Benny Sudakov

Bio: Benny Sudakov is an academic researcher from ETH Zurich. The author has contributed to research in topics: Random graph & Complete graph. The author has an hindex of 50, co-authored 546 publications receiving 10258 citations. Previous affiliations of Benny Sudakov include Institute for Advanced Study & Princeton University.

Pseudo-random Graphs

Abstract: Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs

Finding a large hidden clique in a random graph

Abstract: We consider the following probabilistic model of a graph on n labeled vertices First choose a random graph Gn ,1 r 2 ,and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k This question was posed independently, in various variants, by Jerrum and by Kucera In this paper we present an efficient algorithm for all k ) cn 05 , for ˇ 05 05 any fixed c ) 0, thus improving the trivial case k ) cn log n The algorithm is based on the spectral properties of the graph Q 1998 John Wiley & Sons, Inc Random Struct Alg, 13, 457)466, 1998

Finding a large hidden clique in a random graph

Abstract: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998

The Largest Eigenvalue of Sparse Random Graphs

Abstract: We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.

Acyclic edge colorings of graphs

Abstract: A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ-regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001

Random graphs

Abstract: 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.

Spectra of graphs

Abstract: This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Critical phenomena in complex networks

Abstract: The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

Ising formulations of many NP problems

Abstract: We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems This collects and extends mappings to the Ising model from partitioning, covering and satisfiability In each case, the required number of spins is at most cubic in the size of the problem This work may be useful in designing adiabatic quantum optimization algorithms

Clustering gene expression patterns.

Abstract: Recent advances in biotechnology allow researchers to measure expression levels for thousands of genes simultaneously, across different conditions and over time. Analysis of data produced by such experiments offers potential insight into gene function and regulatory mechanisms. A key step in the analysis of gene expression data is the detection of groups of genes that manifest similar expression patterns. The corresponding algorithmic problem is to cluster multicondition gene expression patterns. In this paper we describe a novel clustering algorithm that was developed for analysis of gene expression data. We define an appropriate stochastic error model on the input, and prove that under the conditions of the model, the algorithm recovers the cluster structure with high probability. The running time of the algorithm on an n-gene dataset is O[n2[log(n)]c]. We also present a practical heuristic based on the same algorithmic ideas. The heuristic was implemented and its performance is demonstrated on simulated data and on real gene expression data, with very promising results.