With the advent of digital computers more than half a century ago, - searchers working in a wide range of scienti?c disciplines have obtained an extremely powerful tool to pursue deep understanding of natural processes in physical, chemical, and biological systems. Computers pose a great ch- lenge to mathematical sciences, as the range of phenomena available for rigorous mathematical analysis has been enormously expanded, demanding the development of a new generation of mathematical tools. There is an explosive growth of new mathematical disciplines to satisfy this demand, in particular related to discrete mathematics. However, it can be argued that at large mathematics is yet to provide the essential breakthrough to meet the challenge. The required paradigm shift in our view should be compa- ble to the shift in scienti?c thinking provided by the Newtonian revolution over 300 years ago. Studies of large-scale random graphs and networks are critical for the progress, using methods of discrete mathematics, probabil- tic combinatorics, graph theory, and statistical physics. Recent advances in large scale random network studies are described in this handbook, which provides a signi?cant update and extension - yond the materials presented in the 'Handbook of Graphs and Networks' published in 2003 by Wiley. The present volume puts special emphasis on large-scale networks and random processes, which deemed as crucial for - tureprogressinthe?eld. Theissuesrelatedtorandomgraphsandnetworks pose very di?cult mathematical questions.

Table of contents

"Chapter 1 Random Graphs and Branching Processes (p. 15-16)

B´ELA BOLLOB´AS and OLIVER RIORDAN

During the past decade or so, there has been much interest in generating and analyzing graphs resembling large-scale real-world networks such as the world wide web, neural networks, and social networks. As these large-scale networks seem to be ‘random’, in the sense that they do not have a transparent, well-de?ned structure, it does not seem too unreasonable to hope to ?nd classical models of random graphs that share their basic properties.

Such hopes are quickly dashed, however, since the classical random graphs are all homogeneous, in the sense that all vertices (or indeed all k-sets of vertices) are a priori equivalent in the model. Most real-world networks are not at all like this, as seen most easily from their often unbalanced (power-law) degree sequences. Thus, in order to model such graphs, a host of inhomogeneous random graph models have been constructed and studied.

In this paper we shall survey a number of these models and the basic results proved about the inhomogeneous sparse (bounded average degree) random graphs they give rise to. We shall focus on mathematically tractable models, which often means models with independence between edges, and in particular on the very general sparse inhomogeneous models of Bollob´as, Janson and Riordan. The ?rst of these encompasses a great range of earlier models of this type; the second, the inhomogeneous clustering model, goes much further, allowing for the presence of clustering while retaining tractability.

We are not only interested in our inhomogeneous random graphs themselves, but also in the random subgraphs obtained by keeping their edges with a certain probability p. Our main interest is in the phase transition that takes place around a certain critical value p0 of p, when the component structure of the random subgraph undergoes a sudden change. The quintessential phase transition occurs in the classical binomial random graph G(n, c/n) as c grows from less than 1 to greater than 1 and, as shown by Erd?os and R´enyi, a unique largest component, the giant component, is born.

A ubiquitous theme of our paper is the use of branching processes in the study of random graphs. This ‘modern’ approach to random graphs is crucial in the study of the very general models of inhomogeneous random graphs mentioned above. To illustrate the power of branching processes, we show how they can be used to reprove sharp results about the classical random graph G(n, c/n), ?rst proved by Bollob´as and Luczak over twenty years ago. When it comes to inhomogeneous models, we shall have time only to sketch the connection to branching processes. Finally, we close by discussing the question of how to tell whether a given model is appropriate in a given situation. This leads to many fascinating questions about metrics for sparse graphs, and their relationship to existing models and potential new models."

Table of contents

Title Page

4

Copyright Page

5

Table of Contents

6

Preface

10

Chapter 1 Random Graphs and Branching Processes

16

1. Introduction

17

2. Models

20

2.1. Classical models

20

2.2. Random graphs with a fixed degree sequence

24

2.3. Inhomogeneous models

27

2.4. Models with independence between edges

28

2.5. A general sparse inhomogeneous model

30

2.6. Independence and clustering

33

2.7. Further models

34

3. The Phase Transition in G(n, p)

35

3.1. Historical remarks

36

3.2. Local behaviour

41

3.3. The giant component

44

3.4. Stronger results for G(n, p)

47

3.5. The subcritical case

51

3.6. The supercritical case

61

4. The Phase Transition in Inhomogeneous Random Graphs

69

4.1. Graphs with a given degree sequence

69

4.2. Robustness of the BA or LCD model

71

4.3. The uniformly grown models and Dubins model

73

4.4. Graphs with independence between edges

77

4.5. Applications of non-Poisson branching processes

80

5. Branching Processes and Other Global Properties

85

5.1. Diameter

85

5.2. The k-core

88

6. Appropriateness of Models: Metrics on Graphs

91

6.1. The edit distance(s): dense case

92

6.2. The subgraph distance: dense case

95

6.3. The cut metric: dense case

97

6.4. The sparse case

99

6.5. New metrics and models

103

References

105

Chapter 2 Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

117

1. Introduction

117

2. The Gilbert Disc Model

118

2.1. Percolation

119

2.2. Connectivity

120

2.3. Coverage

121

2.4. Colouring

123

2.5. Thin strips

125

3. The k-nearest Neighbour Model

128

3.1. Percolation

128

3.2. Connectivity

129

3.3. Sharp thresholds

131

4. Random Tessellations

132

4.1. Random Voronoi Percolation

133

4.2. Random JohnsonMehl Percolation

137

5. Outlook

138

References

139

Chapter 3 Scaling Properties of Complex Networks and Spanning Trees