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.

Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Komlos and Szemeredi, as well as earlier work by the authors. The key new ingredient is the applications of Freiman type inverse theorems and other tools from additive combinatorics.

On the singularity probability of random Bernoulli matrices

Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d=d(n) grows more quickly than . These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 346–363, 2001.

/pdf/random-regular-graphs-of-high-degree-24fvvuymdf.pdf

Random regular graphs of high degree

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n-v copies of H whose vertex sets partition V (G).

In this work, we consider the threshold thH(n) of the property that an Erdos-Renyi random graph (on n points) contains an H-factor. Our results determine thH(n) for all strictly balanced H.

The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k-uniform hypergraph, solving the well-known "Shamir's problem." © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.20224

Factors in random graphs

We study the eigenvalues of the covariance matrix 1/n M∗M of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and having finite C0th moment for some sufficiently large constant C0.

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 C0th moment rather than exponential decay.

Random covariance matrices: Universality of local statistics of eigenvalues

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [9] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d=O(n1/3-e), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author.Besides being perhaps the only algorithm which works for relatively large d in practical time, our result also has a significant theoretical value, as it can be used to derive many properties of uniform random regular graphs.

/pdf/generating-random-regular-graphs-36l22t1qcg.pdf

Van Vu

Papers

On the singularity probability of random Bernoulli matrices

Random regular graphs of high degree

Factors in random graphs

Random covariance matrices: Universality of local statistics of eigenvalues

Generating random regular graphs