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Remco van der Hofstad Aan Mad, Max en Lars het licht in mijn leven Ter nagedachtenis aan mijn ouders die me altijd aangemoedigd hebben Contents List of illustrations ix List of tables xi Preface xiii Course Outline xvii 2.3.1 Examples of Stochastically Ordered Random Variables 2.3.2 Stochastic Ordering and Size-Biased Random Variables 2.3.3 Consequences of Stochastic Domination 2.4 Probabilistic Bounds 2.4.1 First and Second Moment Methods 2.4.2 Large Deviation Bounds 2.4.3 Bounds on Binomial Random Variables v vi Contents 2.5 Martingales 2.5.1 Martingale Convergence Theorem 2.5.2 Azuma−Hoeffding Inequality 2.6 Order Statistics and Extreme Value Theory 2.7 Notes and discussion 2.8 Exercises for Chapter 2viii Contents 8 Preferential Attachment Models 8.1 Motivation for the Preferential Attachment Model 8.2 Introduction of the Model 8.3 Degrees of Fixed Vertices 8.4 Degree Sequences of Preferential Attachment Models 8.5 Concentration of the Degree Sequence 8.6 Expected Degree Sequence 8.6.1 Expected Degree Sequence for Preferential Attachment Trees 8.6.2 Expected Degree Sequence for m ≥ 2 * 8.7 Maximal Degree in Preferential Attachment Models 8.8 Related Results for Preferential Attachment Models 8.9 Related Preferential Attachment Models 8.10 Notes and Discussion 8.11 Exercises for Chapter 8 Appendix Some Facts about Measure and Integration References Index Glossary xviii Course OutlineHere is some more explanation as well as a possible itinerary of a master course on random graphs. We include Volume II (see van der Hofstad (2018+)) in the course outline:Start with the introduction to real-world networks in Chapter 1, which forms the inspiration for what follows. Continue with Chapter 2, which gives the necessary probabilistic tools used in all later chapters, and pick those topics that your students are not familiar with and that are used in the later chapters that you wish to treat. Chapter 3 introduces branching processes, and is used in Chapters 4 and 5, as well as in most of Volume II.After these preliminaries, you can start with the classical Erdős-Rényi random graph as covered in Chapters 4 and 5. Here you can choose the level of detail, and decide whether you wish to do the entire phase transition or would rather move on to the random graphs models for complex networks. It is possible to omit Chapter 5 before moving on.After this, you can make your own choice of topics from the models for real-world networks. There are three classes of models for complex networks that are treated in this book. You can choose how much to treat in each of these models. You can either treat few models and discuss many aspects, or instead discuss many models at a less deep level. The introductory chapters about the three models, Chapter 6 for inhomogeneous random graphs, Chapter 7 for the configuration model, and Chapter 8 for preferential attachment models, provide a basic introduction to them, focussing on their degree structure. These introductory chapters need to be read in order to understand the later chapters about thes...

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Percolation and Random Graphs

Hofstad¹

2009

New Perspectives in Stochastic Geometry

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In this section, we define percolation and random graph models, and survey the features of these models. 1.1 Introduction and notation In this section, we discuss random networks. In Section 1.2, we study perco-lation, which is obtained by independently removing vertices or edges from a graph. Percolation is a model of a porous medium, and is a paradigm model of statistical physics. Think of the bonds in an infinite graph that are not removed as indicating whether water can flow through this part of the medium. Then, the interesting question is whether water can percolate, or, alternatively, whether there is an infinite connected component of bonds that are kept? As it turns out, the answer to this question depends sensitively on the fraction of bonds that are kept. When we keep most bonds, then the kept or occupied bonds form most of the original graph. In particular, an infinite connected component may exist, and if this happens, we say that the system percolates. On the other hand, when most bonds are removed or vacant, then the connected components tend to be small and insignificant. Thus, percolation admits a phase transition. Despite the simplicity of the model, the results obtained up to date are far from complete, and many aspects of percolation, particularly of its critical behavior, are ill understood. In Section 1.2 we shall discuss the basics of percolation, and highlight some important open questions. The key challenge in percolation is to uncover the relation between the percolation critical behavior and the properties of the underlying graph from which we obtain percolation by removing edges. In Section 1.3, we discuss random graphs. While in percolation, the random network considered naturally lives on an infinite graph, in random graph theory one considers random finite graphs. Thus, all random graphs are obtained by removing edges from the complete graph, or by adding edges to an empty graph. An important example of a random graph is obtained by independently removing bonds from a finite graph, which makes it clear that there is a strong link to percolation. However, also other mechanisms are possible to generate a random graph. We shall discuss some of the basics of random graph theory, focussing 2 Percolation and random graphs on the phase transition of the largest connected component and the distances in random graphs. The random graph models studied here are inspired by applications , and we shall highlight real-world networks that these random graphs aim to model to some extent. The fields that this contribution covers, percolation and random graph theory , have attracted tremendous attention in the past decades, and enormous progress has been made. It is impossible to cover all material appearing in the literature, and we believe that one should not aim to do so. Thus, we have strived to cover the main results which have been proved, as well as recent results in which we expect that more progress shall be made in the (near?) future, and we list open problems which we find of interest. We ...

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Cited by 10 publications

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Random Graphs and Complex Networks

Percolation and Random Graphs

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