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

Randomized algorithms have become a central part of the algorithms curriculum based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high- probability estimates on the performance of randomized algorithms. It covers the basic tool kit from the Chernoff-Hoeffding (CH) bounds to more sophisticated techniques like Martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities, and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as CH bounds in dependent settings. The authors emphasize comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

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Concentration of Measure for the Analysis of Randomized Algorithms

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

'Satisfiability (SAT) related topics have attracted researchers from various disciplines: logic, applied areas such as planning, scheduling, operations research and combinatorial optimization, but also theoretical issues on the theme of complexity and much more, they all are connected through SAT. My personal interest in SAT stems from actual solving: The increase in power of modern SAT solvers over the past 15 years has been phenomenal. It has become the key enabling technology in automated verification of both computer hardware and software' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University). 'Bounded Model Checking (BMC) of computer hardware is now probably the most widely used model checking technique. The counterexamples that it finds are just satisfying instances of a Boolean formula obtained by unwinding to some fixed depth a sequential circuit and its specification in linear temporal logic. Extending model checking to software verification is a much more difficult problem on the frontier of current research. One promising approach for languages like C with finite word-length integers is to use the same idea as in BMC but with a decision procedure for the theory of bit-vectors instead of SAT. All decision procedures for bit-vectors that I am familiar with ultimately make use of a fast SAT solver to handle complex formulas' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University). 'Decision procedures for more complicated theories, like linear real and integer arithmetic, are also used in program verification. Most of them use powerful SAT solvers in an essential way. Clearly, efficient SAT solving is a key technology for 21st century computer science. I expect this collection of papers on all theoretical and practical aspects of SAT solving will be extremely useful to both students and researchers and will lead to many further advances in the field' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University).

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Handbook of Satisfiability

are independent random variables which take values either 0 or 1, and Y is a multi-variable polynomial in 's with positive coefficients. We give a condition which guarantees that Y concentrates strongly around its mean even when several variables could have a large effect on Y. Some applications will be discussed.

/pdf/concentration-of-multivariate-polynomials-and-its-29toeerpuq.pdf

Concentration of Multivariate Polynomials and Its Applications

We consider the random 2-satisfiability (2-SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/nα, the problem is known to have a phase transition at αc=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α−(n,δ), α+(n,δ)) such that the probability of satisfiability is greater than 1−δ for α α+. We show that W(n,δ)=(1−Θ(n−1/3), 1+Θ(n−1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+e)n, where e may depend on n as long as |e| is sufficiently small and |e|n1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(−Θ(ne3)) above the window, and goes to one like 1−Θ(n−1|e|−3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001

/pdf/the-scaling-window-of-the-2-sat-transition-jyhr4bq55d.pdf

The scaling window of the 2-SAT transition

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [9] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d=O(n1/3-e), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author.Besides being perhaps the only algorithm which works for relatively large d in practical time, our result also has a significant theoretical value, as it can be used to derive many properties of uniform random regular graphs.

/pdf/generating-random-regular-graphs-36l22t1qcg.pdf

Generating random regular graphs

For every fixedk≥3 there exists a constantc
 k
 with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc
 k
 ND
 −1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc
 3
 ND
 −1/2ln3/2
 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N
 1/2ln3/2
 N), improving results by various authors.

/pdf/nearly-perfect-matchings-in-regular-simple-hypergraphs-nb3b5k6pme.pdf

Nearly perfect matchings in regular simple hypergraphs

Let G be a graph with maximum degree Δ(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfieswhere o(l) goes to zero as Δ(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number:provided g(G) < 4.

https://www.cc.gatech.edu/~vigoda/RandAlgs/Kim.pdf

Jeong Han Kim

Papers

Concentration of Multivariate Polynomials and Its Applications

The scaling window of the 2-SAT transition

Generating random regular graphs

Nearly perfect matchings in regular simple hypergraphs

On Brooks' Theorem for Sparse Graphs