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.

https://people.math.ethz.ch/~sudakovb/small-subgraphs.pdf

Note: Small subgraphs of random regular graphs

John-type theorems for generalized arithmetic progressions and iterated sumsets

l∗A = {a1 + · · ·+ al|ai ∈ Ai, ai = aj} Among the most well-known results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently large odd numbers, and Waring’s conjecture (proved by Hilbert, Hardy and Littlewood, Hua, and many others), which asserts that for any given r, there is a number l such that l∗Nr (N denotes the set of rth powers) contains all sufficiently large positive integers (see [29] for an excellent exposition concerning these results) In recent years, a considerable amount of attention has been paid to the study of finite sumsets Given a finite set A and a positive integer l, the natural analogue of Vinogadov-Waring results is to show that under proper conditions, the sumset lA (l∗A) contains a long arithmetic progression Let us assume that A is a subset of the interval [n] = {1,    , n}, where n is a large positive integer The concrete problem we would like to address is to estimate the minimum length of the longest arithmetic progression in lA (l∗A) as a function of l, n, and |A| We denote this function by f(|A|, l, n) (f∗(|A|, l, n)), following the notation in [13] Many estimates for f(|A|, l, n) have been discovered by Bourgain, Freiman, Halberstam, Green, Ruzsa, and Sarkozy (see Section 3), but most of these results focus on sets with very high density, namely |A| is close to n Estimating f∗(|A|, l, n) seems much harder, and not much was known prior to our study

/pdf/long-arithmetic-progressions-in-sumsets-thresholds-and-xtbfv6bhuh.pdf

Long arithmetic progressions in sumsets: Thresholds and bounds

Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc. 
In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.

A Simple SVD Algorithm for Finding Hidden Partitions

Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$ is polynomial in $n$ with very high probability. The main novelty here is that we allow $N_n$ to have discrete distribution.

Van Vu

Papers

Note: Small subgraphs of random regular graphs

John-type theorems for generalized arithmetic progressions and iterated sumsets

Long arithmetic progressions in sumsets: Thresholds and bounds

A Simple SVD Algorithm for Finding Hidden Partitions

The condition number of a randomly perturbed matrix