An Application of the PageRank Algorithm to NCAA Football Team Rankings

University of Mississippi University of Mississippi

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An Application of the PageRank Algorithm to NCAA Football An Application of the PageRank Algorithm to NCAA Football

Team Rankings Team Rankings

Morgan Majors

University of Mississippi

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An Application of the Pagerank Algorithm to NCAA

Football Team Rankings

Morgan Majors

Abstract

We investigate the use of Google’s PageRank algorithm to rank sports teams. The

PageRank algorithm is used in web searches to return a list of the websites that are of most

interest to the user. The structure of the NCAA FBS football schedule is used to construct

anetworkwithasimilarstructuretotheworldwideweb. Parallelsaredrawnbetweenpages

that are linked in the world wide web with the results of a contest between two sports teams.

The teams under considerat ion here are the members of the 2021 Football Bowl Subdivision.

We achieve a total ordering of th e 2021 FBS teams by applying the PageRank algorithm

to the results of the regular and bowl seasons. A statistical method of correlation is used

to compare the ﬁnal AP ranki n gs with PageRank models based on Margin of Victory and

Total Points Scored.

Dedication

I dedicate my thesis work to my parents, Penn and Ashley Majors. Their constant

support has propelled me to t h e ﬁnish line of thi s experience. They have stood by my side

since day one, and withou t them, I would not be the person I am t oday. The immense love

Ifeelfromthemdailyistrulyoverwhelming.

Aspecialfeelingofgratitudetomyfather,whoseloveformathematicsandsportsIinher-

ited. Some of my favorite memories have been alongside my father at a multitude of Ole Miss

sporting event s. The long nights spent at the dining roo m table while in elementary school

learning basic mathematics are still persistent m em ori e s. His encouragement throughout all

levels of my education has undoubtedly helped me to pursue my ambitions.

Acknowledgements

Iwishtothankmycommitteemembersfortheirgeneroustimeandexpertisespent

helping me make my thesis experience the best it could be. A special thanks to Ja m es

Reid who was always patient, knowledgeable, and encouragi n g when I needed assistance and

advice. This project could not have been completed without him. Thank you to Dr. Reid,

Dr. Sh ep p a r d so n , and Dr. Wilder for agreei n g to serve on my committee.

I would also like to acknowledge my family for the immense love they have always provided

me. My parents, brother, and grandparents have always shown sincere support toward my

academic goals and me. A special thanks to my brother, Penn Earl Majors, for always

pushing me to be a better version of myself. He has be en a constant pillar of support i n my

life. I also would like to thank my grandfather, Penn Majors, who inspired me to pursue

mathematics. As a former mathematics teacher, his wisdom always encouraged me to follow

in his footsteps. My grandmothers, Trudy Bomer and Mary Majors, have been my ultimate

encouragers throu gh out this journey.

I would like to thank everyone who has been i nvolved in this process. All of my former

teachers and professors have played a hand in where I am today. I am eternally grateful to

the Sally McDonnell Barksdale Honors College for the opportuniti es they have provided.

III

Contents

1 Introduction 2

1.1 College Football Rankings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The PageRank Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 NCAA 2021 Football Season . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematics of PageRank 14

2.1 A Graph Model of College Football . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Matrix Algebra and Ei g envectors . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Modiﬁe d PageRank Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Kendall Rank Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Results 23

3.1 Margin of Victory Rankings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Total Score Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Appendix 34

Chapter 1

Introduction

College football is a proﬁtable business with ni net een teams reportin g at least 100 million

dollars in Revenue in 2019 [8]. Hence i t is of intense interest to the Universities participatin g

in College Football t o achieve an accurate ranking of their perform an ce . This thesis considers

an approach to r a nki n g these teams that uses the Google PageRank algorithm for ranking

webpages. The ranking of nodes in networks and sports teams by an eigenv alue approach

is an active area of research (see, for example, [7, 9, 12, 18, 19, 20, 24, 27, 36, 38]). The

use of these ranking method s here is motivated by the 2008 doctoral dis ser t at i o n of Anjela

Yuryevna Govan [19].

We next discuss the organization of the pap er. In Chapter 1 we discuss the history of

College Football rank in g m et h ods. We then discuss the PageRank algorithm of Sergey Brin

and Larry Page that brought organiz at i o n to the world wide web. We next discuss the

widespread ap p l i cati on s of the al gori t hm to areas of academic and scientiﬁc i nterest. We

provide background on the NCAA 2021 Football Bowl Subdivision teams used in this study

to conclude Chapter 1.

In Chapter 2 we give the mathematical background of the results presented here. We

begin with the notion of the Wor l d Wide Web as a network or graph. We then discuss the

matrix theory from mathematics used to produce ranking eigenvectors by our algorithm.

Then the mathematical fo r mulation of the PageRank algori t h m is given. We discuss the

modiﬁcation of the PageRank al g or i t h m which will be used to produce our results to co n cl u d e

Chapter 2.

In Chapter 3, we use the modiﬁed PageRank algorithm to rank the results of teams in the

2021 FBS College Football season using two methods. The ﬁrst method takes into account

the Margin of Victory results of each team to compute its ranking. The second method is

based on the Total Score of each team in each of its games to produce its ranking. We then

compare the Margin of Victory and Total Score rankings with the ﬁnal AP rankings for that

season. Finally, we make some conclusions and observatio n s about the eﬃciency of these

ranking methods.

1.1 College Football Rankings

There are multiple ranking methods for College Football Teams that determine the order

of teams from b est to worst based on results. These ranking methods are important in

determining which teams will play in the most prestigious bowls. The most notable rank-

ing polls are the Associated Press, USA Today’s Coaches, Bowl Championship Series, and

College Football Playo↵ Polls. The ranking of college football teams began in 1936 with the

release of the ﬁrst Associated Press (AP) poll [29]. The ﬁr st AP Poll included only the top

twenty teams, with Minnesota ranking at number one [11]. College football gained national

attention at the time of t h i s ﬁrst poll. Foo t b al l was created in the northeast United States

and was initia l l y dominated by Ivy L eag u e universities. These schools not on l y dominated

the world of football physically, but the rules committee was also made up of Ivy League

alumni, coaches, and players. These schools were considered to have played a superior brand

of football. However, the ﬁrst AP Poll illustrated how teams from other areas of the county

were now superior to the Ivy League teams. Minnesota, a Midwestern team, ﬁnished in the

one spot, with schools from the Deep South, West, and Great Plains ﬁnishing in the top ﬁve

places in the ﬁrst poll [11].

The USA Today Coaches Poll that d eb u t ed in the 1950 season ranked t h e top 20 NCAA

football and b asketball teams. In 1990 the Coaches Poll conformed to the method of the

other p ol l s by ranking the top 2 5 teams. Until the 1973 season, the poll was released only

in December, after the conclusion of the regular season but before the bowl games began.

AscandalregardingcontroversialvotingpracticeswithintheCoachesPollarosein2005,

resulting in ESPN d ro p p i n g its sponsorship of this poll [1]. Th e Coaches Poll became the

USA Today Coaches poll in 2005 [1].

The Bowl Championship Series (BCS) Poll debuted in 1998 [ 29] . Th e BCS Poll was a

revolutionary idea that propelled the popula r i ty of college football. Prior to the BCS Poll,

the idea of a true championship game between the top two teams was uncommon [32]. This

was t h e reason for the creation of the BCS Poll. After naming two teams as Co-Nati on a l

Champions in back-to-back years in the 1990 and 1991 seasons, college football fans wanted

clarity on a true champion. Some e↵orts were made through organizations such as the

Coalition, the Alliance, and the Super Alliance to construct a number 1 versus number 2

National Championship matchup, but none of these organi za t i on s could get every conference

on board. The year of the ﬁrst BCS did exactly what it was created to do, create the matchup

between the two best teams in the country. However, issues still arose such as when the BCS

poll claimed LSU as the 2003 Na t i on a l Champions, but the AP claimed USC as the best

team of the season. These issues led to the birth of the College Football Playo↵. The BCS

Poll bridged the gap between the poll era and the College F ootball Playo↵ Poll [32].

All of the polls d i ↵ er in their ranking strategies, but th e AP, Coaches, and BCS Polls

have one thing in common: the method they use to rank the teams. All of these polls

use a committee, ranging in size, to construct a ranking. This is done by each committee

member casting votes for what team should be placed at 1, and so forth. The team that the

committee member thinks should rank at 1 recei ves 25 points, the 2 spot receives 24 points,

and so on until the team at 25 receives 1 point. Each poll releases a new ranking of teams

1-25 weekly. Today, the AP Poll committee i s made of 63 members, the Coaches Poll is 62,

BCS is 62, and CFP is 13. The CFP Poll di↵ers from the rest because it does not construct

a ranking until well into the season [31].

The ﬁ r st Co l l eg e Football Playo↵ (CFP) Poll was released in 2014 [29]. While other

polls simp l y rank team s for the audience’s en g ag em ent, the CFP Poll has a great deal of

meaning. This poll sin g l e-h a n d ed l y chooses the top four teams that will play for the National

Championship. Another re sponsibility of the CFP Poll is to place teams that did not receive

an invitation to the playo↵s into the New Year’s Bowls. Six games occur on New Year’s Eve

and New Year’s Day. These games are th e Cotton Bowl, Fiesta Bowl, Orange Bowl, Peach

Bowl, Rose Bowl Game, and Sugar Bowl. The two Playo↵ Semiﬁnal games rotate annually

through two of these games [2].

The ranking of college football teams is importa nt for the advancement of the sport and

its nationwide recognition. The top 4 teams in the College Football Playo↵ Poll receive the

chance to compete for a National Champion s h i p . The team ranked at 1 plays against the 4-

ranked team, and the team ranked at 2 plays against the 3-ranked team. The winners of the

two semiﬁnal games face o↵ against one another i n the ﬁnal Championship game. The 2022

National Championship was played on Janu ar y 10, 2022, between the Georgia Bulldogs and

the Alabama Crimson Tide. The ﬁr st National Title by the Georgia Bulldogs since 1980 was

watched by 24.5 million viewers. [10]. This was the most-watched non-NFL sp or t i n g event

in two years, with a 19 percent increase in viewers since the 2021 National Ch a m p i on s h i p

[10]. Along with the national attention that a team receives from being selected into the

College Football Playo↵s, teams receive large payou ts for their selection. The four teams

selected into playo↵s receive a 6 million dol l a r payout from the CFB R evenue Pool[4]. No

additional payout occurs for being one of the last two remaining teams, but all expenses are

covered [4]. Simi l a r l y, popular bowls have a l a r ge payout [4]. With the ﬁnancial implications

of rankings to college football teams, interest in their methodology continues to grow.

1.2 The Pa ge Ra nk Al go ri thm

The origins of the met h ods considered here can be found in the desire to ﬁnd order in the

World Wide Web. The need for such an order arose in the late 20th century. There was

an explosion in the number of webpages that occurred during this time. For example, in

the years 1995, 1996, and 1997 the number of webpages increased from 23, 500 to 250, 601

to 1, 117, 255 (see https://www.internetlivestats.com/total-number-of-websites for

alivecountofthenumberofpagesontheweb,currentlycloseto2billion). SergeyBrin

and Larry Page approached the problem of organizing the web by developing a web search

engine in 1995. They call their web search engine Google. With this Google web search

engine, they sought to place link s to webpages of the greatest interest at the top of a search

query. Brin and Page were computer science doctoral students at Stanford University at

the time (see [22, Chapter 3]). They used their dorm rooms as oﬃces for their business. A

tech report from 1998 [9] announced the development of the search engine. Google’s parent

company Al p h abet is currently valued at over $1.4 trillion [13].

The basis for the Google web search engine is the PageRank algori t h m . The World Wide

Web is organized by hyperlinks. Selecting a hyperlink to webpage Y that appears on webpage

X will send the reader to webpage Y. Each webpage typically contains links to several other

webpages. A webpage will be considered important if a link to it appears on other important

webpages. This reasoning appears to be circular because two webpages might be considered

to be important if each says the other is import ant. However, we show in Chapter 2 that

applying mathematical tools to this reasoning leads to a well-deﬁned ranking of webpages

and therefore of sports teams.

We may think of the web as a network of webpages with links between some of the pages.

Auserofthewebcanbethoughtofas“surﬁng”betweenthesepages.Thenetworkevolves

as the user at a particular page has certain probabilities of traveling to other linked pages.

Markov’s Theory is useful in describing the evolution of such a network. This theory can

be traced back to 19 06 with the work of the Russian mathemat i ci a n A.A. Markov [17]. A

Markov process describes a sequence of possibl e events where the probability of e ach event

only d epends on the state of the pr ev i ous. In order to assign ranks t o webpages with the

ranks being real numbers versus complex numbers, Brin and Page needed a result from

Linear Algebra of Perron [ 3 0] and Perron and Frobenius (see [22, p. 172]). It is important

to assign a real number rank to a web page as the set of real numb ers is ordered, e.g. the

number 0.3 is larger than the number 0.2. The set of complex numbers is not ordered. For

example, there is no way to know whether a complex number such as 1+3i is lar ger than

another complex number, so it is important to obtain ranking numbers that are positive real

numbers.

The Power Method [26], developed by von Mises and Pollaczek-Geiringer, was a key piece

of the puzzl e necessary to make the large matrix co m p u t at i o n s needed to rank webpages using

PageRank. Brin and Page needed to compute eigenvectors of large matrices (see Section 2.2)

to provide their rankings and adaptations of this method of updating ranks iteration by

iteration stabilized on well-deﬁned ranking eigenvecto r s in a computable number o f steps

[22, Chapter 9]. All of these too l s were synthesized by Brin and Page in 1998 to create the

revolutionary PageRank algorithm. At the heart of this algorithm is the beautiful equation

given in 1.2.1.

⇡

= ⇡

(↵S +(1 ↵)E)(1.2.1)

Some consid er this equation to be as important as equations fr o m the theory of relativi ty

such as E = mc

of Einstein and the Energy Wave Equation E = hf suggested by Planck.

Graham Farmelo [16] edited a book of beauti ful equations entitled ”It Must Be Beautiful:

Great Equations of Modern Science” t h at contains Einstein’s and Planck’s equations as well

as Equation 1.2.1 of Brin and Pa ge. We explain the derivation of this equation that un d er l i es

the organization of the world wide web in Section 2.3.

An important alternative algorithm to the PageRank algorithm is the Hypertext Induced

Topic Search algorithm, abbreviated HITS. The HITS algorithm was foun d ed in 1998 by Jon

Kleinberg. [22]. Thus this algori t h m is a contemporary of the PageRank algor i th m . The

HITS algorithm is a sy st em for ranking webpages while considering a webpage’s p o p u l ar i ty.

It is very simi lar to the PageRank algorithm except it relies on the qu er y to rank webpages

and produces two popularity scores for each web page whereas PageRank is not dependent on

the query and only produces one po p u l a ri ty score [28] . Also, HITS operates by associating

webpages with authorities and hubs. An authority is a webpage with many inlinks, and a hub

is a webpage with many outlinks [22]. Authorities and hubs have a d ependent relationship,

as good hubs result in good authorities and good authorities result in good hubs [28].

Methods of ranking webpages remain of consi d er ab l e interest. After t h e founding of

PageRank and HITS, less popular ranking methods were developed. A few years after the

success of PageRank and HITS, in 200 0, a Stochastic Approach to Link Structu r e Analysis

was developed [22]. This stochastic algorith m later became known as SALSA (Stochastic

Approach for Link-Structure Analysis) [15]. This approach has similari t i es to both PageRank

and HITS. It creates both hub and authority scores while being derived from Markov chains.

While it combines som e of the best features from both major algorithms, SALSA has one

major setback [15]. The major dependence on queries fro m SALSA is what keeps this

algorithm from being well-known. This is because the query-dependence results in a ranking

are not unique [22].

Anewwaveofrankingmethodswasintroducedaround2005throughmeta-searchengines

[22]. TraﬃcRank attempted to rank webpages by merging the results from several di↵erent

ranking algorithms [12]. It hoped to provide the most accurate and precise rankings through

the use of a multi-link algorithm. TraﬃcRank encouraged the mindset that the World

Wide Web does not consist of webpages connected by a single road, inste ad , it is billi on s of

webpages connected by billions of connections [22].

Although the PageRank algorithm is primarily used in web-search engines, it is used in

many di↵erent ways throughout society. The PageRank algorithm has been manipulated

and modiﬁed to ﬁt a number of applications. A history of the development of the PageRank

algorithm and its uses are given in Table 1.1 (see [17]). We will discuss the applications of

this algorithm in the next section of the thesis.

Table 1.1: PageRank History

1906 Markov Markov th eor y

1907 Perron Perron theorem

1912 Frobenius Perron-Frobenius theorem

1929 von Mises and Pollaczek-Geiringer Power method

1941 Leontief Econometric model

1949 Seeley Sociometric model

1952 Wei Sport ranking model

1953 Katz Sociometric model

1965 Hubbell Sociometric model

1976 Pinksi and Narin Bibliometric model

1998 Kleinberg HITS

1998 Brin and Page PageRank

1.3 Applications

The PageRank algorithm is primar i l y associated with the Worl d Wide Web and the way

web pages are linked. However, the PageRank algorithm can be applied to many di↵erent

subjects. For example, the PageRank algorithm can be applied within t he ﬁelds of chemistry,

biology, literature, and social networks to organize the complex relationships between data.

In chemistry, PageRank is used to study molecules and their valences throu g h t he stu d y of

change within water molecules linked by hydrogen bonds. Ultimately, it is u se d to determi n e

if the molecules have a pot ential hydrogen bond to a solute molecule [18]. Teleportation

occurs to determine structural di↵erences. The development of PageRank within chemistry

has evolved into the abili ty to provide analytical derivatives and illustrati on s [39]. Through

di↵erent experiments, it can be determined which solvent rearrangement has the most impact

on the reaction pathway [39].

Common u ses of the PageRank algorit hm are in biology such as GeneRank [27], Protein-

Rank [36], and IsoRank[24]. All of these app l i c at i o n s study the dynamics of network data in

bioinformatics. They result in the sharing of locali ze d information about the graph produced

by the PageRank algorithm. The reactions between the genes, proteins, and isotopes are

transformed into interactions between nodes in the network, therefore allowing the PageR-

ank algorithm to be applied to rank objects in the network [18]. In a recent experiment, the

measurement of the graph centrality of the network associated with the PageRank algorithm

was used in the identiﬁcation of cancer genes [14]. Experi m ental data is used to determine

the accuracy of the centrality within biology [14].

A fasci n a ti n g aspect of the PageRank alg o ri t h m is that it is used in the world of literature

through BookRank [37]. Elliot Yates and Louise Dixon in [37] state that the BookRank

algorithm can sh ed light on “ ...the central questions of liter at u r e: What are the most

important books? Which story is most li kely to occur within a novel? And what should I read

next?” BookRank is used to catalog books on the web. This is useful for the organization

of the multitude of written works. The idea of the BookRank algorithm is similar to the use

of PageRank in Google, b u t book titles are exchanged for webpages in thi s algorithm [18].

PageRank is used in social networking [5] through BuddyRank and TwitterRank. The

people interacting on social media platforms act as the nodes, while the relationships formed

on social networking sites act as the edges or links between nodes. It can make predictions

about who an individual might connect with next and evaluate th e centrality of estimating

someone’s social status. Also, it can estimate the potential inﬂuence of a person’s opinion

of the social network. Both of these ranking programs typically use a standard alpha of .85

to perform a reverse application of PageRank [18].

PageRank allows endless possibiliti e s for its uses. Whether they are used in everyday life

or as crucial research opp ortunities, it is undeniable that PageRank has millions of possible

applications. Biology, chemistry, literature, and social networks are just the surface of what

PageRank can do. Although it is typically used in web search engines and mathematics,

PageRank can be used in any subject whose interactions can be model ed by a network.

1.4 NCAA 2021 Football Season

The 2021 NCAA Football S ea so n began with a game between Illinois and Nebraska pl ayed

on August 28, 2021, at 1:20 pm Eastern time an d concluded with a game between Georgia

and Alabama at 8:00 pm Eastern time on January 10, 2022 [3]. NCAA Divisi o n I Footba l l

is categor i zed into two divisions, the Football Bowl Subdivisio n (FBS) and the Football

Championship Subdivision (FCS)[6]. Our research is centered around the 130 teams of the

2021 NCAA FBS se aso n. There are an addit i o n al 128 NCAA FCS teams that participated

in the 2021 season. Game s are pri m a ri l y pl ayed between a pair of FBS teams and a pair of

FCS teams although a limited number of games are played between teams of di↵erent levels.

The 2021 FBS teams are organized into 11 conferences and 7 independent schools as

follows. The Atlantic Coast Con fer ence includes the Atlantic division of Wake Forrest,

Clemson, North Carolina State, Louisville, Flordia State, Boston College, Syr a cu se, and

the Coa st al division of Pitt, Miami (FL), Virgini a, Virginia Tech, North Carolina, Georgia

Tech, and Duke. The American Athletic Conference includes Cincinnati, H ou st on , UCF,

East Carolina, Tulsa, SMU, Memphis, Navy, Temple, South Florida, and Tulane. The

Big 1 2 Conference includes Baylor, Oklahoma State, Oklahoma, Iowa State, Kansas Sta t e,

West Virgini a, Texas Tech, Tex as , Texas Christian, and Kansas. The Big Ten Conference

includes the Ea st division of Michigan, Ohio State, Michigan State, Penn State, Maryland,

Rutgers, Indiana, and the West divi si o n of Iowa, Minnesota, Purdue, Wi sco n si n , Illinois,

Nebraska, and Northwestern. The Conference USA includes the East division of Western

Kentucky, Marshall, O l d Dominion, Middle Tennessee State, Charlotte, Florida Atlant i c,

Florida International, and the West d i v i si on o f UTSA, UA B, Nor th Texas, UTEP, Rice,

Louisiana Tech, an d Southern Mississippi. The Mid-American Conference includes the East

division of Kent Sta te , Miami (OH), Ohio, Bowling Green, Bu↵alo, Akron, and the West

division of Northern Illinois, Central Michigan, Toldeo, Western Michigan, Eastern Michigan,

and Ball State. The Mountain West Conference includes the Mountain division of Utah

State, Air Force, Boise State, Wyoming, Colorado State, and New Mexico, and the West

division of San Di ego State, Fresno State, Nevada, Hawaii, San Jose S t at e, and Nevada-

Las Vegas. The Pac-12 Conference includes the North division of Oregon, Washington

State, Oregon State, California, Washington, and St an fo r d , and the South division of Utah,

UCLA, Arizona State, Colorado, USC, and Arizona. The Southeastern Conference incl u d e s

the East division o f Georgia, Kentucky, Tennessee, South Carolina, Missouri, Florida, and

Vanderbilt, and the West division of Alabama, Ole Miss , Arkansa s, Texas A&M, Mis si ssi p p i

State, Auburn, and LSU. The S u n Belt Conference includes the East division of Appalachian

State, Coastal Carolina, Georgia State, Troy, and Georgia Southern, and the West division

of L o u i si a n a, Texas St at e, South Alaba m a, Louisian a -M on r oe, and Ar kansas State. The

Independent schools that are not in confer en ces are Notre Dame, New Mexico State, Liberty,

Massachusetts, Connecticut, Army, and BYU. [33]. Each team in a conference typically pl ays

most of its games against other teams in the same conference. Each team in the FBS typically

has one or two opponents in the FCS with the remaining games against other FBS team s.

Teams that are successful during the season with a winning record typically play in a Bowl

Game after the conclusion of the regular season. Since there are 130 FBS team s and most

teams play between 11 and 13 games, most pairs of teams do not play each other. This

observation is the reason that College Football Ranking systems are useful to compare pairs

of teams that do not play each other and the relative rankings of paired teams can be in

dispute.

Chapter 2

Mathematics of PageRank

We discuss the mathematical concepts used in this research in this chapter. The PageRank

algorithm uses an interesting mix of tools from a variety of mathematical disciplines. The

research uses models and c on ce p ts from Graph Theory, Matrix Algebra, a n d Statistics. An

understanding of the basic principles of all of these ﬁelds is necessary to develop the simple

premise of the PageRank Algorithm:

Axiom 2.1. The ranking of a team is proporti on al to th e sum of the r an ki n gs of the teams

that it defeats.

In order to examine the implications of Ax i o m 2.1 to team ranking we will begin with a

discussion of Graph Theory.

2.1 A Graph Model of College Football

Graph Theory is an area of mathemat i cal research that is concerned with mod el in g the

interactions between objects of interest in a subject using a graph. The Graph Theory

terminology used here can be found in (see [35]. A graph is a pair G =(V,E), where V is a

nonempty set called the vertex set of G and E is a set called the edge set. The elements of

Alabama

Mississippi

Auburn

TexasAM

Figure 2.1: A four team exa m p l e

E consist of pairs of vertices (the plural of vertex) of V .Agraphiscalleddirectedwhenthe

two vertices of each edge are given an order or equivalently a direction. Consider the di r ect ed

graph G =(V,E) whose diagram is given in Figure 2.1. The vertex set of thi s graph consists

of four elements labeled “Alabama”, “Auburn”, “ Mi ss i ssi p p i ” , and “TexasAM”. There are

edges between each distinct pair of vertices in G such as between the vertices labeled by

Alabama and Aubu r n . Further, the edges have directions indicated by arrows such as the

arrow that points from Auburn to Ala b am a . The graph G represents the results of all

the games played between the Uni versity of Alabama, Auburn University, th e University of

Mississippi, and Texas A&M University in the NCAA 2021 football season. T h e directed

edge from Auburn to Alabama in the gra p h G represents that Alabama defeated Auburn

during this season . Thus this small graph represents that Alabama defeat ed Auburn and

Mississippi, Auburn defeated Mississippi, Mississippi defeated Texas A&M, and Texas A&M

defeated both Auburn and Alabama. One may think of this as vertex Alabama voting for

vertex TexasAM, in the same way, a web page link “votes for” or “suggests” that the web

user visits the linked page.

Assign a vertex to each of the 130 NCAA FBS football teams from the 2021 season, and

Cincinnati

MiamiOH

MurrayState

Indiana

NotreDame

Temple

UCF

Navy

Tulane

Tulsa

SouthFlorida

SMU

EastCarolina

Houston

Marshall

CharlestonSouthern

Memphis

Rice

Grambling

Connecticut

Auburn

NichollsState

ArkansasState

MississippiState

Army

AbileneChristian

NorthTexas

LouisianaTech

TCU

FloridaAM

Akron

Wagner

MorganState

OldDominion

BoiseState

BethuneCookman

Florida

BostonCollege

Colgate

Massachusetts

Missouri

VirginiaTech

GeorgiaTech

Clemson

SouthCarolinaState

Syracuse

FloridaState

Louisville

WakeForest

SouthCarolina

IowaState

Duke

NorthCarolinaAT

Northwestern

Kansas

NorthCarolina

Miami

KennesawState

EasternKentucky

AppalachianState

CentralConnecticutState

NorthCarolinaState

Pittsburgh

GeorgiaState

Virginia

Wofford

Furman

Tennessee

NewHampshire

Ohio

Albany

Liberty

WilliamMary

Illinois

MiddleTennesseeState

Richmond

NorfolkState

Rutgers

Baylor

TexasState

TexasSouthern

WestVirginia

BYU

Texas

Oklahoma

KansasState

TexasTech

OklahomaState

Mississippi

NorthernIowa

UNLV

SouthDakota

WesternCarolina

Nebraska

Oregon

MissouriState

Duquesne

California

Louisiana

StephenFAustin

FIU

LIU

Charlotte

PennState

Minnesota

Idaho

WesternKentucky

Iowa

KentState

ColoradoState

Maryland

Howard

Michigan

WesternMichigan

Washington

NorthernIllinois

Wisconsin

OhioState

MichiganState

YoungstownState

Colorado

Purdue

Fordham

Buffalo

IndianaState

Utah

BallState

Villanova

OregonState

Delaware

EasternMichigan

ArizonaState

GardnerWebb

FloridaAtlantic

GeorgiaSouthern

UTEP

SoutheasternLouisiana

NorthCarolinaCentral

Monmouth

SouthernMississippi

Toledo

NorthwesternState

UTSA

Hampton

UAB

JacksonvilleState

NewMexicoState

NewMexico

Lamar

TennesseeMartin

Miami OH

AirForce

Bucknell

Arizona

UtahState

WashingtonState

IdahoState

USC

Yale

Campbell

Troy

Stanford

Bryant

BowlingGreen

WesternIllinois

CentralMichigan

RobertMorris

StFrancisPA

VirginiaMilitaryInstitute

Central Michigan

Maine

IllinoisState

SanJoseState

Nevada

Lafayette

Wyoming

FresnoState

CalPoly

UCLA

SanDiegoState

Hawaii

PortlandState

HoustonBaptist

Towson

SouthernUtah

NorthDakota

MontanaState

SacramentoState

NorthernColorado

StonyBrook

Vanderbilt

LSU

WeberState

Montana

Alabama

Mercer

Arkansas

Georgia

TexasAM

ArkansasPineBluff

AlabamaState

Samford

Kentucky

LouisianaMonroe

TennesseeChattanooga

McNeeseState

AustinPeay

TennesseeState

SoutheastMissouriState

EasternIllinois

TennesseeTech

SouthAlabama

PrarieViewAm

Elon

CoastalCarolina

CentralArkansas

Citadel

JacksonState

AlcornState

Southern

Figure 2.2: The 2021 NCAA Football Season as a Graph

to each of the FCS teams that played at least one FBS tea m dur i n g the season to form the

vertex set of a graph such as in Figure 2.2. We draw an edge between two vertices when the

corresponding teams played during eith er the regular or bowl season. This is the web-type

network for which we seek an ordering of the vertex set. This graph was produced using the

computer program Mathematica. One can note how th e FCS teams appear in the periphery

of the graph while teams in the interior of the graph are clustered mostly by conference and

geographic location.

2.2 Matrix Algebra and Eigenvectors

We develop the notion of a ranking vector in this section of the thesis. The Linear Algebra

notation used here follows the textbook Leon [23]. The equations given in the table of

Figure 2.2 . 1 arise naturally from Axi o m 2.1 and the graph given in Figur e 2.1. Here the

word “proportional” is modeled by the multiplication of the sums by the real number



For ex am p l e, in the ﬁrst row of the table in Equation 2.2.1 the rank of Alabama contains

the proportionality constant



times the sum of the r an k s of Auburn and Mississippi, the

two teams in this list of four that Alabama defeated. This reasoning for computing rankings

appears to be circular so we will examine the implications of such a computation from the

perspective o f Linear Algebra. Equation 2.2.1 can be written using matrix multiplication as

in shown in Fi g u r e 2.4.

r(Al a bama )=



(0 · r(Alabama)+1· r(Auburn)+1· r(Mississippi)+0· r(TexasA&M ))

r(Auburn)=



(0 · r(Alabama)+0· r(Auburn)+1· r(Mississippi)+0· r(TexasA&M ))

r(Mississippi)=



(0 · r(Alabama)+0· r(Auburn)+0· r(Mississippi)+1· r(TexasA&M ))

r(TexasA&M)=



(1 · r(Alabama)+1· r(Auburn)+0· r(Mississippi)+0· r(TexasA&M ))

(2.2.1)

We let R be the ranking vector and A b e the matrix given in Figure 2 . 3. Then the matrix

equation R =



AR is obtained. So R represents a vector with four rows and one column.

The entries obtained in R will be the ranking score of Alabama, Auburn, Mississippi, and

Texas A&M, respectively, rea d from top to bottom. Th e vector R is called an eigenvector

of the matrix A while th e number  is called an eigenvalue. There is a problem with this

method in ﬁnding the ranking vector R in that the entries may be complex numbers and not

real numbers. The ﬁeld of complex numbers is not ordered as is the ﬁeld of real numbers.

This potential problem is addressed in the next section of th e thesis. For example, the four

eigenvalues obt ai n ed as solutions in  to the equation R = AR are 1.3953, .4604+1.1393i,

.4604  1.1393i,and.4746. There is a unique positive real eigenvalue, 1.3953. The

eigenvector corresponding to the eigenvalue 1.3953 is h.5516,.3213,.4484,.6256i

.Thiswould

correspond to Alabama having the second highest ranking of . 5 5 16 , Auburn having the l owest

ranking of .3213, Mississippi having the second lowest ranking of .4484, and Texas A&M

having the highest ranking of .6256 . It is natural that Alabama and Texas A&M are ranked

higher in this little four-team league as both o f tho se tea ms had two wins. Texas A&M is

ranked higher than Alabama as it defeated Alabama. Both Auburn and Mississippi had one

win and Auburn defeated Mississipp i , but Mississippi is ranked higher than Auburn. The

explanation for this eigenvalue ranking is that Mississippi d efeated the highest-ranked team

Texas A&M. This small example ranking will be generalized to rank large complex networks

such a s that given in Figure 2.2. The eigenvalue ranking presented here is simpler than the

PageRank algorithm presented in the next section of the thesis.

2.3 Modiﬁed Pa g eRa nk A lg or it hm

We next develop the Goog le PageRank algorithm for ranking the relative import an ce of

webpages. Consider the directed graph G given in Figure 2.5. Suppose this gra p h models

the web links of six websites labeled by 1 through 6 (see [22, Chapter 4]). So WebPage 1

has links to We b Pages 2 and 3, for examp l e. Following Axiom 2.1, Brin and Page let r(P

)

denote the rank of page P

.LetB

be the set of pages that point to page P

for each i.So

R =

r(Alabama)

r(Auburn)

r(Mississippi)

r(T exas A&M )

and A =

0110

0010

0001

1100

Figure 2.3: The ranking vector

= {P

} and B

= {P

} for example. So t h ey obtained the equation gi ven in 2.3.1

where |P

| denotes the total number of outlinks from page P

r(P

)

(2.3.1)

So consider the ranking of the page P

by the method of Equation 2.3.1. We obtain r(P

r(P

)

r(P

)

.InordertostartthisprocessweneedranksforP

and P

.Sowelet

each of the six vertices of G have rank

.Thisassignmenthastheadvantagethateachrow

of H will sum to 1. So the rank of P

becomes

(

). In terms of vectors we can write

arowvector⇡

= h

i where the rank i ng of vertex i is found in the ith place of

the vector.

Let ⇡

=⇡(0)

and ⇡

(1)T

=⇡

(0)T

H.Then,weobtainanewrankingvector

⇡

(1)

=h1/18, 5/36, 1/12, 1/4, 5/36, 1/6i.RepeatingthisprocessforK =1, 2,...,weobtain

⇡

(K+1)T

= ⇡

(K)T

H. Now, the su r p r i si n g thing noticed by Brin and Page is that the ran k i n g

vectors converge to a unique vector of positive, real entries. The reason for this convergence

is the Perron-Frobenius Theorem from 1908 [22, p. 172]. The description here is the Power

Method for computing the ranking Eigenvector.

Equation 2.3.1 can caus e a division by 0 if a page has no inl i n k s. This problem can be

solved by adding non-zero entries to H as follows. Let e be the 6⇥1vectorofonesandS be the

matrix Z = ↵S +(1 ↵)

.ThenZ is called the Google Matrix.Thenumber↵ is chosen

between 0 and 1, typically ↵ =0.85 is chosen. This parameter is called the teleportation

index.Soif↵ =0.85, and we surf the web of the graph G with 85% probability, we follow the

r(Alabama)

r(Auburn)

r(Mississippi)

r(T exas A&M )



0110

0010

0001

1100

⇥

r(Alabama)

r(Auburn)

r(Mississippi)

r(T exas A&M )

Figure 2.4: Matrix form of Axiom 2.1

Figure 2.5: PageRank Example Graph G

hyperlink structure of H,andwith15%probability,werandomlyteleporttoanewvertex

with p r ob a b il i ty

.SowemultiplytherankingvectortransposeontheleftofG iteratively

until the ranking vector for the pages is obtained. In general, we use n instead of 6 for a

graph of n vertices. Th i s ranking can b e considered P

“voting” for P

with hal f-credi t as P

has two out nodes and P

“voting” for P

with one-third credit as P

has three out nodes.

Computing rankings for each vertex of the graph G of Figure 2.5 by Equation 2.3.1 we obtain

amatrixH in Figure 2.6 where the rank of each vertex P

is the sum of the entries in column

i of the matrix. The matrix H is call ed the Hyper l i nk matrix of the Graph G.Wecontinueto

iterate the process of applying Equation 2.3.1 where the new rank of each P

is computed by

multiplying the transpose of the current ranking vector by H.Theserankingvectorforthis

example converges to ⇡ = h0.33550.02500.35020.10940.09300.0870i.IfR(i) is the ranking of

node i,thenR(1) = 0.3355, R(2) = 0.0250, R(3) = 0.3502, R(4) = 0.1094, R(5) = 0.0930,

and R(6) = 0.0870 . In this framework, smaller ranked nodes represent websites that are

more likely to be visited. So, the most important nodes in order would be 2, 6, 5, 4 , 1, and

3. One can imagine that Page 2 is the most likely visited page as one surfs this six-page web

as it has inlinks but not outlinks. On average, the nodes on the right side of the graph are

higher ranked than the nodes on the left side of the graph due to the link from Page 3 to

Page 5.

d 123456

101/21/20 0 0

20 0 0 0 0 0

31/31/30 01/30

40 0 0 01/21/2

50 0 01/201/2

60 0 0 1 0 0

Figure 2.6: Hyperlink mat ri x H

01/21/20 0 0

1/61/61/61/61/61/6

1/31/30 01/30

00001/21/2

0001/201/2

000100

Figure 2.7: Stochastic Ma tr i x S,Rowentriessumto1

1/61/61/61/61/61/6

Figure 2.8: The matrix 1/n ee

2.4 Kendall Rank Correlation

We discuss the Kendall Rank Correlation of two data sets in this section of the thesis.

Kendall Rank Correlation is a non-parametric measure of relationships between columns of

ranked data [25]. We use the Kendall Rank Correlation to test the similarities in the ordering

of our data [25].

Let x

, x

, ···, x

and y

, y

, ···, y

be two ranked lists of observations. Then the Kenall

Rank Correlation ⌧ of t h e variables is given in Equation 2.4.1.

⌧ =

number of concordant pairs  number of discordant pairs

number of pairs

(2.4.1)

Here a pair of observations ( x

)and(x

), i<j,isconcordantifthesortorderofx

and x

and the sort order of y

and y

agree, otherwise the pair is said to be discordant.

The number of pairs i s





. Note th a t if the rank i n gs are the sam e, then ⌧ =1whileifthe

rankings are reversed, then ⌧ = 1. The closer the rankings are to each other, the closer ⌧

will be to one. A small example of computing this parameter for the rankings 1, 2, 3, 4and

1, 4, 2, 3 is given next. Here









=6. Theconcordantpairsare12,13,14,and23while

the discordant pairs are 24 and 34. Thus ⌧ =

42

.That⌧ is a positive measure that

the lists agree in order more than they disagree.

We can see the results of the Kendall Ran k in 3.2 and 3.3. The linear reg ress io n line

can be given in the equation Y = a + bX, where X is the explanatory variable and Y is the

dependent variable [34]. Plots above the linear regression li n e are more highly Y-ranked and

plots below th e linear regression line are more highly X-ranked. A graph with mo r e plots

clustered close to the l i n e of regression gives a more consistent evaluation. The plots in 3.2

show a closer relationship to the ﬁnal AP ran k i n gs than 3.3 because of the lesser distance

between the linear regression line and the majority of plots.

Chapter 3

Results

We give a total ordering of all teams i n the FBS based on t h e results of the 2021 seaso n by

two di↵erent methods in this chapter. The methods used are adapted fro m th o se used i n

the 2008 Ph.D. thesis of Anjela Govan at North Carolina State University [19]. Following

Govan’s work and a subsequent paper of Laurie Zack, Ron Lamb, and Sarah Ball [38] that

appeared in the journal Involve we investigate the rankings produced using b oth a Margin

of Victory approach and a Total Points scored in a contest app r o ach. Our rankings are for

the 130 teams that were in the FBS of College football in 2021. The ranking of Gova n and

Zack et. al. was for the 32 teams in the Nation a l Football League. It was a challenge to

compile the data from the large pool of teams in the FBS.

3.1 Margin of Victory Rankings

We present a total ordering of the FBS football teams by using the computer program

Matlab with code as given in Figure 3.1. Note that the l i n e headings “Line 1” etc. were

not part of the code used, they are added for reference’s sake. The matrix R is at the heart

of th e ranking. This matrix had both rows an d columns indexed by the 130 FBS teams in

alphabetic order. For the matrix entry in the column labeled “Ole Miss” and the row lab e l ed

“LSU”, we accounted for the score of the game between LSU and Ole Miss as follows. Ole

Miss won the game 31 to 17 so a 14 was entered i n this entry as that was the Margin of

Victory for Ole Miss over LSU. So each team such as Ole Miss received a positive entry in

its column at each corresponding row where the team won a game against that opponent. If

ateamsuchasOleMisseitherlostagameordidnotplayagameagainsttheteamlabeling

the corresponding row of its column, then a “0” was entered. If two teams played twice,

then the scores were added for both games and a Mar g i n o f Vi ct or y positive number was

entered for the team with the most total points in i ts column. On occasion, some teams

played anoth er team more than once. Remat ches of this type occurred four times during the

season. The most notable example of a rematch was Alabama versus Georgia in the S EC

Championship and in the National Championship. In the SEC Championship, Al a b am a won

by a score of 41-24 with a Margin of Victory of 17. In the National Championship, Georgia

won by a score of 33-18 with a Margin of Victory of 15. Both Margins of Victory scores

were recorded. So returning to Line 1 of Figure 3.1 the matrix R features prominently. This

matrix is 130 rows by 130 columns. The matrix C is merely a square mat r i x of the same

dimension as R with the number 0.1ineachentry. WeaddC to R so that we do not divide

by 0 in the computations that follow as the matrix R has a 0 in the majority of its entries.

Line 2 of the code is necessary to make the matrix called Hv row stochastic where the entries

in each row sum to 1. L i n e 3 sets th e t el eportation index ↵ equ al to 0.85. This is the number

between 0 and 1 commonly used in applica t i on s of the PageRank algorithm. We found that

this ↵ was a good choice based on experiments with choices ranging from 0.5to0.99. We

note that in the matrix R we ignored the result s of matches between FBS and FCS teams as

these did not appreci ab l y a↵ect rankings according to our experiments. In Line 6 we create

the Google matrix Ga.IfweimagineourselvesrandomlytravelingaroundtheMarginof

Victory network, then with prob a b i l i ty ↵ we follow nodes suggested by the numbers in Hv

Line 1: S=sum(R+C,1);

Line 2: Hv=(R+C)./S;

Line 3: alpha = 0.85;

Line 4: e = ones(130,1);

Line 5: v = e/130;

Line 6: Ga = alpha*Hv+(1-alpha)*v*e’;

Line 7: q = rand(130,1); % random initial vector

Line 8: q = q/sum(q); % normalise p to have unit sum

Line 9: for k=1:100

Line 10: q = Ga*q;

Line 11: end

Figure 3.1: Matlab Code for Finding a Pagerank Vector

where teams lose to other teams vote for th e m based on the Margin of Victory, while with

probability 1↵ we randomly jump around the network based on the numbers in the square

matrix v ⇤ e

with each entry as 1/130. The initial vector q produced in Lines 7 and 8 is a

random vector with entries between 0 and 1 that is normalized so that i t s entries summed to

1 which is our initial ra n k i n g. An amazing aspect of this algorithm is that it doesn’t matter

what is our initial ranking. In Line 10 of the algorithm, we iterate q by left multiplying it by

the matrix Ga 100 times. Eventually, the numbers in q stabilize to a ﬁ n al ranking vector.

In Tab l e 3.1 we list the top 25 teams obtained by the algori th m of Figure 3.1. The q value

obtained for each team is listed there. The complete rankings for all 130 teams are given

in the Appendix. In this table, a smaller q value produces a better ranking. Note that the

algorithm could be adapted to a transposed matrix so that the larger q value would produce

a bett er ranking. We also give the Final AP Top 25 in th e last column for comparison’s

sake.

The top 25 in Table3.1 pro d u ced some expected results and some surprising results.

Georgia and Alabama are high l y ranked in both pol l s as expected. Teams such as Notre

Dame an d Coastal Carolina seem to be more h i g h l y ranked than would be expected. Note

that the 2021 coach of Notre Dame, Brian Kelly, received a large increase in salary to join LSU

Team q-Value AP Top 25

1 Notre Dame 0.00194679056358042 Georgia

2 Georgia 0.00196000393177937 Alabama

3 Alabama 0.00196797504119794 Michigan

4 Coastal Carolina 0.00196988141542532 Ohio State

5 Cincinnati 0.00198379033469467 Baylor

6 Oklahoma State 0.00200420842507026 Oklahoma State

7 Michigan 0.00202229517821375 Cincinn at i

8 Oklahoma 0.00206023741216896 Notre Dame

9 Clemson 0.00207774962269084 Oklahoma

10 Baylor 0.0021016193527964 Michigan State

11 Ohio State 0.00220550673438769 Ole Miss

12 Air Force 0.00229666661999109 Arkansas

13 BYU 0.00229756915826271 Utah

14 Pi t t sbu r g 0.00230684479094631 Pitt

15 Michigan State 0.0023120444199844 NC State

16 Penn State 0.00241098386284439 Clemson

17 Ole Miss 0.00246464202733623 Minnesota

18 Arkansas 0.00248728348591432 Wisconsin

19 East Carolina 0.00250162674755332 Purdue

20 NC State 0.00250988269888323 Tennessee

21 Tex as A&M 0.00252239514451533 Kentucky

22 Boise State 0.00253635971904086 Iowa

23 Wake Forest 0.00255176433875926 Wake Forrest

24 Kentucky 0.00255453226853235 Utah State

25 SMU 0.00260051664523215 San Diego State

Table 3.1: Margin of Victory

after the 2021 season. Similarly, the coach of Coastal Carol i n a, Jamey Chad well, received a

large increase in salary to join Lib erty after the 2022 season. The quality of these two teams’

seasons was noticed by the coll e ge footba l l community. Now we discuss a statist i ca l method

for comparing the PageRank list with the AP List. Note that we compared all 130 teams in

both lists.

3.2 Total Score Ranking

We constructed the Total Score Rank i n g very similarly to Margin of Victory. This method

di↵ers from Margin of Victory because it uses two weighted arrows to represent each team’s

score rather than one for th e Margin of Victory. [38]. Using the same example in Chapter

3.1, Ole Miss defeated LSU 31 to 17. For the Total Score Ranking, a 31 would be placed in

the matrix entry in the column labeled ”O l e Miss” and the row labeled ”LSU”. Similarly, a

17 would be placed in the matrix entry in the column labeled ”LSU” and the row labeled

”Ole Miss”. This way Ole Miss was positively recorded for scoring 31 points and defeating

LSU and LSU was recorded for scoring 17 points. If a team such as Ole Miss did not play a

game against the team labeling the corresponding row of its column, then a ”0” was entered.

If two teams played twice, then the total number of points scored by t hat team was added

together and placed in the corresponding matrix entry of the row and column. The same

code used in the Margin of Victory computations was u sed for Tot a l Score Rankings, se e

below. The only change that was made was the matrix R.Weimporteda130rowby130

column matrix with the Total Score entries. Our code for the Matlab computation of the

Margin of Victory Ran k i n g used q.Weusedp to note the di↵erence in the matrix impo r te d .

Eventually, the numbers in p stabilize to a ﬁnal ranking vector.

In Tabl e 3.2 we li st the top 2 5 teams obtained by the algorithm of Figur e 3.4. The p value

obtained for each team is listed there. The complete rankings for all 130 teams are given

Figure 3.2: Margin of Victory Kendall’s Tau

in the appendix. In this table, a smaller p value produces a better ranking. Note that the

algorithm could be applied to a transposed matrix so that the larger p value would produce

a bett er ranking. We also give the Final AP Top 25 in th e last column for comparison’s

sake.

The top 25 in Table 3.2 produced some ex pected resul t s and some sur p r i si n g results.

Georgia and Alabama are hig h l y ranked in both polls as expected. Teams such as Liberty

and Texas A& M seem to be m or e highly ranked than would be ex pected. Note t h a t the

head coach of the Liberty Fla m s, Hugh Freeze, received a large increase in salary to join

the Auburn Tigers after the 2022 season. Ji mbo Fisher, the head co ach of the Tex as A&M

Aggies, received a four-year contract extension following th e 2020 season that raised his

annual salary by 1.5 milli o n dollars [21]. The conclus ion s for Total Score di↵ er from the

results of Margin of Victory. This is because not all of the teams shown in 3.2 represent

teams with a heavy win season. While many of the teams listed in 3.2 did win the majority

of their games, having a winning schedule was not an exclusive factor, as it was in Margin

of Victory. The teams listed in 3.2 scored many points, whether they were victorious in the

game or not. Teams in the Margin of Victory table outscored their opponents the most,

while teams in the Total Score table scored many points, regardless of a win or loss.

The top 25 in Table3.2 pro d u ced some expected results and some surprising results.

Georgia and Alabama are hig h l y ranked in both polls as expected. Teams such as Liberty

were n o t as expe ct ed to be ranked highly. Note that the head coach of the Liberty Flames,

Hugh Freeze, received a large salary i n cr ea se by joining the Auburn Tigers afte r the 2022

season. Jimbo Fisher, the head coach of the Texas A&M Aggies, received a four-year contract

extension following the 2 0 20 season that raise d his annual salary by 1.5 million dollars [21].

The conclusions for Tota l Score di↵er from the results of Margin of Victory. This is because

not all of the teams shown in 3.2 represent team s with a heavy win season. While m any o f

the teams listed in 3.2 did win the majority of their games, having a winning schedule was

Figure 3.3: Total Score Kendall’s Tau

Line 1: S=sum(H+C,1);

Line 2: Hv=(H+C)./S;

Line 3: alpha = 0.85;

Line 4: e = ones(130,1);

Line 5: v = e/130;

Line 6: Ga = alpha*Hv+(1-alpha)*v*e’;

Line 7: p = rand(130,1); % random initial vector

Line 8: p = p/sum(p); % normalise p to have unit sum

Line 9: for k=1:100

Line 10: p = Ga*p;

Line 11: end

Figure 3.4: Matlab Code for Finding a Pagerank Vector

Team p-Value AP Top 25

1 Liberty 0.00304746969912415 Georgia

2 Georgia 0.00381204199025146 Alabama

3 Wisconsin 0.00407542269923828 Michigan

4 Penn State 0.00417366519034118 Ohio State

5 Clemson 0.00419166990380779 Baylor

6 TexasA& M 0.0042417589958973 Oklahoma State

7 Nebraska 0.00473118334230752 Cincinnati

8 Oklahoma State 0.00486888996498965 Notre Dame

9 Michigan 0.00488610520420468 Oklahoma

10 NC State 0.00497641969495136 Michigan State

11 Alabama 0.00521606093022588 Ole Miss

12 Purdue 0.00530534261905736 Arkansas

13 Iowa 0.00533300262674794 Utah

14 Minnesota 0.00534577785603451 Pitt

15 Kansas State 0.00536983025416077 NC State

16 Iowa State 0.0054653187130986 Clemson

17 Aubur n 0.00563143171890239 Minnesota

18 Syracuse 0.00566076793675136 Wisconsin

19 Illinois 0.00574882147723986 Purdue

20 Arkansas 0.00575824184598954 Tennessee

21 Florida Stat e 0.00576672770807803 Kentucky

22 Ole Miss 0.00592753339050079 Iowa

23 Michigan State 0.0059897626199456 Wake Forrest

24 Californi a 0.00601905226057784 Utah State

25 Florida 0.0061003127503729 San Diego State

Table 3.2: Total Score

not an exclusive factor, as it was in Margin of Victory. The teams listed in 3.2 scored many

points, whether they were victorious in the game or not. Teams in the Marg i n of Victory

table outscored their opponents the most, while teams in the Total Score table scored many

points, regardless of a win or loss.

3.3 Conclusions

The Google PageRank alg or i t h m proved to be an accurate and e↵ect i ve tool for providing

rankings that corre l at ed well with th e Final AP ranking s. The Margin of Victory method

provided a better correlation wit h the AP ranki n g s than did the Total Score rankings. Ma t h -

ematica was an e↵ective tool for illu st r ati n g t h e network invol ved in the algorithm. The

combination of Matlab and the power method is a better tool for computing matrix eigen-

values of large matrices such as the 130 by 130 NCAA FBS season results. The FBS versus

FCS matchups did not appreciably a↵ect the rankings of the teams and were ignored. This

observation was veriﬁed computationally and supported by previous research [20]. Future

directions for research could be the e↵ect of ﬁrst downs, turnovers, yards gained, passing

yards, running yards, and many other game parameters to see which factors contribute the

most to team success.

The metho d of applying the PageRank al gor i t hm to rank sports teams can be directed

to many future research opportunities. This method can be used to produce and compute

rankings for any sport. In a similar approach, this ideology could be applied to any factor

regarding sports. For example, time of possession, turnovers, total yardage, yards gained

by pass and run, points gained by ﬁeld goals, time of possession, and many other factors

could be applied to the PageRank met h od. Another way to further research the ranking of

sports teams would be the changing of th e value of ↵. Through o u t our resea rch, we used a

standard ↵ of 0.85 to evaluate the teleportation index. The value of ↵ could range from 0

to 1. Changing the value of ↵ allows for a di↵erent ranking vector to be obtained.

Chapter 4

Appendi x

A complete list of the Margin of Victory PageRank rankings of the 202 1 FBS footb al l season

is given next.

Air Force Falcons 0.002297 Akron Zips 0.061325 Alabama Crimson Tide 0.001968 Ap-

palachian State Mountaineers 0.002951 Arizona Wildcat s 0. 01 574 0 Ari zo n a St a te Su n D ev -

ils 0.004497 Arkansas Razorbacks 0.002487 Arkansas State Red Wolves 0.015871 Army

Black Knights 0.003912 Au b u r n Tigers 0.003000 Ball State Cardinals 0.008654 Baylor Bear s

0.002102 Boise State Broncos 0.002536 Boston College Eagles 0.004283 Bowling Green

Falcons 0.057965 Bu↵alo Bulls 0.028531 BYU Cougars 0.002298 California Golden Bears

0.014806 Central Michigan Chippewas 0.004611 Charlotte 49ers 0.013100 Cincinnati Bearcats

0.001984 Clemson Tigers 0.002078 Coastal Carolina Chanticleers 0.001970 Colorado Buf-

faloes 0.012612 Colorad o State Rams 0.009721 Duke Blue Devils 0.012657 East Carolin a

Pirates 0.002502 Eastern Michigan Eagles 0.012743 FIU Panthers 0.030598 Florida G at o r s

0.003968 Flo ri d a Atlantic Ow l s 0.006102 Florid a State Seminol es 0.002742 Fresno State Bull-

dogs 0.002873 Georgia Bulldogs 0.001960 Georgia Southern Eagles 0.009894 Georgia State

Panthers 0.005741 Georgia Tech Yellow Jackets 0.004574 Hawaii Rainbow Warriors 0.009230

Houston Cougars 0.002779 Illinois Fighting Illini 0.004247 Indiana Hoosiers 0.007545 Iowa

Hawkeyes 0. 0 02 6 54 Iowa State Cyclones 0.005514 Kansas Jayhawks 0.014448 Kansas State

Wildcats 0.002845 Kent State Golden Flashes 0.008974 Kentucky Wi l dc at s 0.002555 Liberty

Flames 0.004807 Louisiana Ragin’ Cajuns 0.00310 3 Louisiana-Monroe Warhawks 0.018261

Louisiana Tech Bulldogs 0.013464 Louisville Cardinals 0.002998 LSU Tigers 0.003185 Mar-

shall Thundering Herd 0.00359 6 Maryland Terrapins 0.004653 Memphis Tigers 0 . 00 4 63 7

Miami (FL) Hurricanes 0.002730 Miami (OH) RedHawks 0.004 82 5 Michigan Wolverines

0.002022 Michigan State Spartans 0.002312 Middle Tennessee Blue Raiders 0.006750 Min-

nesota Golden Gophers 0.009449 Mississippi State Bulldogs 0.004004 Missouri Tigers 0.003702

Navy Midshipmen 0.005613 Nebraska Cornhuskers 0.003237 Nevada Wolf Pack 0.003568

New Mexico Lobos 0.012356 New Mexico State Aggies 0.015929 North Carolina Tar Heels

0.006411 North Carolina State Wolfpack 0.002510 North Texas Mean Green 0.010420 North-

ern Illinois Huskies 0.00 4 43 0 Northwestern Wildcats 0.011050 Notre Dam e Fighting Irish

0.001947 Ohio Bobcats 0.031091 Ohio State Buckeyes 0.002206 Oklahoma Sooners 0.002060

Oklahoma State Cowboys 0.002004 Old Dominion Monarchs 0.004932 Ole Miss Rebels

0.002465 Oregon Ducks 0. 00 44 5 4 Oregon State Beavers 0.0058 5 8 Penn State Nittany Lion s

0.002411 Pittsburgh Panthers 0.002307 Purdue Boilerm a kers 0.002760 Rice Owls 0.0116 68

Rutgers Scarlet Knights 0.007779 San Diego State Aztecs 0. 00 30 1 9 San Jose S t at e Spar-

tans 0.009861 SMU Mustangs 0.0026 01 South Alabama Jaguars 0.01218 4 South Carolina

Gamecocks 0.003553 South Florida Bul l s 0.006962 Southern Miss Golden Eagles 0.018827

Stanford Car d i n al 0.010281 Syracuse Orange 0.004115 TCU Horned Frogs 0.005330 Tem-

ple Owls 0.018647 Tennessee Volunteers 0.003038 Texas Lon g h or n s 0. 00 9 09 6 Texas A&M

Aggies 0.002522 Texas State Bobcats 0.0 1077 0 Texas Tech Red Raiders 0.006355 Toledo

Rockets 0.004997 Troy Trojans 0.01 160 9 Tulane Green Wave 0.005181 Tulsa Gol d en Hu r ri -

cane 0.002877 UAB Blazers 0.006338 UCF Knights 0.003262 UCLA Bruins 0.002863 UConn

Huskies 0.026607 UMass Minutemen 0.022084 UNLV Rebels 0.008076 USC Trojans 0.008906

UTEP Miners 0.003932 UTSA Road r u n n er s 0.004133 Utah Utes 0.002646 Utah State Aggies

0.005562 Vanderbilt Commodores 0.010507 Vir g i n i a Cavaliers 0.004298 Virginia Tech Hokies

0.005800 Wake Forest Demon Deacons 0.002552 Washington Huski es 0.004219 Washington

State Cougars 0 . 00 2 64 7 West Virginia Mountaineers 0.003823 West er n Kentucky Hilltop-

pers 0.006111 Western Michigan Broncos 0.005934 Wisconsin Badgers 0.002887 Wyoming

Cowboys 0.012582

A complete list of th e Points Scored PageRank rankings is given below.

Air Force Falcons 0.006603 Akron Zips 0.013724 Alabama Crimson Tide 0.005241 Ap-

palachian State Mountaineers 0.007499 Arizona Wildcat s 0. 00 789 8 Ari zo n a St a te Su n D ev -

ils 0.006329 Arkansas Razorbacks 0.005779 Arkansas State Red Wolves 0.012095 Army

Black Knights 0.007541 Au b u r n Tigers 0.005650 Ball State Cardinals 0.009814 Baylor Bear s

0.006459 Boise State Broncos 0.006511 Boston College Eagles 0.006776 Bowling Green

Falcons 0.010981 Bu↵alo Bulls 0.010814 BYU Cougars 0.007602 California Golden Bears

0.006032 Central Michigan Chippewas 0.012192 Charlotte 49ers 0.010518 Cincinnati Bearcats

0.006180 Clemson Tigers 0.004215 Coastal Carolina Chanticleers 0.009016 Colorado Buf-

faloes 0.006502 Colorad o State Rams 0.008345 Duke Blue Devils 0.009839 East Carolin a

Pirates 0.006889 Eastern Michigan Eagles 0.009999 FIU Panthers 0.007864 Florida G at o r s

0.006119 Flo ri d a Atlantic Ow l s 0.006928 Florid a State Seminol es 0.005791 Fresno State Bull-

dogs 0.007081 Georgia Bulldogs 0.003839 Georgia Southern Eagles 0.009574 Georgia State

Panthers 0.009059 Georgia Tech Yellow Jackets 0.007608 Hawaii Rainbow Warriors 0.010284

Houston Cougars 0.008187 Illinois Fighting Illini 0.005771 Indiana Hoosiers 0.006566 Iowa

Hawkeyes 0. 0 05 3 57 Iowa State Cyclones 0.005490 Kansas Jayhawks 0.009831 Kansas State

Wildcats 0.005392 Kent State Golden Flashes 0.013347 Kentucky Wi l dc at s 0.006186 Liberty

Flames 0.003070 Louisiana Ragin’ Cajuns 0.00713 5 Louisiana-Monroe Warhawks 0.009357

Louisiana Tech Bulldogs 0.009366 Louisville Cardinals 0.006815 LSU Tigers 0.006599 Mar-

shall Thundering Herd 0.00685 5 Maryland Terrapins 0.007592 Memphis Tigers 0 . 00 9 48 2

Miami (FL) Hurricanes 0.006310 Miami (OH) RedHawks 0.008 86 5 Michigan Wolverines

0.004910 Michigan State Spartans 0.006014 Middle Tennessee Blue Raiders 0.007790 Min-

nesota Golden Gophers 0.005361 Mississippi State Bulldogs 0.006659 Missouri Tigers 0.008643

Navy Midshipmen 0.007442 Nebraska Cornhuskers 0.004760 Nevada Wolf Pack 0.008679

New Mexico Lobos 0.008127 New Mexico State Aggies 0.010140 North Carolina Tar Heels

0.007635 North Carolina State Wolfpack 0.005000 North Texas Mean Green 0.007913 North-

ern Illinois Huskies 0.01 1 97 2 Northwestern Wildcats 0.006766 Notre Dam e Fighting Irish

0.006568 Ohio Bobcats 0.010453 Ohio State Buckeyes 0.006285 Oklahoma Sooners 0.007265

Oklahoma State Cowboys 0.004892 Old Dominion Monarchs 0.008413 Ole Miss Rebels

0.005949 Oregon Ducks 0. 00 80 7 0 Oregon State Beavers 0.0079 4 3 Penn State Nittany Lion s

0.004200 Pittsburgh Panthers 0.007238 Purdue Boilerm a kers 0.005332 Rice Owls 0.0090 68

Rutgers Scarlet Knights 0.006489 San Diego State Aztecs 0. 00 70 0 2 San Jose S t at e Spar-

tans 0.007742 SMU Mustangs 0.0081 27 South Alabama Jaguars 0.00922 6 South Carolina

Gamecocks 0.006317 South Florida Bul l s 0.008463 Southern Miss Golden Eagles 0.007797

Stanford Car d i n al 0.008242 Syracuse Orange 0.005685 TCU Horned Frogs 0.007999 Tem-

ple Owls 0.011338 Tennessee Volunteers 0.007035 Texas Lon g h or n s 0. 00 8 13 5 Texas A&M

Aggies 0.004266 Texas State Bobcats 0.0 1062 9 Texas Tech Red Raiders 0.007797 Toledo

Rockets 0.008604 Troy Trojans 0.00 835 0 Tulane Green Wave 0.007789 Tulsa Gol d en Hu r ri -

cane 0.008222 UAB Blazers 0.007027 UCF Knights 0.007679 UCLA Bruins 0.007636 UConn

Huskies 0.009882 UMass Minutemen 0.010104 UNLV Rebels 0.008571 USC Trojans 0.008731

UTEP Miners 0.007895 UTSA Road r u n n er s 0.008369 Utah Utes 0.006886 Utah State Aggies

0.008797 Vanderbilt Commodores 0.009217 Vir g i n i a Cavaliers 0.007191 Virginia Tech Hokies

0.007171 Wake Forest Demon Deacons 0.007518 Washington Huski es 0.006219 Washington

State Cougars 0 . 00 7 74 7 West Virginia Mountaineers 0.006717 West er n Kentucky Hilltop-

pers 0.009320 Western Michigan Broncos 0.009984 Wisconsin Badgers 0.004102 Wyoming

Cowboys 0.009729

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