For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

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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.

This paper surveys various results about Markov chains on general (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.

ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...

this paper, a special Metropolis-Hastings type algorithm, Metropolized independent sampling, proposed firstly in Hastings (1970), is studied in full detail. The eigenvalues and eigenvectors of the corresponding Markov chain, as well as a sharp bound for the total variation distance between the n-th updated distribution and the target distribution, are provided. Furthermore, the relationship between this scheme, rejection sampling, and importance sampling are studied with emphasizes on their relative efficiencies. It is shown that Metropolized independent sampling is superior to rejection sampling in two aspects: asymptotic efficiency and ease of computation. Key Words: Coupling, Delta method, Eigen analysis, Importance ratio. 1 1 Introduction

Decentralized distributed systems such as peer-to-peer systems are particularly vulnerable to sybil attacks, where a malicious user pretends to have multiple identities (called sybil nodes). Without a trusted central authority, defending against sybil attacks is quite challenging. Among the small number of decentralized approaches, our recent SybilGuard protocol [43] leverages a key insight on social networks to bound the number of sybil nodes accepted. Although its direction is promising, SybilGuard can allow a large number of sybil nodes to be accepted. Furthermore, SybilGuard assumes that social networks are fast mixing, which has never been confirmed in the real world. This paper presents the novel SybilLimit protocol that leverages the same insight as SybilGuard but offers dramatically improved and near-optimal guarantees. The number of sybil nodes accepted is reduced by a factor of Θ ( √ n), or around 200 times in our experiments for a million-node system. We further prove that SybilLimit’s guarantee is at most a log n factor away from optimal, when considering approaches based on fast-mixing social networks. Finally, based on three large-scale real-world social networks, we provide the first evidence that real-world social networks are indeed fast mixing. This validates the fundamental assumption behind SybilLimit’s and SybilGuard’s approach. 1.

CONTENTS 0 PREFACE 3 1 BASICS OF PROBABILITY THEORY 5 2 MARKOV CHAINS 12 3 COMPUTER SIMULATION OF MARKOV CHAINS 19 4 IRREDUCIBLE AND APERIODIC MARKOV CHAINS 25 5 STATIONARY DISTRIBUTIONS 30 6 REVERSIBLE MARKOV CHAINS 41 7 MARKOV CHAIN MONTE CARLO 46 8 FAST CONVERGENCE OF MCMC ALGORITHMS 54 9 APPROXIMATE COUNTING 63 10 THE PROPP--WILSON ALGORITHM 74 11 PROPP--WILSON WITH READ-ONCE RANDOMNESS 82 12 SIMULATED ANNEALING 88 13 FURTHER READING 96 1 2 0 PREFACE The first version of these lecture notes was composed for a new course on randomized algorithms at Chalmers University of Technology, in the spring semester 2000. The amount of material contained therein was not quite enough for an entire course, and the idea was to use these notes together with the book "Randomized Algorithms" by Motwani & Raghavan [MR]. The current version of 2001 has now, with the addition of three more chapters (Chapters 8, 9 and 11), expanded into something more "course-sized", bu

The area-interaction process and the continuum random-cluster model are characterised in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen-Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.

this paper, we attempt to describe various mathematical techniques which have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible. Acknowledgements. I thank Eric Belsley for comments and corrections, and thank Persi Diaconis for introducing me to this subject and teaching me so much. 1. Introduction and motivation.