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

Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

Approximation Algorithms

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

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Expander graphs and their applications

Motivation: Our goal is to construct a model for genetic regulatory networks such that the model class: (i) incorporates rule-based dependencies between genes; (ii) allows the systematic study of global network dynamics; (iii) is able to cope with uncertainty, both in the data and the model selection; and (iv) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes. Results: We introduce Probabilistic Boolean Networks (PBN) that share the appealing rule-based properties of Boolean networks, but are robust in the face of uncertainty. We show how the dynamics of these networks can be studied in the probabilistic context of Markov chains, with standard Boolean networks being special cases. Then, we discuss the relationship between PBNs and Bayesian networks—a family of graphical models that explicitly represent probabilistic relationships between variables. We show how probabilistic dependencies between a gene and its parent genes, constituting the basic building blocks of Bayesian networks, can be obtained from PBNs. Finally, we present methods for quantifying the influence of genes on other genes, within the context of PBNs. Examples illustrating the above concepts are presented throughout the paper.

Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks

Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets. >

The influence of variables on Boolean functions

The following is an extended abstract for two papers, one written by Kahn and Szemeredi, the other written by Friedman, which have been combined at the request of the STOC committee. The introduction was written jointly, the second section by Kahn and Szemeredi, and the third by Friedman, Let G be a d-regular (i.e. each vertex has degree d) undirected graph on n nodes. It’s adjacency matrix is symmetric, and therefore has real eigenvalues Ar = d 2 x2 >_ *-. >_ X, with IX,] 5 d. Graphs for which X2 and

On the second eigenvalue of random regular graphs

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ∣I∣ for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0963548301004631

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

Let f(d) be the smallest number so that every set in R d of diameter I can be partitioned into f(d) sets of diameter smaller than 1. Borsuk's conjecture was that f(d) = d + 1. We prove that f(d) ≥ (1.2)√ d for large d

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A counterexample to Borsuk's conjecture

LetX be a probability space and letf: Xn → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byIf(k), as follows: Foru=(u1,u2,…,un−1) ∈Xn−1 consider the setlk(u)={(u1,u2,...,uk−1,t,uk,…,un−1):t ∈X}.\(I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).\)

Jeff Kahn

Papers

The influence of variables on Boolean functions

On the second eigenvalue of random regular graphs

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

A counterexample to Borsuk's conjecture

The influence of variables in product spaces