Course on the Web Graph provides a comprehensive introduction to state-of-the-art research on the applications of graph theory to real-world networks such as the web graph. It is the first mathematically rigorous textbook discussing both models of the web graph and algorithms for searching the web.
After introducing key tools required for the study of web graph mathematics, an overview is given of the most widely studied models for the web graph. A discussion of popular web search algorithms, e.g. PageRank, is followed by additional topics, such as applications of infinite graph theory to the web graph, spectral properties of power law graphs, domination in the web graph, and the spread of viruses in networks.
The book is based on a graduate course taught at the AARMS 2006 Summer School at Dalhousie University. As such it is self-contained and includes over 100 exercises. The reader of the book will gain a working knowledge of current research in graph theory and its modern applications. In addition, the reader will learn first-hand about models of the web, and the mathematics underlying modern search engines.
This book is published in cooperation with Atlantic Association for Research in the Mathematical Sciences (AARMS).
Readership: Graduate students and research mathematicians interested in graph theory, applied mathematics, probability, and combinatorics.
Author(s): Anthony Bonato
Series: Graduate Studies in Mathematics 89
Publisher: American Mathematical Society
Year: 2008
Language: English
Commentary: Fully Bookmarked, Covers
Pages: xi+184
Chapter 1. Graphs and Probability 1
§1.1. Introduction 1
§1.2. Graph Theory 2
§1.3. Probability Theory 9
Exercises 14
Chapter 2. The Web Graph 19
§2.1. Introduction 19
§2.2. Other Real-World Self-Organizing Networks 28
Exercises 31
Chapter 3. Random Graphs 33
§3.1. Introduction 33
§3.2. What is a Random Graph? 34
§3.3. Expectation and the First Moment Method 44
§3.4. Variance and the Second Moment Method 47
§3.5. Martingales and Concentration 50
Exercises 54
Chapter 4. Models for the Web Graph 59
§4.1. Introduction 59
§4.2. On-Line Web Graph Models 61
§4.3. Future Challenges in Modelling the Web Graph 92
Exercises 94
Chapter 5. Searching the Web 97
§5.1. Introduction 97
§5.2. An Overview of Search Engines 98
§5.3. Adjacency Matrices and the Perron-Frobenius Theorem 99
§5.4. Markov Chains 103
§5.5. PageRank 105
§5.6. HITS 110
§5.7. SALSA 113
§5.8. Further Analysis of Web Ranking Algorithms 115
Exercises 117
Chapter 6. The Infinite Web 121
§6.1. Introduction 121
§6.2. The Infinite Random Graph 124
§6.3. Representations and Properties of R 127
§6.4. Limits of Copying Models 132
§6.5. Limits of Preferential Attachment Models 142
§6.6. The n-Ordered Graphs and Their Limits 145
Exercises 153
Chapter 7. New Directions in Internet Mathematics 157
§7.1. Introduction 157
§7.2. Eigenvalues of Power Law Graphs 158
§7.3. Modelling Viruses on the Web 160
§7.4. Dominating Sets in the Web Graph 162
Exercises 168
Bibliography 171
Index 181
