight from our website into graph form
automatically. The code for this also saves an image of the graph and prints the
corresponding features.
4.2 Limitations
Due to the nature of this study, there are inherent limitations to what we could do and
what we could prove. While universities are required by law to release their graduation
rates, they are not required to break this down by major. Without graduation rates from
other institutions’ electrical engineering departments, it is difficult to show
mathematically that the features we are using correlate to graduation rates. Acquiring
these from the schools studied and perhaps several more universities, we could create a
regression model that could tell us the amount of variance, if any, these features were
explaining. At this time, the only information we have from these schools is the average
number of hours engineering students graduate with. From our small sample size, this
27
seems to correlate with the features we are calculating. If graduation rates become
available in the future, this should be revisited. There would also be an issue with that
data if it were to become available. Reporting of graduation rates by department would
depend on when students officially enter a program. At some universities, students
declare a major as soon as they enter. In others, students must take prerequisites before
they are allowed into a program. This would widely affect reported graduation rates
since many dropouts occur early on in a program, often before a student declares a major.
Care would need to be taken to find institutions that report in the same manner. Without
that data, the metric for curricular difficulty/complexity was built from weighting our
original digraphs using our features and student data.
4.3 Curriculum Changes
One way to see the possible effect of curricular difficulty is to perform a longitudinal
study on a changing curriculum. The Computer Engineering program at UNM has
proposed changes to reduce the number of hours by eliminated and combining classes.
The program would go from 128 hours to 120. Figure 12 shows the original curriculum
and figure 13 shows the updated curriculum. Table 4 shows the differences in the graph
metrics that were calculated.
28
Figure 12: The University of New Mexico Computer Engineering Department's current curriculum graph.
Figure 13: The University of New Mexico Computer Engineering Department's proposed 120 hour
curriculum graph.
The differences come from the reduction of two classes and the merging of two others.
The classes being removed are a technical elective, which has no connectivity in the
29
graph, and the Introduction to Computer Architecture class, which previously had a
degree of 4. Also, Applied Ordinary Differential Equations is being combined with
Linear Algebra to create one 4-hour class from two 3 hour classes.
The original curriculum has an edge to node ratio of 1.0(41 edges, 41 nodes), a
maximum in-degree of 5 (Circuit Analysis 1), a maximum out-degree of 6 (Calculus 2),
and the longest path for this curriculum is 8. The new curriculum lowers the edge to
node ratio to .895 (34 edges, 38 nodes), the maximum out-degree to 5, and the longest
path to 7. The maximum in-degree is the only major feature that stays the same.
Program
# of
hours
Edges/node
Max
Indegree
Max
Outdegree
Longest
Path
Bottle
necks
Diff.
Metric
CompE_128
128
1.0
5
6
8
3
8.011
CompE_120
120
.895
5
5
7
2
6.992
Table 4: Comparison of features between the computer engineering curriculum and the proposed, 120-hour
computer engineering curriculum.
The difficulty metric was also calculated between the two curricula, with the new
program lowering this metric by a full point. The fail rate for the new math class was
taken as a combination of the fail rates of the two classes it is replacing.
The Electrical Engineering program has also proposed changes to their curriculum.
The changes include removing Chemistry from the program and using the before
mentioned combination Linear Algebra/Differential equations class. Figure 14 shows the
new graph for the proposed curriculum and Table 5 shows the calculated metrics for each
program.
30
Figure 14: Proposed curriculum for Electrical Engineering with 120 hours.
Program
# of
hours
Edges/node
Max
Indegree
Max
Outdegree
Longest
Path
Bottle
necks
Diff.
Metric
EE_128
128
.83
4
6
7
2
7.68
EE_120
120
.77
4
4
6
2
6.75
Table 5: Features and difficulty metric calculated between UNM's EE curriculum and proposed curriculum.
The proposed curriculum lowers the difficulty metric by close to a point. It also lowers
the longest path by 1 and the maximum out-degree by 2. The edge-to-node degree of .77
is the lowest of all the curricula studied.
Each of the proposed curricula for Computer and Electrical Engineering give students
a better chance of graduating with the fewest number of hours. They have done this by
eliminating redundant information and combining classes to teach only what is necessary
to understand upper level engineering classes. The lower connectivity of the graphs
31
should benefit students who may fail a class, but won’t be set back a whole semester or
year. The fail rate of important classes can also be looked at to lower the difficulty
metric and raise student success.
32
APPENDIX A – CODE
Example Code for creating an rgl graph – University of Houston
require 'rgl/adjacency'
require 'rgl/dot'
require 'rgl/base'
require 'rgl/bidirectional'
require 'rgl/condensation'
require 'rgl/connected_components'
require 'rgl/enumerable_ext'
require 'rgl/graphxml'
require 'rgl/implicit'
require 'rgl/mutable'
require 'rgl/rdot'
require 'rgl/topsort'
require 'rgl/transitiv_closure'
require 'rgl/transitivity'
require 'rgl/traversal'
$g = RGL::DirectedAdjacencyGraph[]
$g.add_vertex 'HIST 1377 - US to 1877'
$g.add_vertex 'POLS 1336 - US & TX Constitutions'
$g.add_vertex 'ENGL 1303 - First Year Writing 1'
$g.add_vertex 'ECE 1100 - Intro to ECE'
$g.add_vertex 'MATH 1431 - Calculus 1'
$g.add_vertex 'CHEM 1372 - Chem for Engr'
$g.add_vertex 'CHEM 1117 - Chem for Engr Lab'
$g.add_vertex 'HIST 1378 - US since 1877'
$g.add_vertex 'ENGL 1304 - First Year Writing 2'
$g.add_vertex 'ECE 1331 - Comp. & Prob. solv.'
$g.add_vertex 'MATH 1432 - Calculus 2'
$g.add_vertex 'PHYS 1321 - Univ. Phys. 1'
$g.add_vertex 'POLS 1337 US Govt.'
$g.add_vertex 'ECE 2100 - Circuits Lab'
$g.add_vertex 'ECE 2300 - Circuit Analysis'
$g.add_vertex 'MATH 2433 - Calculus 3'
$g.add_vertex 'PHYS 1322 - Univ. Phys. 2'
$g.add_vertex 'MATH 3321 - Engineering Math'
$g.add_vertex 'Visual and Perf. Arts Core'
$g.add_vertex 'ENGI 2304 - Technical Comm'
$g.add_vertex 'ECE 3331 - Prog. Applic. in ECE'
$g.add_vertex 'ECE 3337 - EE Analysis'
$g.add_vertex 'ECE 2317 - Applied Elec. & Magnetism'
$g.add_vertex 'Humanities Core'
$g.add_vertex 'ECE 3441 - Digital Logic Design'
$g.add_vertex 'ECE 3155 - Electronics Lab'
33
$g.add_vertex 'ECE 3355 - Electronics'
$g.add_vertex 'ECE 3364 - Circuits & Systems'
$g.add_vertex 'ECE 3317 - Applied EM Waves'
$g.add_vertex 'ECE 4436 - Microprocessors'
$g.add_vertex 'ECE Elective 1'
$g.add_vertex 'ECE 4339 - Solid State Devices and Lab'
$g.add_vertex 'INDE 2333 - Engr Statistics'
$g.add_vertex 'ENGI 2334 - Thermo'
$g.add_vertex 'ECON 2304 - Microecon. Principles'
$g.add_vertex 'ECE 4335 - ECE Sys Design 1'
$g.add_vertex 'ECE Elective and Lab 1'
$g.add_vertex 'ECE Elective 2'
$g.add_vertex 'MECE 3400 - Intro to Mechanics'
$g.add_vertex 'ECE 4336 - ECE Sys. Design 2'
$g.add_vertex 'ECE Elective and Lab 2'
$g.add_vertex 'ECE Elective 3'
$g.add_vertex 'ECE Elective and Lab 3'
$g.add_edge 'ENGL 1303 - First Year Writing 1','ENGL 1304 - First Year Writing 2'
$g.add_edge 'MATH 1431 - Calculus 1','MATH 1432 - Calculus 2'
$g.add_edge 'MATH 1431 - Calculus 1','PHYS 1321 - Univ. Phys. 1'
$g.add_edge 'ECE 1100 - Intro to ECE','ECE 2300 - Circuit Analysis'
$g.add_edge 'MATH 1431 - Calculus 1','ECE 1331 - Comp. & Prob. solv.'
$g.add_edge 'CHEM 1372 - Chem for Engr','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'CHEM 1372 - Chem for Engr','ENGI 2334 - Thermo'
$g.add_edge 'ENGL 1304 - First Year Writing 2','ENGI 2304 - Technical Comm'
$g.add_edge 'ECE 1331 - Comp. & Prob. solv.','ENGI 2304 - Technical Comm'
$g.add_edge 'ECE 1331 - Comp. & Prob. solv.','ECE 3331 - Prog. Applic. in ECE'
$g.add_edge 'ENGL 1304 - First Year Writing 2','ECE 2300 - Circuit Analysis'
$g.add_edge 'ECE 1331 - Comp. & Prob. solv.','ECE 2300 - Circuit Analysis'
$g.add_edge 'ECE 1331 - Comp. & Prob. solv.','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'MATH 1432 - Calculus 2','ECE 2300 - Circuit Analysis'
$g.add_edge 'MATH 1432 - Calculus 2','MATH 2433 - Calculus 3'
$g.add_edge 'MATH 1432 - Calculus 2','INDE 2333 - Engr Statistics'
$g.add_edge 'MATH 1432 - Calculus 2','PHYS 1322 - Univ. Phys. 2'
$g.add_edge 'MATH 1432 - Calculus 2','MATH 3321 - Engineering Math'
$g.add_edge 'PHYS 1321 - Univ. Phys. 1','PHYS 1322 - Univ. Phys. 2'
$g.add_edge 'PHYS 1321 - Univ. Phys. 1','MECE 3400 - Intro to Mechanics'
$g.add_edge 'ECE 2100 - Circuits Lab','ECE 3441 - Digital Logic Design'
$g.add_edge 'ECE 2100 - Circuits Lab','ECE 3355 - Electronics'
$g.add_edge 'ECE 2300 - Circuit Analysis','ECE 3331 - Prog. Applic. in ECE'
$g.add_edge 'ECE 2300 - Circuit Analysis','ECE 3441 - Digital Logic Design'
$g.add_edge 'ECE 2300 - Circuit Analysis','ECE 3337 - EE Analysis'
$g.add_edge 'MATH 2433 - Calculus 3','ENGI 2334 - Thermo'
$g.add_edge 'MATH 2433 - Calculus 3','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'PHYS 1322 - Univ. Phys. 2','ENGI 2334 - Thermo'
$g.add_edge 'PHYS 1322 - Univ. Phys. 2','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'MATH 3321 - Engineering Math','ECE 3331 - Prog. Applic. in ECE'
34
$g.add_edge 'MATH 3321 - Engineering Math','ECE 3337 - EE Analysis'
$g.add_edge 'CHEM 1117 - Chem for Engr Lab','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'ENGI 2304 - Technical Comm','ECE 3355 - Electronics'
$g.add_edge 'ENGI 2304 - Technical Comm','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECE 3331 - Prog. Applic. in ECE','ECE 4436 - Microprocessors'
$g.add_edge 'ECE 3337 - EE Analysis','ECE 3355 - Electronics'
$g.add_edge 'ECE 2317 - Applied Elec. & Magnetism','ECE 3364 - Circuits & Systems'
$g.add_edge 'ECE 2317 - Applied Elec. & Magnetism','ECE 3355 - Electronics'
$g.add_edge 'ECE 2317 - Applied Elec. & Magnetism','ECE 3441 - Digital Logic Design'
$g.add_edge 'ECE 2317 - Applied Elec. & Magnetism','ECE 3317 - Applied EM Waves'
$g.add_edge 'ECE 3441 - Digital Logic Design','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECE 3155 - Electronics Lab','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECE 3155 - Electronics Lab','ECE 4339 - Solid State Devices'
$g.add_edge 'ECE 3355 - Electronics','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECE 3355 - Electronics','ECE 4339 - Solid State Devices'
$g.add_edge 'INDE 2333 - Engr Statistics','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECE 4335 - ECE Sys Design 1','ECE 4336 - ECE Sys. Design 2'
$g.add_edge 'MATH 1431 - Calculus 1','ECE 1100 - Intro to ECE'
$g.add_edge 'CHEM 1372 - Chem for Engr','CHEM 1117 - Chem for Engr Lab'
$g.add_edge 'ECE 1100 - Intro to ECE','ECE 1331 - Comp. & Prob. solv.'
$g.add_edge 'MATH 1432 - Calculus 2','PHYS 1321 - Univ. Phys. 1'
$g.add_edge 'ECE 2300 - Circuit Analysis','ECE 2100 - Circuits Lab'
$g.add_edge 'MATH 2433 - Calculus 3','ECE 2300 - Circuit Analysis'
$g.add_edge 'MATH 2433 - Calculus 3','PHYS 1322 - Univ. Phys. 2'
$g.add_edge 'PHYS 1322 - Univ. Phys. 2','ECE 2300 - Circuit Analysis'
$g.add_edge 'MATH 3321 - Engineering Math','ECE 2300 - Circuit Analysis'
$g.add_edge 'MATH 3321 - Engineering Math','ECE 2317 - Applied Elec. & Magnetism'
$g.add_edge 'MATH 3321 - Engineering Math','MECE 3400 - Intro to Mechanics'
$g.add_edge 'ECE 3337 - EE Analysis','ECE 3317 - Applied EM Waves'
$g.add_edge 'ECE 3337 - EE Analysis','ECE 3364 - Circuits & Systems'
$g.add_edge 'ECE 2100 - Circuits Lab','ECE 3364 - Circuits & Systems'
$g.add_edge 'ECE 2317 - Applied Elec. & Magnetism','ECE 3337 - EE Analysis'
$g.add_edge 'ECE 3441 - Digital Logic Design','ECE 4436 - Microprocessors'
$g.add_edge 'ECE 4436 - Microprocessors','ECE 4335 - ECE Sys Design 1'
$g.add_edge 'ECON 2304 - Microecon. Principles','ECE 4335 - ECE Sys Design 1'
#Graph length
puts $g.num_vertices
puts $g.num_edges
#Calculate the number of edges per vertice
puts Float($g.num_edges)/Float($g.num_vertices)
#Calculate the max out_degree of the graph
a = []
for i in 0..$g.vertices.size-1 do
a[i] = $g.out_degree($g.vertices[i])
35
end
puts a.max
#Calculate the max in_degree of the graph
r = $g.reverse
b = []
for i in 0..r.vertices.size-1 do
b[i] = r.out_degree(r.vertices[i])
end
puts b.max
$g.write_to_graphic_file('jpg')
36
Example code for pulling a graph from the UNM degree site
require 'open-uri'
require 'json'
require 'active_support/all'
url_address = "http://unm-wild-weasel.herokuapp.com/undergrad_programs/170.json"
output_file = "parser_output.dat"
output = File.open( output_file, "a" )
open(url_address) do |f|
json_string = f.read
parsed_json = JSON.parse(json_string)
parsed_json['degree_requirements'].each do |dr|
output <<
"(#{dr['name']})"
unless dr['prerequisites'].size == 0
dr['prerequisites'].each do |p|
output << "(#{p['name']} , #{dr['name']})"
end
end
end
end
output.close
data = open('parser_output.dat', &:read)
#Store all vertices in a vector
$v = data.scan(/\([^,)(]*\)/).flatten
#Store all edges in a vector
$e = data.scan(/[^()]*,[^)]*\)/).flatten
$g = RGL::DirectedAdjacencyGraph[]
#Transfer vertices to graph form
for i in 0..$v.size-1 do
$v[i].slice! ")"
$v[i].slice!(0)
$v[i].strip!
$g.add_vertex($v[i])
end
#Transfer edges to graph form
for j in 0..$e.size-1 do
$e[j].slice! ")"
$q1 = $e[j].split(',')
37
$q1.flatten
$q1[1].slice!(0)
$q1[0].strip!
$q1[1].strip!
$g.add_edge($q1[0],$q1[1])
puts $q1[0]
puts $q1[1]
end
puts $g.num_vertices
#Graph length
puts $g.num_edges
#Calculate the number of edges per vertice
puts Float($g.num_edges)/Float($g.num_vertices)
#Calculate the max out_degree of the graph
a = []
for i in 0..$g.vertices.size-1 do
a[i] = $g.out_degree($g.vertices[i])
end
puts a.max
#Calculate the max in_degree of the graph
r = $g.reverse
b = []
for i in 0..r.vertices.size-1 do
b[i] = r.out_degree(r.vertices[i])
end
puts b.max
$a = $g.topsort_iterator.to_a
$g.write_to_graphic_file('jpg')
38
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