Random Graphs and Complex Networks

TLDR: This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.

Abstract: This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

Figures

Figure 7.4 The pairing of half-edges in the configuration model. In this figure, n = 6, and d1 = 3, d2 = 2, d3 = 1, d4 = 4, d5 = d6 = 2. The first half-edge incident to vertex 2 is paired to the second half-edge incident to vertex 4, while the second half-edge incident to vertex 2 is paired to the second half-edge incident to vertex 5.

Figure 7.5 The pairing of half-edges in the configuration model. In this figure, n = 6, and d1 = 3, d2 = 2, d3 = 1, d4 = 4, d5 = d6 = 2. The third half-edge incident to vertex 4 is paired to the half-edge incident to vertex 3, while the third half-edge incident to vertex 1 is paired to the last half-edge incident to vertex 6, and the fourth half-edge incident to vertex 4 is paired to the last half-edge incident to vertex 5.

Figure 5.3 The degree distribution of ERn(λ/n) with n = 100, λ = 2.

Figure 5.4 The degree distribution of ERn(λ/n) with n = 10,000, λ = 1 (left), and the probability mass function of a Poisson random variable with parameter 1 (right).

Figure 1.11 The average number of friends of friends of individuals of a fixed degree. “Random” indicates the average degree, while “Diagonal” indicates the line where the average degrees of neighbors of an individual equals that of the individual.

Table 1.2 Evolution of the Facebook diameter. Note: it denotes the Italian subgraph of Facebook, se the Swedish, us the United States and fb the entire Facebook Graph

Full text

RANDOM GRAPHS AND COMPLEX NETWORKS
Volume I
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
1 Introduction 1
1.1 Motivation 1
1.2 Graphs and Their Degree and Connectivity Structure 2
1.3 Complex Networks: the Infamous Internet Example 5
1.4 Scale-free, Highly Connected & Small-World Graph Sequences 11
1.4.1 Scale-Free Graph Sequences 11
1.4.2 Highly Connected Graph Sequences 14
1.4.3 Small-World Graph Sequences 15
1.5 Further Network Statistics 16
1.6 Other Real-World Network Examples 20
1.6.1 Six Degrees of Separation and Social Networks 20
1.6.2 Facebook 22
1.6.3 Kevin Bacon Game and the Internet Movie Data Base 25
1.6.4 Erd˝os Numbers and Collaboration Networks 28
1.6.5 The World-Wide Web 31
1.6.6 The Brain 39
1.7 Tales of Tails 40
1.7.1 Old Tales of Tails 40
1.7.2 New Tales of tails 42
1.7.3 Power laws, Their Estimation, and Criticism 43
1.7.4 Network Data 44
1.8 Random Graph Models For Complex Networks 45
1.9 Notation 49
1.10 Notes and Discussion 50
1.11 Exercises for Chapter 1 52
Part I Preliminaries 53
2 Probabilistic Methods 55
2.1 Convergence of Random Variables 55
2.2 Coupling 60
2.3 Stochastic Ordering 63
2.3.1 Examples of Stochastically Ordered Random Variables 64
2.3.2 Stochastic Ordering and Size-Biased Random Variables 65
2.3.3 Consequences of Stochastic Domination 66
2.4 Probabilistic Bounds 67
2.4.1 First and Second Moment Methods 67
2.4.2 Large Deviation Bounds 68
2.4.3 Bounds on Binomial Random Variables 69
v

Frequently asked questions

Q1. What are the contributions in "Random graphs and complex networks" ?

In this first chapter, the authors give an introduction to random graphs and complex networks. The authors discuss examples of real-world networks and their empirical properties, and give a brief introduction to the kinds of models that they investigate in the book. Further, the authors introduce the key elements of the notation used throughout the book. 

Q2. What are the future works mentioned in the paper "Random graphs and complex networks" ?

Since the clustering 1. 5 Further Network Statistics 17 coefficient only depends on the local neighborhoods of vertices, these random graph models are often closely related to the original models. 1. 5 Further Network Statistics 19 Since network community structures are not clearly defined, such methodologies are often ill defined, but this only makes them more interesting and worthy to study. Then, the authors can interpret ρG as the correlation coefficient of the random variables X and Y ( see Exercise 1. 5 ). 

Q3. What is the key ingredient in the investigation of the degrees in preferential attachment models?

Proposition 8.4 is a key ingredient in the investigation of the degrees in preferential attachment models, and is used in many related results for other models. 

Q4. What is the reason why the ERn(p) is the closely related to random?

In fact, ERn(p) for the entire regime of p ∈ [0, 1] can be understood using coalescent processes, for which the multiplicative coalescent is most closely related to random graphs. 

Q5. Why does the author expect that certain events become more likely when p increases?

Because of the monotone nature of ERn(p) one expects that certain events become more likely, and random variables grow larger, when p increases. 

Q6. Why did the increase in computational power have a profound impact on the empirical studies of large networks?

Due to the increased computational power, large data sets can now easily be stored and investigated, and this has had a profound impact on the empirical studies of large networks. 

Q7. How can one check whether a power law is present for a degree sequence?

When all degrees in a network are observed, one can visually check whether a power law is present for the degree sequence by making the log-log plot of the frequency of occurrence of vertices with degree k versus k, and verifying whether this is close to a straight line. 

Q8. What is the probability that the configuration model yields a simple graph?

In this section, the authors investigate the probability that the configuration model yields a simple graph, i.e., the probability that the graph produced in the configuration model has no self-loops nor multiple edges. 

Q9. How do the authors prove the coupling result for the Chung-Lu random graph?

In this section, the authors prove a coupling result for the Chung-Lu random graph, where the edge probabilities are given byp(CL)ij = wiwj `n ∧ 1, (6.8.1)where again`n = ∑ i∈[n] wi. (6.8.2)The authors denote the resulting graph by CLn(w).