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.

This paper studies the symmetry of random regular graphs and random graphs. Our main result shows that for all 3 ≤ d ≤ n - 4 the random d-regular graph on n vertices almost surely has no nontrivial automorphisms. This answers an open question of N. Wormald [13].

On the asymmetry of random regular graphs and random graphs

Choose n random, independent points in R d according to the standard normal distribution. Their convex hull K n is the Gaussian random polytope. We prove that the volume and the number of faces of K n satisfy the central limit theorem, settling a well-known conjecture in the field.

/pdf/central-limit-theorems-for-gaussian-polytopes-4nfk0ngfi4.pdf

Central limit theorems for Gaussian polytopes

Let ξ be a real random variable with mean zero and variance one and A ={a 1; …; a n } be a multi-set in R d . The random sum 

$$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$

where ξ i are iid copies of ξ is of fundamental importance in probability and its applications.

/pdf/small-ball-probability-inverse-theorems-and-applications-fz91s888n5.pdf

Small Ball Probability, Inverse Theorems, and Applications

In this paper, we consider the ensemble of $n \times n$ Wigner Hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{ \ell k}$ are given by $h_{\ell k} = n^{-1/2} ( x_{\ell k} + \sqrt{-1} y_{\ell k} )$, where $x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random variables with mean zero and variance $1/2$, $y_{\ell\ell}=0$ and $x_{\ell \ell}$ have mean zero and variance $1$. We assume the distribution of $x_{\ell k}, y_{\ell k}$ to have subexponential decay. In \cite{ERSY2}, four of the authors recently established that the gap distribution and averaged $k$-point correlation of these matrices were \emph{universal} (and in particular, agreed with those for GUE) assuming additional regularity hypotheses on the $x_{\ell k}, y_{\ell k}$. In \cite{TVbulk}, the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the $x_{\ell k}, y_{\ell k}$. In this short note we observe that the arguments of \cite{ERSY2} and \cite{TVbulk} can be combined to establish universality of the gap distribution and averaged $k$-point correlations for all Wigner matrices (with subexponentially decaying entries), with no extra assumptions.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2010/0017/0004/MRL-2010-0017-0004-a007.pdf

Bulk universality for Wigner hermitian matrices with subexponential decay

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite $C_0$th moment for some sufficiently large constant $C_0$. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions. As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite $C_0$th moment rather than exponential decay.

Van Vu

Papers

On the asymmetry of random regular graphs and random graphs

Central limit theorems for Gaussian polytopes

Small Ball Probability, Inverse Theorems, and Applications

Bulk universality for Wigner hermitian matrices with subexponential decay

Random covariance matrices: Universality of local statistics of eigenvalues