The Mathematics Behind and Applications of Google's PageRank
In 1998 Sergey Brin and Larry Page, the founders of Google, changed the landscape of internet search by introducing their PageRank algorithm for improving how web-searches are performed. The PageRank algorithm applies numerical linear algebraic techniques and exploits the link structure of the internet to rank web pages based on order of importance and relevance. Scientists have since used algorithms heavily influenced by PageRank to study a wide array of topics including: ranking sports teams, calculating research impact factors , predicting outcomes of cancer patients , and finding genes linked to adverse drug reactions .
In this project, students will study the mathematics behind Google's PageRank algorithm. Participants will employ properties of stochastic matrices and learn how the Perron-Frobenius Theorem can be used to guarantee the existence of a unique positive solution x (the PageRank vector) of the now famous eigenvector-eigenvalue problem, Gx=x, ||x||=1, under appropriate conditions on the matrix G. We will use the Power Method to numerically approximate the PageRank vector. While gaining the necessary background to employ the PageRank algorithm, students will also come up with their own novel way to use the algorithm to model some system and use their model to make decisions based upon the results of the algorithm. With a diverse group of student modelers, we may also be able to start to address some of the social injustice issues that surround the current state of algorithms, including Google's search engine. This project will not only prepare students for future Markov Chain based mathematical modeling opportunities, but by working on projects in applied numerical linear algebra, students access a gateway to other problems using linear algebra including the growing field of artificial intelligence. A first course in linear algebra is preferred but not required for students in this program. Completion of Calculus II is the minimum requirement for participation.
Short Bio
Dr. Padraic Taylor is currently an associate professor at Youngstown State University, where he has been working in the Department of Mathematics and Statistics since the fall 2006 semester. He earned his PhD in May 2006 from North Carolina State University, under the guidance of Dr. Jesus Rodriguez. His dissertation was titled ``On the Solvability of Nonlinear Discrete Multipoint Boundary Problems." This work involves finding a link between the nonlinear problem and the solution space of the associated linear problem by using a projection scheme known as the Lyapunov-Schmidt technique. Dr. Taylor has guided over 38 students through senior capstone projects, Master's Theses, group poster projects, and individual research projects. All of these projects have resulted in presentations at local, regional, or national conferences; with several students winning awards for the quality of their work and presentations. Except for the senior projects and Master's Theses, the majority of Dr. Taylor's experience mentoring undergraduates has been with freshmen in his ``Accelerated Honors Calculus" course or research teams mixed with freshmen and upperclassmen as part of the Choose Ohio First Scholarship program. Some of the successes of these freshmen include an award winning oral presentation at MAA MathFest in Washington, D.C. in August 2015; an award winning poster presentation at the Northeast Ohio Choose Ohio First Research Poster Conference at Kent State University in spring 2016; two separate teams of freshmen earning ``Meritorious" (approximately top 8%) honors in COMAP's international mathematical modeling competition (spring 2014, spring 2019); and a team of freshmen eaning an ``Outstanding" award at SCUDEM in October 2018 at Slippery Rock University.
