“Random” random matrix products

The paper deals with compositions of independent random bundle maps whose distributions form a stationary process which leads to study of Markov processes in random environments. A particular case of this situation is a product of independent random matrices with stationarily changing distributions. We obtain results concerning invariant filtrations for such systems, positivity and simplicity of the largest Lyapunov exponent, as well as central limit theorem type results. An application to random harmonic functions and measures is also considered. Continuous time versions of these results, which yield applications to linear stochastic differential equations in random environments, are also discussed. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

Individual customer’s behaviour in networks with state-dependent arrival rates

We consider Jackson networks with state-dependent arrival and service rates which show product form or nearly product form steady-states and come up with examples of load-dependent admission control. For these networks we prove an arrival theorem for external as well as for internal arrivals. In case of open tandem systems with state-independent service rates we compute the joint distribution of the sojourn times of a customer in the nodes and the distribution of the customer’s end-to-end-delay. This revised version was published online in June 2006 with corrections to the Cover Date.     

Limit theorems for random permanents with exchangeable structure

Permanents of random matrices extend the concept of U-statistics with product kernels. In this paper, we study limiting behavior of permanents of random matrices with independent columns of exchangeable components. Our main results provide a general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries. The method of the proofs is based on a Hoeffding-type orthogonal decomposition of a random permanent function. The decomposition allows us to relate asymptotic behavior of permanents to that of elementary symmetric polynomials based on triangular arrays of rowwise independent rv's.     

Diameters of random circulant graphs

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n ^1/k , and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.     

Product form solution for exponential G-networks with dependent service and completion of service of killed customers

Queueing networks with negative customers (G-networks), Poisson flow of positive customers, multi-server exponential nodes, and dependent service at the different nodes are studied. Every customer arriving at the network is defined by a set of random parameters: customer route, the length of customer route, customer volume and his service time at each route stage as well. A killed positive customer is removed at the last place in the queue and quits the network just after his remaining service time will be elaborated. For such G-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form.     

Quasirandom Rumor Spreading on Expanders

Randomized rumor spreading is an efficient way to distribute information in networks. Recently, a quasirandom version of this protocol has been proposed. It was proven that it works equally well or even better in many settings.In this work, we exhibit a natural expansion property for networks, which ensures that quasirandom rumor spreading informs all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erdős-Rényi random graphs.     

Stability of random impulsive coupled systems on networks with Markovian switching

Abstract

This article is intended to study global asymptotical stability in probability for random impulsive coupled systems on networks with Markovian switching. Two cases are considered. (1) Continuous dynamics are stable while impulses are unstable; (2) impulses are stable while continuous dynamics are unstable. To begin with, based on Lyapunov method as well as graph-theoretic technique, several new stability criteria in two cases are derived, that are, the Lyapunov-type criteria and the coefficients-type criteria. Then main results are used for a class of random impulsive coupled oscillators. Finally, the effectiveness of the obtained results is verified by numerical simulations.     

Product forms for availability models

This paper shows and illustrates that product form expressions for the steady state distribution, as known for queueing networks, can also be extended to a class of availability models. This class allows breakdown and repair rates from one component to depend on the status of other components. Common resource capacities and repair priorities, for example, are included. Conditions for the models to have a product form are stated explicitly. This product form is shown to be insensitive to the distributions of the underlying random variables, i.e. to depend only on their means. Further it is briefly indicated how queueing for repair can be incorporated. Novel product form examples are presented of a simple series/parallel configuration, a fault tolerant database system and a multi-stage interconnection network.     

Heuristics for the Stochastic Eulerian Tour Problem

The Stochastic Eulerian Tour Problem (SETP) seeks the Eulerian tour of minimum expected length on an undirected Eulerian graph, when demand on the arcs that have to be serviced is probabilistic. The SETP is NP-hard and in this paper, we develop three constructive heuristics for this problem. The first two are greedy tour construction heuristics while the third is a sub-tour concatenation heuristic. Our experimental results show that for grid networks, the sub-tour concatenation heuristic performs well when the probability of service of each edge is greater than 0.1. For Euclidean networks, as the number of edges increases, the second heuristic performs the best among the three. Also, the expected length of our overall best solution is lower than the expected length of a random tour by up to 10% on average for grid networks and up to 2% for Euclidean networks.     

Wick calculus for nonlinear Gaussian functionals

This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variables.     

Vulnerability of complex networks

We consider normalized average edge betweenness of a network as a metric of network vulnerability. We suggest that normalized average edge betweenness together with is relative difference when certain number of nodes and/or edges are removed from the network is a measure of network vulnerability, called vulnerability index. Vulnerability index is calculated for four synthetic networks: Erdős–Rényi (ER) random networks, Barabási–Albert (BA) model of scale-free networks, Watts–Strogatz (WS) model of small-world networks, and geometric random networks. Real-world networks for which vulnerability index is calculated include: two human brain networks, three urban networks, one collaboration network, and two power grid networks. We find that WS model of small-world networks and biological networks (human brain networks) are the most robust networks among all networks studied in the paper.     

On the integral with respect to the tensor product of two random measures

A Fubini-type formula for the integral with respect to the tensor product of two random measures is established in an intrinsic way. This permits one to consider a convolution product. The results are applied to a stationary continuous random function (which may be multiplicatively written with two stationary components) and to principal component analysis in the frequency domain.     

Rare event probabilities in stochastic networks

The paper is dealing with estimation of rare event probabilities in stochastic networks. The well known variance reduction technique, called Importance Sampling (IS) is an effective tool for doing this. The main idea of IS is to simulate the random system under a modified set of parameters, so as to make the occurrence of the rare event more likely. The major problem of the IS technique is that the optimal modified parameters, called reference parameters to be used in IS are usually very difficult to obtain. Rubinstein (Eur J Oper Res 99:89–112, 1997) developed the Cross Entropy (CE) method for the solution of this problem of IS technique and then he and his collaborators applied this for estimation of rare event probabilities in stochastic networks with exponential distribution [see De Boer et al. (Ann Oper Res 134:19–67, 2005)]. In this paper, we test this simulation technique also for medium sized stochastic networks and compare its effectiveness to the simple crude Monte Carlo (CMC) simulation. The effectiveness of a variance reduction simulation algorithm is measured in the following way. We calculate the product of the necessary CPU time and the estimated variance of the estimation. This product is compared to the same for the simple Crude Monte Carlo simulation. This was originally used for comparison of different variance reduction techniques by Hammersley and Handscomb (Monte Carlo Methods. Methuen & Co Ltd, London, 1967). The main result of the paper is the extension of CE method for estimation of rare event probabilities in stochastic networks with beta distributions. In this case the calculation of reference parameters of the importance sampling distribution requires numerical solution of a nonlinear equation system. This is done by applying a Newton–Raphson iteration scheme. In this case the CPU time spent for calculation of the reference parameter values cannot be neglected. Numerical results will also be presented. This work was supported by grant from the Hungarian National Scientific Research Grant OTKA T047340.     

Effect of clustering on attack vulnerability of interdependent scale-free networks

In order to deeply understand the complex interdependent systems, it is of great concern to take clustering coefficient, which is an important feature of many real-world systems, into account. Previous study mainly focused on the impact of clustering on interdependent networks under random attacks, while we extend the study to the case of the more realistic attacking strategy, targeted attack. A system composed of two interdependent scale-free networks with tunable clustering is provided. The effects of coupling strength and coupling preference on attack vulnerability are explored. Numerical simulation results demonstrate that interdependent links between two networks make the entire system much more fragile to attacks. Also, it is found that clustering significantly increases the vulnerability of interdependent scale-free networks. Moreover, for fully coupled network, disassortative coupling is found to be most vulnerable to random attacks, while the random and assortative coupling have little difference. Additionally, enhancing coupling strength can greatly enhance the fragility of interdependent networks against targeted attacks. These results can not only improve the deep understanding of structural complexity of complex systems, but also provide insights into the guidance of designing resilient infrastructures.     

Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions

The paper provides a condition for differentiability as well as an equivalent criterion for Lipschitz continuity of singular normal distributions. Such distributions are of interest, for instance, in stochastic optimization problems with probabilistic constraints, where a comparatively small (nondegenerate-) normally distributed random vector induces a large number of linear inequality constraints (e.g. networks with stochastic demands). The criterion for Lipschitz continuity is established for the class of quasi-concave distributions which the singular normal distribution belongs to.     

Survey on applications of semi-tensor product method in networked evolutionary games

Semi-tensor product (STP) method of matrices has received more and more attention from the communities of both engineering and economics in recent years. This paper presents a comprehensive survey on the applications of STP method in the theory of networked evolutionary games. In the beginning, some preliminary results on STP method are recalled. Then, the applications of STP method in many kinds of networked evolutionary games, such as general networked evolutionary games, networked evolutionary games with finite memories, networked evolutionary games defined on finite networks, and random networked evolutionary games, are reviewed. Finally, several research problems in the future are predicted.     

Due date assignment for multistage assembly systems

This paper is concerned with the study of the constant due-date assignment policy in a multistage assembly system. The multistage assembly system is modeled as an open queueing network. It is assumed that the product order arrives according to a Poisson process. In each service station, there is either one or infinite machine with exponentially distributed processing time. The transport times between every pair of service stations are independent random variables with generalized Erlang distributions. It is assumed that each product has a penalty cost that is some linear function of its due-date and its actual completion time. The due date is found by adding a constant to the time that the order arrives. This constant value is the constant lead time that a product might expect between time of placing the order and time of delivery. By applying the longest path analysis in queueing networks, we obtain the distribution function of manufacturing lead time. Then, the optimal constant lead time is computed by minimizing the expected aggregate cost per product. Finally, the results are verified by Monte Carlo simulation.     

Data association based on optimization in graphical models with application to sensor networks

We propose techniques based on graphical models for efficiently solving data association problems arising in multiple target tracking with distributed sensor networks. Graphical models provide a powerful framework for representing the statistical dependencies among a collection of random variables, and are widely used in many applications (e.g., computer vision, error-correcting codes). We consider two different types of data association problems, corresponding to whether or not it is known a priori which targets are within the surveillance range of each sensor. We first demonstrate how to transform these two problems to inference problems on graphical models. With this transformation, both problems can be solved efficiently by local message-passing algorithms for graphical models, which solve optimization problems in a distributed manner by exchange of information among neighboring nodes on the graph. Moreover, a suitably reweighted version of the max–product algorithm yields provably optimal data associations. These approaches scale well with the number of sensors in the network, and moreover are well suited to being realized in a distributed fashion. So as to address trade-offs between performance and communication costs, we propose a communication-sensitive form of message-passing that is capable of achieving near-optimal performance using far less communication. We demonstrate the effectiveness of our approach with experiments on simulated data.     

Markov network processes with product form stationary distributions

This study concerns the equilibrium behavior of a general class of Markov network processes that includes a variety of queueing networks and networks with interacting components or populations. The focus is on determining when these processes have product form stationary distributions. The approach is to relate the marginal distributions of the process to the stationary distributions of “node transition functions” that represent the nodes in isolation operating under certain fictitious environments. The main result gives necessary and sufficient conditions on the node transition functions for the network process to have a product form stationary distribution. This result yields a procedure for checking for a product form distribution and obtaining such a distribution when it exits. An important subclass of networks are those in which the node transition rates have Poisson arrival components. In this setting, we show that the network process has a product form distribution and is “biased locally balanced” if and only if the network is “quasi-reversible” and certain traffic equations are satisfied. Another subclass of networks are those with reversible routing. We weaken the known sufficient condition for such networks to be product form. We also discuss modeling issues related to queueing networks including time reversals and reversals of the roles of arrivals and departures. The study ends by describing how the results extend to networks with multi-class transitions. This revised version was published online in June 2006 with corrections to the Cover Date.