A Course on the Web Graph (Graduate Studies in Mathematics, 89)

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. C*-approximation Theory Has Provided The Foundation For Many Of The Most Important Conceptual Breakthroughs And Applications Of Operator Algebras. This Book Systematically Studies (most Of) The Numerous Types Of Approximation Properties That Have Been Important In Recent Years: Nuclearity, Exactness, Quasidiagonality, Local Reflexivity, And Others. Moreover, It Contains User-friendly Proofs, Insofar As That Is Possible, Of Many Fundamental Results That Were Previously Quite Hard To Extract From The Literature. Indeed, Perhaps The Most Important Novelty Of The First Ten Chapters Is An Earnest Attempt To Explain Some Fundamental, But Difficult And Technical, Results As Painlessly As Possible. The Latter Half Of The Book Presents Related Topics And Applications--written With Researchers And Advanced, Well-trained Students In Mind. The Authors Have Tried To Meet The Needs Both Of Students Wishing To Learn The Basics Of An Important Area Of Research As Well As Researchers Who Desire A Fairly Comprehensive Reference For The Theory And Applications Of C*-approximation Theory.--publisher's Description. Fundamental Facts -- Nuclear And Exact C* -algebras: Definitions, Basic Facts And Examples -- Tensor Products -- Constructions -- Exact Groups And Related Topics -- Amenable Traces And Kirchberg's Factorization Property -- Quasidiagonal C* -algebras -- Af Embeddability -- Local Reflexivity And Other Tensor Product Conditions -- Summary And Open Problems -- Simple C* -algebras -- Approximation Properties For Groups -- Weak Expectation Property And Local Lifting Property -- Weakly Exact Von Neumann Algebras -- Classification Of Group Von Neumann Algebras -- Herrero's Approximation Problem -- Counterexamples In K- Homology And K- Theory -- Appendices: A. Ultrafilters And Ultraproducts -- B. Operator Spaces, Completely Bounded Maps And Duality -- C. Lifting Theorems -- D. Positive Definite Functions, Cocycles And Schoenberg's Theorem -- E. Groups And Graphs -- F. Bimodules Over Von Neumann Algebras. Nuclear And Exact C*-algebras : Definitions, Basic Facts, And Examples -- Tensor Products -- Constructions -- Exact Groups And Related Topics -- Amenable Traces And Kirchberg's Factorization Property -- Quasidiagonal C*-algebras -- Af Embeddability -- Local Reflexivity And Other Tensor Production Conditions -- Simple C*-algebras -- Approximation Properties For Groups -- Weak Expectation Property And Local Lifting Property -- Weakly Exact Von Neumann Algebras -- Classification Of Group Von Neumann Algebras -- Herrero's Approximation Problem -- Counterexamples In K-homology And K-theory -- Appendix A : Ultrafilters And Ultraproducts -- Appendix B : Operator Spaces, Completely Bounded Maps, And Duality -- Appendix C : Lifting Theorems -- Appendix D : Positive Definite Functions, Cocycles, And Schoenberg's Theorem -- Appendix E : Groups And Graphs -- Appendix F : Bimodules Over Von Neumann Algebras. Nathaniel P. Brown, Narutaka Ozawa. Includes Bibliographical References (p. 493-502) And Index. 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 "A 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."--Publisher's description

A 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.

$\mathrm{C}^•$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications—written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\mathrm{C}^•$-approximation theory. $\textrm{C}^*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\textrm{C}^*$-approximation theory. $\textrm{C}^*$-approxmation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important such as: nuclearity, exactness, quasidiagonality, and local reflexivity. Presents a comprehensive introduction to research on the applications of graph theory to real-world networks such as web graph. This book discusses both models of the web graph and algorithms for searching the web.