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

Let $H$ be a fixed graph and $\mathcal{G}$ a subcritical graph class. In this paper we show that the number of occurrences of $H$ (as a subgraph) in a uniformly at random graph of size $n$ in $\mathcal{G}$ follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations. As a case study, we get explicit expressions for the number of triangles and cycles of length four for the family of series-parallel graphs.

Subgraph statistics in subcritical graph classes

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. 
First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. 
Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions.

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

Variable-to-variable (VV) codes are very attractive yet not well understood data compression schemes. In 1972, Khodak claimed to provide upper and lower bounds for the achievable redundancy rate, however, he did not offer explicit construction of such codes. In this paper, we first present a constructive and transparent proof of Khodak's result showing that for memoryless sources there exists a code with the average redundancy bounded by D -5/3, where D is the average delay (e.g., the average length of a dictionary entry). We also describe an algorithm that constructs a VV length code with a small redundancy rate for large D. Then, we discuss several generalizations. We prove that the worst case redundancy does not exceed D -4/3. Furthermore, we provide similar upper bound for Markov sources (of order 1). Finally, we consider bounds that are valid for almost all memoryless and Markov sources for which the set of exceptional source parameters has zero measure. In particular, for all memoryless sources outside this exceptional class, we prove there exists a VV code with the average redundancy rate bounded by D -1-m/3+ epsiv and the worst case redundancy rate bounded by D -1-m/3+ epsiv, where m is the cardinality of the alphabet. We complete our analysis with a lower bound showing that for all VV codes the average and the worst case redundancy rates are at least D -2m-1- epsiv for almost all memoryless sources in the sense that the set of exceptional source parameters has zero measure. We prove these results using techniques of Diophantine approximations.

/pdf/on-the-construction-of-explicit-khodak-s-code-and-its-34p4b551y8.pdf

On the Construction of (Explicit) Khodak's Code and Its Analysis

For a sequence ofm urns we investigate how the number of urns satisfying a certain condition (e.g., being empty) evolves in time when after each time unit a ball is thrown. We show for a variety of urn models that this process (suitably normalized) converges weakly to a Gaussian process.

Michael Drmota

Papers

Subgraph statistics in subcritical graph classes

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

On the Construction of (Explicit) Khodak's Code and Its Analysis

A unified presentation of some urn models

Möbius orthogonality for the Zeckendorf sum-of-digits function