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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

/pdf/prediction-learning-and-games-31z1lqjpec.pdf

Prediction, learning, and games

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

http://www.dmg.tuwien.ac.at/drmota/reinf5.pdf

Reinforced weak convergence of stochastic processes

The purpose of this paper is to provide a central limit theorem for the number of occurrences of double triangles in random planar maps. This is the first result of this kind that goes beyond face counts of given valency. The method is based on generating functions, an involved combinatorial decomposition scheme that leads to a system of catalytic functional equations and an analytic extension of the Quadratic Method to systems of equations.

https://drops.dagstuhl.de/opus/volltexte/2018/8912/pdf/LIPIcs-AofA-2018-19.pdf/

The Number of Double Triangles in Random Planar Maps

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/pdf/irregularities-of-continuous-distributions-1tn0vc6ldr.pdf

Irregularities of continuous distributions

Pattern occurrences in random planar maps

Two decision-makers A and B observe sequentially a given permutation of n uniquely rankable options. A and B have one choice each (without recall) and both must make a choice. At each step only the relative ranks are known, and A has the priority of choice. At the end the (absolute) ranks are compared and the winner is the one who has chosen the better rank. Extending results by Enns and Ferenstein [6] and Berry et al. [1] this article gives, for both A and B , the optimal strategy and the corresponding winning probabilities. We show in particular that the limiting winning probabilities for A and B
do exist, which closes a most important gap in the work of previous authors. This also provides an algorithm for numerically computing the limiting value of these probabilities. Although our proof is analytic in a strong sense, it is interesting to see that it would have been very hard to assemble it without the help of computer algebra. The reason is that the functions we have to investigate display subranges of indices which contrast considerably with respect to error terms when certain terms are replaced by approximations, and that computations were very helpful to locate those ranges where a particularly fine tuning of error estimates turned out to be indispensable.

Michael Drmota

Papers

Reinforced weak convergence of stochastic processes

The Number of Double Triangles in Random Planar Maps

Irregularities of continuous distributions

Pattern occurrences in random planar maps

The complete solution of the competitive rank selection problem