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

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

https://deim.urv.cat/~alexandre.arenas/publicacions/pdf/review.pdf

Synchronization in complex networks

This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists.

Introduction to the non-asymptotic analysis of random matrices.

Let H be a (k+1)-uniform, D-regular hypergraph on n vertices and let (H) be the minimum number of vertices left uncovered by a matching in H. Cj(H), the j-codegree of H, is the maximum number of edges sharing a set of j vertices in common. We prove a general upper bound on (H), based on the codegree sequence C2 (H), C3 (H),…. Our bound improves and generalizes many results on the topic, including those of Grable, Alon, Kim, and Spencer, and Kostochka and Rodl. It also leads to a substantial improvement in several applications. The key ingredient of the proof is the so-called polynomial technique, which is a new and useful tool to prove concentration results for functions with large Lipschitz coefficient. This technique is of independent interest. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 29–63, 2000

New bounds on nearly perfect matchings in hypergraphs: higher codegrees do help

Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. 
In this paper we consider the threshold $th_{H} (n)$ of the property that an Erdős-Renyi random graph (on $n$ points) contains an $H$-factor. Our results determine $th_{H} (n)$ for all strictly balanced $H$. 
The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random $k$-uniform hypergraph, solving the well-known "Shamir's problem."

Factors in random graphs

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

/pdf/roots-of-random-polynomials-with-coefficients-of-polynomial-is0ko9erw5.pdf

Roots of random polynomials with coefficients of polynomial growth

We prove the analogue of Turan's Theorem in random graphs with edge probability p(n) > n-1/(k-1.5). With probability 1 - o(1), one needs to delete approximately 1/k-1-fraction of the edges in a random graph in order to destroy all cliques of size k.

Turán's theorem in sparse random graphs

Additive combinatorics is a subfield of combinatorics, and so it is no surprise that graph theory plays an important role in this theory. Graph theory has already made an implicit appearance in previous chapters, most notably in the proof of the Balog–Szemeredi–Gowers theorem (Theorem 2.29). However there are several further ways in which graph theoretical tools can be utilized in additive combinatorics. We will only discuss a representative sample of these applications here. First we discuss Turan's theorem , which shows that sparse graphs contain large independent sets, and which is useful for constructing sum-free sets. Next we give a very brief tour of Ramsey theory , which allows one to find monochromatic structures in colored graphs (or other colored objects), in particular allowing one to find monochromatic progressions in any coloring of the integers ( van der Waerden's theorem ). Then we use some results about connectivity of dense graphs to establish the Balog–Szemeredi–Gowers theorem , which relates partial sum sets to complete sum sets and which has already been exploited in Chapter 2. Finally, we use the theory of commutative directed graphs to establish the Plunnecke inequalities , which are perhaps the sharpest inequalities known for sum sets and which strengthen several of the results already established in Chapter 2. In Chapter 10 and Chapter 11 we shall discuss one final graph-theoretical tool, the Szemeredi regularity lemma , which has had many applications in several areas of discrete mathematics, but which in additive combinatorics has had an especially crucial role in the study of arithmetic progressions in dense sets.

Van Vu

Papers

New bounds on nearly perfect matchings in hypergraphs: higher codegrees do help

Factors in random graphs

Roots of random polynomials with coefficients of polynomial growth

Turán's theorem in sparse random graphs

Additive Combinatorics: Graph-theoretic methods