Konstantin Avrachenkov - Profile on Academia.edu (original) (raw)

Papers by Konstantin Avrachenkov

We study a TCP-like linear-increase multiplicative-decrease ow control mechanism. We consider con... more We study a TCP-like linear-increase multiplicative-decrease ow control mechanism. We consider congestion signals that arrive in batches according to a Poisson process. We focus on the case when the transmission rate cannot exceed a certain maximum value. The distribution of the transmission rate in steady state as well as its moments are determined. Our model is particularly useful to study the behavior of TCP, the congestion control mechanism in the Internet. Burstiness of packet losses is captured by allowing congestion signals to arrive in batches. By a simple transformation, the problem can be reformulated in terms of an equivalent M G 1 queue, where the transmission rate in the original model corresponds to the workload in the`dual' queue. The service times in the queueing model are not i.i.d., and they depend on the workload in the system.

Performance Evaluation Methodolgies and Tools, Oct 22, 2007

We study power control in optimization and game frameworks. In the optimization framework there i... more We study power control in optimization and game frameworks. In the optimization framework there is a single decision maker who assigns network resources and in the game framework users share the network resources according to Nash equilibrium. The solution of these problems is based on so-called water-filling technique, which in turn uses bisection method for solution of non-linear equations for Lagrange multipliers. Here we provide a closed form solution to the water-filling problem, which allows us to solve it in a finite number of operations. Also, we produce a closed form solution for the Nash equilibrium in symmetric Gaussian interference game with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the game. In addition to its mathematical beauty, the explicit solution allows one to study limiting cases when the crosstalk coefficient is either small or large. We provide an alternative simple proof of the convergence of the Iterative Water Filling Algorithm. Furthermore, it turns out that the convergence of Iterative Water Filling Algorithm slows down when the crosstalk coefficient is large. Using the closed form solution, we can

Proceedings of the 6th International Workshop on Algorithms and Models for the Web-Graph

IEEE ACM Transactions on Networking, Apr 1, 2005

In this paper, we present a model for TCP/IP congestion control mechanism. The rate at which data... more In this paper, we present a model for TCP/IP congestion control mechanism. The rate at which data is transmitted increases linearly in time until a packet loss is detected. At this point, the transmission rate is divided by a constant factor. Losses are generated by some exogenous random process which is assumed to be stationary ergodic. This allows us to account for any correlation and any distribution of inter-loss times. We obtain an explicit expression for the throughput of a TCP connection and bounds on the throughput when there is a limit on the window size. In addition, we study the effect of the Timeout mechanism on the throughput. A set of experiments is conducted over the real Internet and a comparison is provided with other models that make simple assumptions on the inter-loss time process. The comparison shows that our model approximates well the throughput of TCP for many distributions of inter-loss times.

arXiv (Cornell University), Dec 4, 2006

The choice of the PageRank damping factor is not evident. The Google's choice for the value c = 0... more The choice of the PageRank damping factor is not evident. The Google's choice for the value c = 0.85 was a compromise between the true reflection of the Web structure and numerical efficiency. However, the Markov random walk on the original Web Graph does not reflect the importance of the pages because it absorbs in dead ends. Thus, the damping factor is needed not only for speeding up the computations but also for establishing a fair ranking of pages. In this paper, we propose new criteria for choosing the damping factor, based on the ergodic structure of the Web Graph and probability flows. Specifically, we require that the core component receives a fair share of the PageRank mass. Using singular perturbation approach we conclude that the value c = 0.85 is too high and suggest that the damping factor should be chosen around 1/2. As a by-product, we describe the ergodic structure of the OUT component of the Web Graph in detail. Our analytical results are confirmed by experiments on two large samples of the Web Graph.

Springer eBooks, 2012

Our goal is to quickly find top k lists of nodes with the largest degrees in large complex networ... more Our goal is to quickly find top k lists of nodes with the largest degrees in large complex networks. If the adjacency list of the network is known (not often the case in complex networks), a deterministic algorithm to find the top k list of nodes with the largest degrees requires an average complexity of O(n), where n is the number of nodes in the network. Even this modest complexity can be very high for large complex networks. We propose to use the random walk based method. We show theoretically and by numerical experiments that for large networks the random walk method finds good quality top lists of nodes with high probability and with computational savings of orders of magnitude. We also propose stopping criteria for the random walk method which requires very little knowledge about the structure of the network.

Journal of Statistical Physics, Jul 5, 2018

We analyse a mean-field model of Personalized PageRank on the Erdős-Rényi random graph containing... more We analyse a mean-field model of Personalized PageRank on the Erdős-Rényi random graph containing a denser planted Erdős-Rényi subgraph. We investigate the regimes where the values of Personalized PageRank concentrate around the mean-field value. We also study the optimization of the damping factor, the only parameter in Personalized PageRank. Our theoretical results help to understand the applicability of Personalized PageRank and its limitations for local graph clustering.

Journal of Scientific Computing, Aug 30, 2022

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide ra... more Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (positive or negative), and at each iteration, some nodes receive 'green light' to distribute their wealth or debt proportionally to the transition probabilities of the Markov chain; the stationary probability of a node is computed as a ratio of the cash distributed by this node to the total cash distributed by all nodes together. Our method includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, Gauss-Southwell, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive scheduling strategies for the green-light, that achieve convergence rate faster than state-of-the-art algorithms.

Performance de Compound TCP en pr�sence de pertes al�atoires

Abstract: We analyze with NS simulations the aggregated packet arrival process into a bottleneck ... more Abstract: We analyze with NS simulations the aggregated packet arrival process into a bottleneck queue generated by multiplexed TCP flows. We explain qualitatively the shape of the packet interarrival time distribution. In particular, we provide conditions under which the distribution of the inter packet arrivals is close to exponential and show how this condition scales when the network capacity becomes large. In addition, we analyze the structure of the autocorrelation function of times between packet arrivals. For the case of a packet arrival ...

We explore the dependence structure in the sampled sequence of large networks. We consider random... more We explore the dependence structure in the sampled sequence of large networks. We consider randomized algorithms to sample the nodes and study extremal properties in any associated stationary sequence of characteristics of interest like node degrees, number of followers or income of the nodes in Online Social Networks etc, which satisfy two mixing conditions. Several useful extremes of the sampled sequence like kth largest value, clusters of exceedances over a threshold, first hitting time of a large value etc are investigated. We abstract the dependence and the statistics of extremes into a single parameter that appears in Extreme Value Theory, called extremal index (EI). In this work, we derive this parameter analytically and also estimate it empirically. We propose the use of EI as a parameter to compare different sampling procedures. As a specific example, degree correlations between neighboring nodes are studied in detail with three prominent random walks as sampling techniques.

We develop a generalized optimization framework for graph-based semi-supervised learning. The fra... more We develop a generalized optimization framework for graph-based semi-supervised learning. The framework gives as particular cases the Standard Laplacian, Normalized Laplacian and PageRank based methods. We have also provided new probabilistic interpretation based on random walks and characterized the limiting behaviour of the methods. The random walk based interpretation allows us to explain differences between the performances of methods with different smoothing kernels. It appears that the PageRank based method is robust with respect to the choice of the regularization parameter and the labelled data. We illustrate our theoretical results with two realistic datasets, characterizing different challenges: Les Miserables characters social network and Wikipedia hyper-link graph. The graph-based semi-supervised learning classifies the Wikipedia articles with very good precision and perfect recall employing only the information about the hyper-text links.

ArXiv, 2015

Online social networks (OSN) contain extensive amount of information about the underlying society... more Online social networks (OSN) contain extensive amount of information about the underlying society that is yet to be explored. One of the most feasible technique to fetch information from OSN, crawling through Application Programming Interface (API) requests, poses serious concerns over the the guarantees of the estimates. In this work, we focus on making reliable statistical inference with limited API crawls. Based on regenerative properties of the random walks, we propose an unbiased estimator for the aggregated sum of functions over edges and proved the connection between variance of the estimator and spectral gap. In order to facilitate Bayesian inference on the true value of the estimator, we derive the asymptotic posterior distribution of the estimate. Later the proposed ideas are validated with numerical experiments on inference problems in real-world networks.

The ANZIAM Journal, 2020

The population dynamics for the replicator equation has been well studied in continuous time, but... more The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.

We describe a systematic procedure to calculate the resolvent operator for a linear pencil on Ban... more We describe a systematic procedure to calculate the resolvent operator for a linear pencil on Banach space and thereby simplify, unify and extend known methods for resolution and representation of marginally stable time series. We pay particular attention to those time series commonly known as unit root processes. The new method uses infinite-length Jordan chains to find the key spectral separation projections which enable separation and solution of the fundamental equations for the Laurent series coefficients of the resolvent. It is then possible to define the desired Granger-Johansen representation for the time series. The method remains valid when the resolvent has an isolated essential singularity at unity.

2019 24th Conference of Open Innovations Association (FRUCT), 2019

We consider the large graphs as the object of study and deal with the problem of escape probabili... more We consider the large graphs as the object of study and deal with the problem of escape probability estimation. Generally, the required characteristic cannot be calculated analytically and even numerically due to the complexity and large size of the investigation object. The purpose of this paper is to offer the effective method for estimating the probability that the random walk on graph rst enters a node b before returning into starting node a. Regenerative properties of the random walk allow using an accelerated method for the cycles simulation based on the splitting technique. The results of numerical experiments con rm the advantages of the proposed method.

2016 IEEE 55th Conference on Decision and Control (CDC), Dec 1, 2016

We describe a test statistic based on the L 1norm of the eigenvectors of a modularity matrix to d... more We describe a test statistic based on the L 1norm of the eigenvectors of a modularity matrix to detect the presence of an embedded Erdős Rényi (ER) subgraph inside a larger ER random graph. An embedded subgraph may model a hidden community in a large network such as a social network or a computer network. We make use of the properties of the asymptotic distribution of eigenvectors of random graphs to derive the distribution of the test statistic under certain conditions on the subgraph size and edge probabilities. We show that the distributions differ sufficiently for well defined ranges of subgraph sizes and edge probabilities of the background graph and the subgraph. This method can have applications where it is sufficient to know whether there is an anomaly in a given graph without the need to infer its location. The results we derive on the distribution of the components of the eigenvector may also be useful to detect the subgraph nodes.

2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Jul 1, 2019

We consider caching of VoD contents in a cellular network in which each base station is equipped ... more We consider caching of VoD contents in a cellular network in which each base station is equipped with a cache. Videos are partitioned into chunks according to a layered coding mechanism and the goal is to place chunks in caches such that the expected utility is maximized. The utility depends on the quality at which a user is requesting a file and the chunks that are available. We impose alpha-fairness across files and qualities. We develop a distributed asynchronous algorithm for deciding which chunks to store in which cache.

Flow Aware Congestion Avoidance Method and System