We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Probability and Measure

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.

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Random Graphs and Complex Networks

We analyse the strong approximation of the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero by a positivity preserving drift-implicit Euler-type method. As an error criterion, we use the p th mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process, the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with Lipschitz coefficients, despite the fact that the CIR process has a non-Lipschitz diffusion coefficient.

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An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process

Dans cet article, nous etudions l’approximation d’une mesure de probabilite $\mu$ sur $\mathbb{R}^{d}$ par sa mesure empirique $\hat{\mu}_{N}$, interpretee comme quantification aleatoire. Comme critere d’erreur, nous considerons une moyenne de metrique de Wasserstein d’ordre $p$. Dans le cas $2p<d$, nous etablissons des bornes superieures et inferieures ameliorees pour l’erreur, une formule haute resolution. De plus, nous donnons une estimation universelle a base de moments, nommeee estimation du type Pierce. En particulier, nous prouvons que, sous de faibles hypotheses, la quantification par des mesures empiriques est d’ordre optimal.

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Constructive quantization: Approximation by empirical measures

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional Levy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Levy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the Levy measure around zero. In particular, one can derive upper bounds from the Blumenthal--Getoor index of the Levy process. In the case where the Blumenthal--Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the Levy process does not incorporate a Wiener process or if the Blumenthal--Getoor index $\beta$ is larger than $\frac{4}{3}$, then the upper bound is of order $\tau^{-({4-\beta})/({6\beta})}$ when the runtime $\tau$ tends to infinity. Whereas in the case, where $\beta$ is in $[1,\frac{4}{3}]$ and the Levy process has a Gaussian component, we obtain bounds of order $\tau^{-\beta/(6\beta-4)}$. In particular, the error is at most of order $\tau^{-1/6}$.

Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction

https://www.mathematik.uni-marburg.de/~dereich/Preprints/quad_levy.pdf

A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

Steffen Dereich

Papers

An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process

Constructive quantization: Approximation by empirical measures

Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction

A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

Constructive quantization: approximation by empirical measures