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 are going to discuss recent progress on many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

/pdf/recent-progress-in-combinatorial-random-matrix-theory-xz6tr7dizu.pdf

Recent progress in combinatorial random matrix theory

We investigate the local distribution of roots of random functions of the form $F_n(z)= \sum_{i=1}^n \xi_i \phi_i(z) $, where $\xi_i$ are independent random variables and $\phi_i (z) $ are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics. 
We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $\xi_i$ are gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers. 
Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment of all of them. 
Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erd{o}s and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series.

Roots of random functions: A general condition for local universality

We demonstrate that the problem of training neural networks with small (average) squared error is computationally intractable. Consider a data set of M points (Xi, Yi), i = 1,2, ..., M, where Xi are input vectors from Rd, Yi are real outputs (Yi ∈ R). For a network f0 in some class F of neural networks, (1/M) Σi=1M (f0(Xi)- Yi)2)1/2 - inff∈F(1/M) Σi=1M (f(Xi) - Yi)2)1/2 is the (avarage) relative error occurs when one tries to fit the data set by f0. We will prove for several classes F of neural networks that achieving a relative error smaller than some fixed positive threshold (independent from the size of the data set) is NP-hard.

/pdf/on-the-infeasibility-of-training-neural-networks-with-small-3881zgpqd2.pdf

On the Infeasibility of Training Neural Networks with Small Squared Errors

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications.

Random matrices: tail bounds for gaps between eigenvalues

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 the Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give sub-exponential approximation ratios with high probability.

Van Vu

Papers

Recent progress in combinatorial random matrix theory

Roots of random functions: A general condition for local universality

On the Infeasibility of Training Neural Networks with Small Squared Errors

Random matrices: tail bounds for gaps between eigenvalues

Concentration of random determinants and permanent estimators