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

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

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

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

A wealth of protein and DNA sequence data is being generated by genome projects and other sequencing efforts. A crucial barrier to deciphering these sequences and understanding the relations among them is the difficulty of detecting subtle local residue patterns common to multiple sequences. Such patterns frequently reflect similar molecular structures and biological properties. A mathematical definition of this "local multiple alignment" problem suitable for full computer automation has been used to develop a new and sensitive algorithm, based on the statistical method of iterative sampling. This algorithm finds an optimized local alignment model for N sequences in N-linear time, requiring only seconds on current workstations, and allows the simultaneous detection and optimization of multiple patterns and pattern repeats. The method is illustrated as applied to helix-turn-helix proteins, lipocalins, and prenyltransferases.

/pdf/detecting-subtle-sequence-signals-a-gibbs-sampling-strategy-56bu9neqln.pdf

Detecting Subtle Sequence Signals: A Gibbs Sampling Strategy for Multiple Alignment

The elements of many classical combinatorial structures can be naturally decomposed into components Permutations can be decomposed into cycles, polynomials over a finite field into irreducible factors, mappings into connected components In all of these examples, and in many more, there are strong similarities between the numbers of components of different sizes that are found in the decompositions of `typical' elements of large size For instance, the total number of components grows logarithmically with the size of the element, and the size of the largest component is an appreciable fraction of the whole This book explains the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition The discussion is conducted in the language of probability, enabling the theory to be developed under rather general and explicit conditions; for the finer conclusions, Stein's method emerges as the key ingredient The book is thus of particular interest to graduate students and researchers in both combinatorics and probability theory

Logarithmic Combinatorial Structures: A Probabilistic Approach

Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable result is due to Chen (1975). The method also provides an upper bound on the total variation distance to the Poisson distribution, and succeeds in cases where third and higher moments blow up. This paper presents Chen's results in a form that is easy to use and gives a multivariable extension, which gives an upper bound on the total variation distance between a sequence of dependent indicator functions and a Poisson process with the same intensity. A corollary of this is an upper bound on the total variation distance between a sequence of dependent indicator variables and the process having the same marginals but independent coordinates.

/pdf/two-moments-suffice-for-poisson-approximations-the-chen-2irpxe1jnb.pdf

Two Moments Suffice for Poisson Approximations: The Chen-Stein Method

The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applicability and utility of the Chen-Stein method. These examples include birthday coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.

https://projecteuclid.org/download/pdf_1/euclid.ss/1177012015

Poisson Approximation and the Chen-Stein Method

The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applica- bility and utility of the Chen-Stein method. These examples include birth- day coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.

https://www.jstor.org/stable/pdfplus/2245366.pdf

Poisson Approximation and the Chen-Stein ethod

Consider a system of particles moving on the integers with a simple exclusion interaction: each particle independently attempts to execute a simple symmetric random walk, but any jump which would carry a particle to an already occupied site is suppressed. For the system running in equilibrium, we analyze the motion of a tagged particle. This solves a problem posed in Spitzer's 1970 paper "Interaction of Markov Processes." The analogous question for systems which are not one-dimensional, nearest-neighbor, and either symmetric or one-sided remains open. A key tool is Harris's theorem on positive correlations in attractive Markov processes. Results are also obtained for the rightmost particle in the exclusion system with initial configuration $Z^-$, and for comparison systems based on the order statistics of independent motions on the line.

https://projecteuclid.org/download/pdf_1/euclid.aop/1176993602

Richard Arratia

Papers

Logarithmic Combinatorial Structures: A Probabilistic Approach

Two Moments Suffice for Poisson Approximations: The Chen-Stein Method

Poisson Approximation and the Chen-Stein Method

Poisson Approximation and the Chen-Stein ethod

The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on $Z$