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Generalizations of Polya’s urn problem
 Annals of Combinatorics
, 2003
"... Abstract. We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1 − p, place the b ..."
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Abstract. We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1 − p, place the ball in an existing bin, such that the probability the ball is placed in a bin is proportional to m γ,wheremis the number of balls in that bin. For p = 0, the number of bins is fixed and finite, and the behavior of the process depends on whether γ is greater than, equal to, or less than 1. We survey the known results and give new proofs for all three cases. We then consider the case p>0. When γ = 1, this is equivalent to the socalled preferential attachment scheme which leads to power law distribution for bin sizes. When γ>1, we prove that a single bin dominates, i.e., as the number of balls goes to infinity, the probability converges to 1 that any new ball either goes into that bin or creates a new bin. When p>0andγ<1, we show that under the assumption that certain limits exist, the fraction of bins having m balls shrinks exponentially as a function of m. We then discuss further generalizations and pose several open problems.
A Stochastic Model for the Evolution of the Web
 Computer Networks
, 2002
"... Recently several authors have proposed stochastic models of the growth of the Web graph that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the "rich get richer" phenomenon. However, these models fail to explain several d ..."
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Recently several authors have proposed stochastic models of the growth of the Web graph that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the "rich get richer" phenomenon. However, these models fail to explain several distributions arising from empirical results, due to the fact that the predicted exponent is not consistent with the data. To address this problem, we extend the evolutionary model of the Web graph by including a nonpreferential component, and we view the stochastic process in terms of an urn transfer model. By making this extension, we can now explain a wider variety of empirically discovered powerlaw distributions provided the exponent is greater than two. These include: the distribution of incoming links, the distribution of outgoing links, the distribution of pages in a Web site and the distribution of visitors to a Web site. A byproduct of our results is a formal proof of the convergence of the standard stochastic model (first proposed by Simon).
Conjunctive search for one and two identical targets
 JOURNAL OF EXPERIMENTAL PSYCHOLOGY: HUMAN PERCEPTION AND PERFORMANCE
, 1989
"... The assumptions of feature integration theory as a blind, serial, selfterminating search (SSTS) mechanism are extended to displays containing 2 identical targets.The SSTS predicts no differences in negativeresponse displays, which require an exhaustive search of the display. Quantitative predictio ..."
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The assumptions of feature integration theory as a blind, serial, selfterminating search (SSTS) mechanism are extended to displays containing 2 identical targets.The SSTS predicts no differences in negativeresponse displays, which require an exhaustive search of the display. Quantitative predictions are confirmed for the positive responses, but not for the negatives, suggesting that the SSTS model is incorrect. Two possible explanations for the results in the negative conditions, differential search rates and early quitting in the negatives, are rejected. It is suggested that using any selfterminating search mechanism will lead to difficulty in interpreting the results, including accounts for which the search is parallel over small groups of items. A resourcelimited parallel model, which is based on the diffusion model of Ratcliff (1978), appears to fit the data well.
Recovery time of dynamic allocation processes
 IN PROCEEDINGS OF THE 10TH ANNUAL ACM SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES, PUERTO VALLARTA, MEXICO, 28 JUNE–2
, 1998
"... Many distributed protocols arising in applications in online load balancing and dynamic resource allocation can be modeled by dynamic allocation processes related to the “balls into bin” problems. Traditionally the main focus of the research on dynamic allocation processes is on verifying whether a ..."
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Cited by 12 (3 self)
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Many distributed protocols arising in applications in online load balancing and dynamic resource allocation can be modeled by dynamic allocation processes related to the “balls into bin” problems. Traditionally the main focus of the research on dynamic allocation processes is on verifying whether a given process is stable, and if so, on analyzing its behavior in the limit (i.e., after sufficiently many steps). Once we know that the process is stable and we know its behavior in the limit, it is natural to analyze its recovery time, which is the time needed by the process to recover from any arbitrarily bad situation and to arrive very closely to a stable (i.e., a typical) state. This investigation is important to provide assurance that even if at some stage the process has reached a highly undesirable state, we can predict with high confidence its behavior after the estimated recovery time. In this paper we present a genera / framework to study the recovery time of discretetime dynamic allocation processes. We model allocation processes by suitably chosen ergodic Markov chains. For a given Markov chain we apply path coupling arguments to bound its convergence rates to the stationary distribution, which directly yields the estimation of the recovery time of the corresponding allocation process. Our coupling approach provides in a relatively simple way an accurate prediction of the recovery time. In particular, we show that our method can be applied to significantly improve estimations of the recovery time for various allocation processes related to allocations of balls into bins, and for the edge orientation problem studied before by Ajtai et al.
Measures of Distinctness for Random Partitions and Compositions of an Integer
, 1997
"... This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts ..."
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This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts
Improving ParallelDisk Buffer Management using Randomized Writeback
 Proc. Int’l Conf. Parallel Processing
, 1998
"... We address the problems of I/O scheduling and buffer management for general reference strings in a parallel I/O system. Using the standard parallel disk model with D disks and a shared I/O buffer of size M, we study the performance of online algorithms that use bounded global Mblock lookahead. We ..."
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We address the problems of I/O scheduling and buffer management for general reference strings in a parallel I/O system. Using the standard parallel disk model with D disks and a shared I/O buffer of size M, we study the performance of online algorithms that use bounded global Mblock lookahead. We introduce the concept of writeback whereby blocks are dynamically relocated between disks during the course of the computation. Writeback allows the layout to be altered to suit different access patterns in different parts of the reference string. We show that any boundedlookahead online algorithm that uses purely deterministic policies must have a competitive ratio of (D). We show how to improve the performance by using randomization, and present a novel algorithm, RANDWB, using a randomized writeback scheme. RANDWB has a competitive ratio of ( p D), which is the best achievable by any online algorithm with only global Mblock lookahead. If the initial layout of data on the disks is uniformly random, RANDWB has a competitive ratio of (log D). 1.
Probabilistic Pursuits on the Grid
"... this paper continue to hold when the lag \Delta is not held constant, but is allowed to vary from one ant to the next. We could also allow for the chasing ant to be guided by an ant other than the one immediately ahead. To achieve the asymptotic results, we need only ensure that eventually the curre ..."
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this paper continue to hold when the lag \Delta is not held constant, but is allowed to vary from one ant to the next. We could also allow for the chasing ant to be guided by an ant other than the one immediately ahead. To achieve the asymptotic results, we need only ensure that eventually the current ant is many generations removed from the first one. Also we need to have \Delta 2 infinitely often at each stage of the walk. The results discussed in this paper can be generalized to three (or more) dimensional space. The probability of A n+1 moving along each axis will, in this case, be proportional to the projection of the vector A n \Gamma A n+1 along this axis. Ants obeying the probabilistic pursuit model have the property of moving, on the average, in the same direction as a continuous pursuit. However, their speed is not constant since it depends on the location of the chaser relative to the target. To overcome this problem, for purposes of approximating continuous pursuit, one might consider the following Euclidean probabilistic rule of pursuit:
Bins And Balls: Large Deviations Of The Empirical Occupancy Process
 KWABENA DOKUAMPONSAH AND PETER M ÖRTERS
, 2000
"... . In the random allocation model, balls are sequentially inserted at random into n exchangeable bins. The occupancy score of a bin denotes the number of balls inserted in this bin. The (random) distribution of occupancy scores denes the object of this paper: the empirical occupancy measure which is ..."
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. In the random allocation model, balls are sequentially inserted at random into n exchangeable bins. The occupancy score of a bin denotes the number of balls inserted in this bin. The (random) distribution of occupancy scores denes the object of this paper: the empirical occupancy measure which is a probability measure over the integers. This measurevalued random variable packages many useful statistics. This paper characterizes the Large Deviations of the ow of empirical occupancy measures when n goes to innity while the number of inserted balls remains proportional to n. The main result is a Sanovlike theorem for the empirical occupancy measure when the set of probability measures over the integers is endowed with metrics that are slightly stronger than the total variation distance. Thanks to a coupling argument, this result applies to the degree distribution of sparse random graphs. 1. Introduction Consider the following classical model in random combinatorics. At each time ...
A stochastic evolutionary model exhibiting powerlaw behaviour with an exponential cutoff
 in the Condensed Matter Archive, condmat/0209463
, 2005
"... Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed sto ..."
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Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein and email networks. Such networks may in fact exhibit an exponential cutoff in the powerlaw scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after, for example, a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will be discarded. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a powerlaw distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model by analysing the protein yeast network, whose distribution is known to follow a power law with an exponential cutoff. 1