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 Mn =(źij)1≤i,j≤n be a real symmetric random matrix in which the upper-triangular entries źij, i

Random matrices have simple spectrum

The independence number of a graph and its chromatic number are known to be hard to approximate. Due to recent complexity results, unless coRP = NP, there is no polynomial time algorithm which approximates any of these quantities within a factor of n
1−∈ for graphs on n vertices. We show that the situation is significantly better for the average case. For every edge probability p = p(n) in the range n
−1/2+∈ ≤ p ≤ 3/4, we present an approximation algorithm for the independence number of graphs on n vertices, whose approximation ratio is O((np)1/2/log n) and whose expected running time over the probability space G(n, p) is polynomial. An algorithm with similar features is described also for the chromatic number. A key ingredient in the analysis of both algorithms is a new large deviation inequality for eigenvalues of random matrices, obtained through an application of Talagrand's inequality.

Approximating the independence number and the chromatic number in expected polynomial time

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 [10] 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.

The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs.

Generating Random Regular Graphs

For fixed $$m > 1$$
 , we study the product of $$m$$
 independent $$N \times N$$
 elliptic random matrices as $$N$$
 tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $$1$$
 , to the $$m$$
 -th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Gotze–Tikhomirov (On the asymptotic spectrum of products of independent random matrices, available at http://arxiv.org/abs/1012.2710
 ) and O’Rourke–Soshnikov (J Probab 16(81):2219–2245, 2011) concerning the product of independent iid random matrices.

Products of Independent Elliptic Random Matrices

Let $\\eta_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset $\\bv$ of $n$ numbers $v_1, ..., v_n$, the \\emph{concentration probability} $\\P_1(\\bv)$ of $\\bv$ is defined as $\\P_1(\\bv) := \\sup_{x} \\P(v_1 \\eta_1+ ... v_n \\eta_n=x)$. A classical result of Littlewood-Offord and Erd\\H os from the 1940s asserts that if the $v_i $ are non-zero, then this probability is at most $O(n^{-1/2})$. Since then, many researchers obtained better bounds by assuming various restrictions on $\\bv$. \nIn this paper, we give an asymptotically optimal characterization for all multisets $\\bv$ having large concentration probability. This allow us to strengthen or recover several previous results in a straightforward manner.

Van Vu

Papers

Random matrices have simple spectrum

Approximating the independence number and the chromatic number in expected polynomial time

Generating Random Regular Graphs

Products of Independent Elliptic Random Matrices

A sharp inverse Littlewood-Offord theorem: Littlewood-Offord Theorem