Complex Graphs and Networks

Contents Preface vii Chapter 1. Graph Theory in the Information Age 1 1.1. Introduction 1 1.2. Basic definitions 3 1.3. Degree sequences and the power law 6 1.4. History of the power law 8 1.5. Examples of power law graphs 10 1.6. An outline of the book 17 Chapter 2. Old and New Concentration Inequalities 21 2.1. The binomial distribution and its asymptotic behavior 21 2.2. General Chernoff inequalities 25 2.3. More concentration inequalities 30 2.4. A concentration inequality with a large error estimate 33 2.5. Martingales and Azuma's inequality 35 2.6. General martingale inequalities 38 2.7. Supermartingales and Submartingales 41 2.8. The decision tree and relaxed concentration inequalities 46 Chapter 3. A Generative Model — the Preferential Attachment Scheme 55 3.1. Basic steps of the preferential attachment scheme 55 3.2. Analyzing the preferential attachment model 56 3.3. A useful lemma for rigorous proofs 59 3.4. The peril of heuristics via an example of balls-and-bins 60 3.5. Scale-free networks 62 3.6. The sharp concentration of preferential attachment scheme 64 3.7. Models for directed graphs 70 Chapter 4. Duplication Models for Biological Networks 75 4.1. Biological networks 75 4.2. The duplication model 76 4.3. Expected degrees of a random graph in the duplication model 77 4.4. The convergence of the expected degrees 79 4.5. The generating functions for the expected degrees 83 4.6. Two concentration results for the duplication model 84 4.7. Power law distribution of generalized duplication models 89 Chapter 5. Random Graphs with Given Expected Degrees 91 5.1. The Erd˝ os-Rényi model 91 5.2. The diameter of G n,p 95 iii iv CONTENTS 5.3. A general random graph model 97 5.4. Size, volume and higher order volumes 97 5.5. Basic properties of G(w) 100 5.6. Neighborhood expansion in random graphs 103 5.7. A random power law graph model 107 5.8. Actual versus expected degree sequence 109 Chapter 6. The Rise of the Giant Component 113 6.1. No giant component if w < 1? 114 6.2. Is there a giant component if˜w > 1? 115 6.3. No giant component if˜w < 1? 116 6.4. Existence and uniqueness of the giant component 117 6.5. A lemma on neighborhood growth 126 6.6. The volume of the giant component 129 6.7. Proving the volume estimate of the giant component 131 6.8. Lower bounds for the volume of the giant component 136 6.9. The complement of the giant component and its size 138 6.10. …