In the past 20 years, there has been a lot of work concerning modeling real networks such as Internet, WWW, social networks, biological networks, etc. In our talk, we will mainly concentrate on models devoted to incorporating properties of the web graph. We will give a survey of different such models including those of Bollob'as--Riordan, Buckley--Osthus, Cooper--Frieze, etc. We will also present some new related mathematical results obtained by our group in Yandex, Moscow Institute of Physics and Technology, Moscow…

Consider the following situation. Alice knows string x and Bob knows string y. There is a limited capacity information channel from Alice to Bob. Alice wants to transmit a message that is sufficient for Bob to recover x. What is a minimum capacity that allows her to do it? Obviously, there is a theoretical minimum: the amount of information in x that is not contained in y. Astonishing enough, this is a precise estimate: Alice can produce an optimal code…

We consider the random graph model $H_{a,m}^n$. An explicit construction of this model was suggested by Buckley and Osthus. For any fixed positive integer $m$ and positive real $a$ we define a random graph $H_{a,m}^n$ in the following way. The graph $H_{a,1}^1$ consists of one vertex and one loop. Given a graph $H_{a,1}^{t-1}$ we transform it into $H_{a,1}^{t}$ by adding one new vertex and one new random edge $(t,i)$, where $i in {1, dots, t }$ and $Prob(i=t) = frac{a}{(a+1)t-1}$,…

DNA sequencing is the basic workhorse of modern biology and medicine. Shotgun sequencing is the dominant technique used: many randomly located short fragments called reads are extracted from the DNA sequence, and these reads are assembled to reconstruct the original DNA sequence. Despite major effort by the research community, the DNA sequencing problem remains unresolved from a practical perspective: it is currently not known how to produce a good assembly of most read data-sets. By drawing an analogy between the…

We study particle systems corresponding to highly connected queueing networks. We examine the validity of the so-called Poisson Hypothesis (PH), which predicts that the Markov process, describing the evolution of such particle system, started from a reasonable initial state, approaches the equilibrium in time independent of the size of the network. This is indeed the case in many situations. However, there are networks for which the relaxation process slows down. This behavior reflects the fact that the corresponding infinite system…

We study the sequences of Markov processes defining the evolution of a general symmetric queueng network with the number of nodes tending to infinity. These sequences have limits which are nonlinear Markov processes. Time evolution of these nonlinear Markov processes are sometimes simpler and easier to analyze than the evolution of corresponding prelimit Markov processes on finite graphs. This fact give the possibility to find good approximation for the behavior of symmetric queueing networks on large graphs. Examples will be…

The goal of the talk is to describe a class of solvable random growth models of one and two-dimensional interfaces. The growth is local (distant parts of the interface grow independently), it has a smoothing mechanism (fractal boundaries do not appear), and the speed of growth depends on the local slope of the interface. The models enjoy a rich algebraic structure that is reflected through closed determinantal formulas for the correlation functions. Large time asymptotic analysis of such formulas reveals…

Sparse Principal Component Analysis (SPCA) is a remarkably useful tool for practitioners who had been relying on ad-hoc thresholding methods. Our analysis aims at providing a a framework to test whether the data at hand indeed contains a sparse principal component. More precisely we propose an optimal test procedure to detect the presence of a sparse principal component in a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known…

We study a multi-server model with n flexible servers and rn queues, connected through a fixed bipartite graph, where the level of flexibility is captured by the average degree, d(n), of the queues. Applications in content replication in data centers, skill-based routing in call centers, and flexible supply chains are among our main motivations. We focus on the scaling regime where the system size n tends to infinity, while the overall traffic intensity stays fixed. We show that a large…

Low-rank matrix regularization is an important area of research in statistics and machine learning with a wide range of applications. For example, in the context of Missing data imputation arising in recommender systems (e.g. the Netflix Prize), DNA microarray and audio signal processing, among others, the object of interest is a matrix (X) for which the training data comprises of a few noisy linear measurements. The task is to estimate X, under a low rank constraint and possibly additional affine…