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.

We prove a general recursive inequality concerning /spl mu//sup */(R), the asymptotic (least) density of the best binary covering codes of radius R. In particular, this inequality implies that /spl mu//sup */(R)/spl les/e/spl middot/(RlogR+logR+loglogR+2), which significantly improves the best known density 2/sup R/R/sup R/(R+1)/R!. Our inequality also holds for covering codes over arbitrary alphabets.

/pdf/covering-codes-with-improved-density-42n4kt23q6.pdf

Covering codes with improved density

A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$. A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan. 
Let $G$ be a graph with largest eigenvalue $\lambda_1$ and let $\rho$ be the spectral radius of its universal cover. Friedman (2003) proved that every "new" eigenvalue of a random lift of $G$ is $O(\rho^{1/2}\lambda_1^{1/2})$ with high probability, and conjectured a bound of $\rho+o(1)$, which would be tight by results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved Friedman's bound to $O(\rho^{2/3}\lambda_1^{1/3})$. For $d$-regular graphs, where $\lambda_1=d$ and $\rho=2\sqrt{d-1}$, this translates to a bound of $O(d^{2/3})$, compared to the conjectured $2\sqrt{d-1}$. 
Here we analyze the spectrum of a random $n$-lift of a $d$-regular graph whose nontrivial eigenvalues are all at most $\lambda$ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is $O((\lambda \vee \rho) \log \rho)$. This result is tight up to a logarithmic factor, and for $\lambda \leq d^{2/3-\epsilon}$ it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical $n$-lift of a Ramanujan graph is nearly Ramanujan.

Spectra of lifted Ramanujan graphs

We demonstrate that the problem of training neural networks with small mean-squared error is computationally intractable. This answers a question posed by Jones (1997).

On the infeasibility of training neural networks with small mean-squared error

We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean As an application, we show that Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give subexponential approximation ratios with high probability A positive answer to the main conjecture of the paper would lead to polynomial approximation ratios in the above problem

Concentration of Random Determinants and Permanent Estimators

We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focusing in particular on the Four Moment Theorem and its applications.

Van Vu

Papers

Covering codes with improved density

Spectra of lifted Ramanujan graphs

On the infeasibility of training neural networks with small mean-squared error

Concentration of Random Determinants and Permanent Estimators

Random matrices: The Four Moment Theorem for Wigner ensembles