Random Graphs and Complex Networks

Abstract

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...

Average nearest neighbor degrees in scale-free networks

“…degree sequences, formally defined as follows [13]. Let D be a positive integer-valued random variable, with cumulative distribution function F and probability density function f .…”

“…The RCM can be applied successfully only if the probability to obtain a simple graph converges to a nonzero value as n → ∞. It is well-known (see [13,Chapter 7]) that this is indeed the case if and only if D has finite second moment. Another way to obtain a simple graph is to simply remove all self-loops and replace all multiple edges between i and j by a single edge.…”

“…Another way to obtain a simple graph is to simply remove all self-loops and replace all multiple edges between i and j by a single edge. This model, called the erased configuration model (ECM), generates a simple graph with correct asymptotic degree distribution whenever D n = IID(D) and D has a finite mean (see [13,Theorem 7.10]). …”

Limit theorems for the weights and the degrees in anN-interactions random graph model

“…Random graphs are used to describe real-life networks. For overviews of random graph models and their properties see [1][2][3]. It is known that many real-life networks (e.g.…”

“…A rigorous definition of the preferential attachment model was given in [5], where a mathematical proof of the power law degree distribution was presented for d Ä n 1=15 (d is the degree, n is the number of steps). For the recent development of the topic see [3,6]. Besides the degree distribution, other characteristics are also worth studying.…”