Understanding the Google PageRank Algorithm

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Authors

Madeline Renee Boss
Samhitha Devi Kunadharaju
University of Texas at Austin
Course: CS 323E – Elements of Scientific Computing

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Overview

This project implements and analyzes the Google PageRank algorithm using both the iterative update $x_{k+1} = Hx_k$ and the dominant eigenvector of the hyperlink matrix $H$. Two example web networks are evaluated, and both methods converge to the same steady-state PageRank vector.

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Included Files

File	Description
pagerank_analysis.ipynb	Jupyter/Colab notebook with hyperlink matrix construction, iterative computation, eigenvector solution, and plots
PageRank_Paper.pdf	Full paper explaning PageRank, linear algebra background, and results
convergence-6.png	PageRank Convergence Behavior for Six-Page Web Network
convergence-8.png	PageRank Convergence Behavior for Eight-Page Web Network
requirements.txt	List of dependencies (NumPy, Matplotlib, SciPy)

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Key Ideas

• The web is modeled as a directed graph, producing a column-stochastic matrix.
• PageRank is the eigenvector of H with eigenvalue 1.
• Iterative updates and eigenvector computation give the same result (Theorem 4.9).
• A page is important when important pages link to it, not simply by having many links.

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Results (Summary)

• 6-page network: Pages 4, 6, and 5 retain all rank; pages 1–3 converge to zero.
• 8-page network: All pages have nonzero rank; Page 4 is the most influential.

How to Run

pip install -r requirements.txt