Urn Model Research Articles

Asymptotic results of a multiple-entry reinforcement process

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions {Pt}t≥1, where Pt is a partition of {1,2,…,Rt}. At each time t, R numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block’s cardinality, independently. We prove results concerning the asymptotic cardinality of a given block and central limit theorems for associated fluctuations about this asymptotic cardinality: these are proved both for a fixed block and for the maximum among all blocks. We also prove that with probability one, a single block eventually takes and maintains the leadership in cardinality. Depending on the way one sees this partition process, one can translate our results to Balls and Bins processes, Generalized Chinese Restaurant Processes, Generalized Urn models and Preferential attachment random graphs.

Exactly solvable urn models under random replacement schemes and their applications

AbstractWe examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers $q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $\pi_q(r)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to $\pi_q(r)$. For several natural graph properties, such as $k$-connectivity, minimum degree at least $k$, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.

Urns & Tubes

Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An urn contains coloured balls, which can be drawn with probabilities proportional to the numbers of balls of each colour. For each colour a tube is assumed. These tubes have different sizes (lengths). The idea is that after drawing a ball from the urn it is dropped in the tube of the corresponding colour. We consider two associated probability distributions. The first-full distribution on colours gives for each colour the probability that the corresponding tube is full first, before any of the other tubes. The negative distribution on natural numbers captures for a numberkthe probability that all tubes are full for the first time afterkdraws.This paper uses multisets to systematically describe these first-full and negative distributions in the urns and tubes setting, in fully multivariate form, for all three standard drawing modes (multinomial, hypergeometric, and Pólya).

Random walk with multiple memory channels.

A class of one-dimensional, discrete-time random walk models with memory, termed "random walk with n memory channels" (RWnMC), is proposed. In these models the information of n (n∈Z) previous steps from the walker's entire history is needed to decide a future step. Exact calculation of the mean and variance of position of the RW2MC (n=2) has been done, which shows that it can lead to asymptotic diffusive and superdiffusive behavior in different parameter regimes. A connection between RWnMC and a Pólya-type urn model evolving by drawing n balls at a time has also been reported. This connection for the RW2MC is discussed in detail and suggests the applicability of RW2MC in many population dynamics models with multiple competing species.

Statistical test for an urn model with random multidrawing and random addition

We complete the study of the model introduced in Crimaldi et al., (2022). It is a two-color urn model with multiple drawing and random (non-balanced) time-dependent reinforcement matrix. The number of sampled balls at each time-step is random. We identify the exact rates at which the number of balls of each color grows to +∞ and define two strongly consistent estimators for the limiting reinforcement averages. Then we prove a Central Limit Theorem, which allows to design a statistical test for such averages.

Distributions of cherries and pitchforks for the Ford model

Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating the evolutionary forces underlying the observed phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for the Ford model, the alpha model introduced by Daniel Ford. It is a one-parameter family of random phylogenetic tree models which includes the proportional to distinguishable arrangement (PDA) and the Yule models, two tree models commonly used in phylogenetics. Based on a non-uniform version of the extended Pólya urn models in which negative entries are permitted for their replacement matrices, we obtain the strong law of large numbers and the central limit theorem for the joint distribution of these two statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics. This leads to exact formulas for their means and higher order asymptotic expansions of their second moments, which allows us to identify a critical parameter value for the correlation between these two statistics. That is, when the number of tree leaves is sufficiently large, they are negatively correlated for 0≤α≤1/2 and positively correlated for 1/2<α<1.

The estimation in Pólya–Eggenberger urn model with a delay

In this article, we introduce a new model derived from Pólya–Eggenberger urn model. This model is defined by considering a delay in collecting information. The mathematical formulation of this model is done through four parameters; the number of balls in the first structure (N 0), the number of white balls in the first structure (W 0), the number of rewarded balls of the color of the ball withdrawn (a), and the length of the delay (i). As the first attempt to deal with point estimation in this model, we consider at any time one of the parameters separately as unknown conditioned to knowing the other three parameters, and find its estimation. Accordingly, we introduce a sufficient estimator for this model, and found on it, obtain the maximum likelihood estimators for each of the four parameters. In addition, moment estimators for N 0 and W 0 are calculated. Also, for the other parameters, we obtain estimators based on the correlation coefficient of consecutive withdrawals. To evaluate the performance of the obtained estimators and compare their accuracy, we apply five simulations of the delayed Pólya urn model. The simulations have been done with the software Matlab R2015b. According to the simulation study, the estimators obtained from the method of moments are preferable to maximum likelihood estimators.

A master equation for power laws.

We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker-Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson's Law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log-log plots in the frequency and rank distributions.

Combining Euler series and series from continuations of the Bernoulli scheme

This article describes a new approach to constructing numerical series which converge to an irrational number, a two series combination being proposed. The first series is constructed from the probabilistic interpretation of summing numerical series, which is the authors’ continuation of the Bernoulli scheme. Our ideas are based on the rejection of finiteness and independence of trails. A probabilistic urn model allows us to get a convergent numerical series as well as a series coinciding with the elements of Leibniz harmonic triangle diagonals. The second series used in the considered approach is presented by different Euler series which are extensions of the Basel problem. The combination of two numerical series makes it possible to sum unknown numerical series in closed form. A new reading of the Leibniz harmonic triangle and Euler series variations as well as an approach to finding probabilistic urn models of the summing series are proposed. It is symbolic that the paper considers the ways of connecting the “Basel series” and the series obtained from the continuation of the Bernoulli scheme taking into account that Jacob and Johann Bernoulli worked at the University of Basel. The adoption of the method of combining two numerical series enables to obtain results which have not been previously published in scientific literature.

Noisy voter model: Explicit expressions for finite system size.

Urn models are classic stochastic models that have been used to describe a diverse kind of complex systems. Voter and Ehrenfest's models are very well-known urn models. An opinion model that combines these two models is presented in this work and it is used to study a noisy voter model. In particular, at each temporal step, an Ehrenfest's model step is done with probability α or a voter step is done with probability 1-α. The parameter α plays the role of noise. By performing a spectral analysis, it is possible to obtain explicit expressions for the order parameter, susceptibility, and Binder's fourth-order cumulant. Recursive expressions in terms of the dual Hahn polynomials are given for first passage and return distributions to consensus and the equal coexistence of opinions. In the cases where they follow power-law distributions, their exponents are computed. This model has a pseudocritical noise value that depends on the system size; a discussion about thermodynamic limits is given.

Polya urn model-based performance assessment of PSC bridges: prestress loss consideration

A methodology for performance assessment of prestressed concrete (PSC) girder bridge system, based on strain monitoring in limited number of girders, is proposed in this paper. The methodology uses Polya urn model for determining probabilities of the bridge system being in different condition states with respect to loss of prestress. Performability measure is used for describing the performance of the bridge system. A condition state is assigned for the bridge system from a predefined set of condition states. The time for detailed inspection is determined as the time instant at which the performability of the bridge system becomes less than the target/required performance level. Performance assessment of a bridge system with one hundred PSC girders is considered for illustrating the methodology. The obtained values of condition state probabilities and performability for the considered scenarios (i.e., different number of monitored girders with prestress loss exceeding the allowable value) suggest that the methodology is able to consider the value of available information.

Interacting urns on a finite directed graph

AbstractWe introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

From the multiterm urn model to the self-exciting negative binomial distribution and Hawkes processes.

This paper considers a multiterm urn process that has a correlation in the same term and temporal correlation. .The objective is to clarify the relationship between the urn model and the Hawkes process. Correlation in the same term is represented by the Pólya urn model and the temporal correlation is incorporated by introducing the conditional initial condition. In the double scaling limit of this urn process, the self-exciting negative binomial distribution process, which is a marked Hawkes process, is obtained. In the standard continuous limit, this process becomes the Hawkes process, which has no correlation in the same term. The difference is the variance of the intensity function in that the phase transition from the steady to the nonsteady state can be observed. The critical point, at which the power-law distribution is obtained, is the same for the Hawkes and the urn processes. These two processes are used to analyze empirical data of financial default to estimate the parameters of the model. For the default portfolio, the results produced by the urn process are superior to those obtained with the Hawkes process and confirm self-excitation.

The rescaled Pólya urn: local reinforcement and chi-squared goodness-of-fit test

AbstractMotivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.

Transport and nonequilibrium phase transitions in polygonal urn models.

We study the deterministic dynamics of N point particles moving at a constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback control on the particles, inverting the horizontal component of their velocities when their number in the channel exceeds a fixed threshold. Such a bounce-back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed for both rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted.

Almost sure convergence of randomized urn models with application to elephant random walk

We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and L1 convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d. replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athreya and Ney (1972) requiring Llog+L moments. For more general adaptive sequence of replacement matrices, a little more than Llog+L condition is required. Similar results based on L1 moment assumption alone has been considered independently and in parallel in Zhang (2018). Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in d dimension with random memory.

Asymptotics of the overflow in urn models

AbstractConsider a finite or infinite collection of urns, each with capacity r, and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain r balls. When $r=1$ , this is the number of balls landing in non-empty urns, which has been studied in the past. Our aim here is to use martingale methods to study the asymptotics of the overflow in the general situation, i.e. for arbitrary r. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics.

Contagious statistical distributions: k-connections and applications in infectious disease environments.

Contagious statistical distributions are a valuable resource for managing contagion by means of k–connected chains of distributions. Binomial, hypergeometric, Pólya, uniform distributions with the same values for all parameters except sample size n are known to be strongly associated. This paper describes how the relationship can be obtained via factorial moments, simplifying the process by including novel elements. We describe the properties of these distributions and provide examples of their real–world application, and then define a chain of k–connected distributions, which generalises the relationship among samples of any size for a given population and the Pólya urn model.

Nonlinear and unbalanced urn models with two types of strategies: a stochastic approximation point of view

AbstractIn this paper, we treat a nonlinear and unbalanced $2$-color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.