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

The main purpose of this paper is to discuss the asymptotic behaviour of the difference s
 q,k(P(n)) - k(q-1)/2 where s
 q,k (n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q
 1-ary and q
 2-ary digital expansions where q
 1 and q
 2 are coprime.

Functional limit theorems for digital expansions

A digital search tree (DST) -- one of the most fundamental data structures on words -- is a digital tree in which keys (strings, words) are stored directly in (internal) nodes Such trees find myriad of applications from the popular Lempel-Ziv'78 data compression scheme to distributed hash tables The profile of a DST measures the number of nodes at the same distance from the root; it is a function of the number of stored strings and the distance from the root Most parameters of DST (eg, height, fill-up) can be expressed in terms of the profile However, from the inception of DST, the analysis of the profile has been elusive and it has become a prominent open problem in the area of analysis of algorithms We make here the first, but decisive, step towards solving this problem We present a precise analysis of the average profile when stored strings are generated by a biased memoryless source The main technical difficulty of analyzing the profile lies in solving a sophisticated recurrence equation We present such a solution for the Poissonized version of the problem (ie, when the number of stored strings is generated by a Poisson distribution) in the Mellin transform domain To accomplish it, we introduce a novel functional operator that allows us to express the solution in an explicit form, and then using analytic algorithmics tools to extract the asymptotic behavior of the profile This analysis is surprisingly demanding but once it is carried out it reveals unusually intriguing and interesting behavior The average profile undergoes several phase transitions when moving from the root to the longest path At first, it resembles a full tree until it abruptly starts growing polynomially and it oscillates in this range Our results are derived by methods of analytic algorithmics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis and uniform asymptotic analysis

Un)expected behavior of digital search tree profile

Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1 − o(1) asymptotically equal to c log n, where c a 2.529 is determined explicitly. A similar result is also true for random 2-connected planar graphs.Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.

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The maximum degree of random planar graphs

Mathematics and Computer Science III

The main goal of this paper is to determine the asymptotic behavior of the number X_n of cut-vertices in random planar maps with n edges It is shown that X_n/n → c in probability (for some explicit c>0) For so-called subcritial subclasses of planar maps like outerplanar maps we obtain a central limit theorem, too

https://drops.dagstuhl.de/opus/volltexte/2020/12040/pdf/LIPIcs-AofA-2020-10.pdf/

Michael Drmota

Papers

Functional limit theorems for digital expansions

Un)expected behavior of digital search tree profile

The maximum degree of random planar graphs

Mathematics and Computer Science III

Cut Vertices in Random Planar Maps.