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.

Theorems on extremal set systems with intersections of restricted cardinality are probably among the most well-known and powerful theorems in extremal combinatorics. In this paper, we discuss the so-call ”weak” version of some theorems of this type, when the restricted intersection property is weakened by the possible existence of some (maybe many) intersections having ”bad” sizes. In particular, we give a tight upper bound for the weak version of the ”odd town” problem and non-uniform Fisher’s inequality. The second problem leads to an extremal set theoretic characterization of Hadamard’s matrices. We also give a tight bound for the weak version of the ”even town” problem, raised by Erdős. This bound turns out to be an useful tool to handle systems with restricted multi-intersections.

Extremal Set Systems with Weakly Restricted Intersections

Consider a random matrix of the form , where Mn is a Wigner matrix and Dn is a real deterministic diagonal matrix (Dn is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of Wn for a general class of Wigner matrices Mn and diagonal matrices Dn. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

Universality of local eigenvalue statistics in random matrices with external source

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) > d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0963548302005199

A Sharp Threshold for Network Reliability

On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes

Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$. Under normal circumstances, it is clear that this problem is underdetermined. However, in the case when $X$ is sparse and random, Spielman, Wang and Wright showed that one can recover both $A$ and $X$ efficiently from $Y$ with high probability, given that $p$ (the number of samples) is sufficiently large. Their method works for $p \ge C n^2 \log^ 2 n$ and they conjectured that $p \ge C n \log n$ suffices. The bound $n \log n$ is sharp for an obvious information theoretical reason. 
In this paper, we show that $p \ge C n \log^4 n$ suffices, matching the conjectural bound up to a polylogarithmic factor. The core of our proof is a theorem concerning $l_1$ concentration of random matrices, which is of independent interest. 
Our proof of the concentration result is based on two ideas. The first is an economical way to apply the union bound. The second is a refined version of Bernstein's concentration inequality for the sum of independent variables. Both have nothing to do with random matrices and are applicable in general settings.

Van Vu

Papers

Extremal Set Systems with Weakly Restricted Intersections

Universality of local eigenvalue statistics in random matrices with external source

A Sharp Threshold for Network Reliability

On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes

Dictionary Learning with Few Samples and Matrix Concentration