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...

We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 − p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c (G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3 , where λ is fixed and positive. We show that for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c . In particular, we show that the largest cluster inside a scaling window of size |p − p c | = Θ(Ω −1 V −1/3 ) is of size Θ(V 2/3 ), while below this scaling window, it is much smaller, of order O(ǫ −2 log(V ǫ 3 )), with ǫ = Ω(p c − p). We also obtain an upper bound O(Ω(p − p c )V ) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p − p c )). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph.The latter is readily verified for several graphs with vertex set {0, 1, . . . , r − 1} n , including the Hamming cube on an alphabet of r letters (the n-cube, for r = 2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n > 6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount. We analyze the configuration model with degree power-law exponent $\tau>2$, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent $\tau-1>1$, or has even thinner tails ($\tau=\infty$). In this model, the degrees have a finite first moment, while the variance is finite for $\tau>3$, but infinite for $\tau\in(2,3)$. We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to $\alpha\log{n}$, where $\alpha\in(0,1)$ for $\tau\in(2,3)$, while $\alpha>1$ for $\tau>3$. Here $n$ denotes the size of the graph. For $\tau\in (2,3)$, it is known that the graph distance between two randomly chosen connected vertices is proportional to $\log \log{n}$ [Electron. J. Probab. 12 (2007) 703--766], that is, distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [J. Math. Phys. 49 (2008) 125218] of showing that $\log{n}$ is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees ($\tau\in[1,2)$) is studied in [Extreme value theory, Poisson--Dirichlet distributions and first passage percolation on random networks (2009) Preprint] where it is proved that the hopcount remains uniformly bounded and converges in distribution.Comment: Published in at http://dx.doi.org/10.1214/09-AAP666 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper we study a random graph with N nodes, where node j has degree D j and. We assume that 1 − F (x) ≤ cx −τ +1 for some τ > 3 and some constant c > 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance.The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N , when the base of the logarithm equalsThis confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.

We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distributions that are widely used in statistics and other fields. This definition allows the distribution to deviate from a pure power law arbitrarily but without affecting the power-law tail exponent. We then identify three estimators of these exponents that are proven to be statistically consistent-that is, converging to the true value of the exponent for any regularly varying distribution-and that satisfy some additional niceness requirements. In contrast to estimators that are currently popular in network science, the estimators considered here are based on fundamental results in extreme value theory, and so are the proofs of their consistency. Finally, we apply these estimators to a representative collection of synthetic and real-world data. According to their estimates, real-world scale-free networks are definitely not as rare as one would conclude based on the popular but unrealistic assumption that real-world data comes from power laws of pristine purity, void of noise and deviations.

We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent \tau. We investigate the case where $\tau\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting times of a "thinned" L\'{e}vy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case \tau>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erd\H{o}s-R\'{e}nyi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/11-AOP680 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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