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.

The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.

/pdf/from-the-littlewood-offord-problem-to-the-circular-law-ipfwwx962k.pdf

From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e
 −
 t
 232
 s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces.

/pdf/on-the-concentration-of-eigenvalues-of-random-symmetric-2t9molkm7s.pdf

On the concentration of eigenvalues of random symmetric matrices

In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Furedi and Komlos and also correct an incomplete argument in their proof.

Spectral norm of random matrices

In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Furedi and Komlos.

Let Qn denote a random symmetric (n×n)-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that Qn is nonsingular with probability 1-O(n-1/8+δ) for any fixed δ>0. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices

Van Vu

Papers

From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices

On the concentration of eigenvalues of random symmetric matrices

Spectral norm of random matrices

Spectral norm of random matrices

Random symmetric matrices are almost surely nonsingular