Urn Model Research Articles

Revisit of a Diaconis urn model

Let G be a finite Abelian group of order d. We consider an urn in which, initially, there are labeled balls that generate the group G. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees, 1994). Siegmund and Yakir (2005) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group G. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.

Social Influence in Signed Networks

Social influence has been widely discussed in various disciplines due to its important sociological significance. However, the dynamics of social influence in signed networks have nonetheless received fairly little attention. In this article, we propose a generalized Pólya urn model that considers the effect of negative relationships and is capable of comprehending the specific mechanisms of homophily and xenophobia in the dynamics. Based on the mathematical deduction, we find that the signed network guides social influence in a trend toward equality. Simulation shows that a higher effect or larger proportion of negative relationships may break the self-reinforcement and make the market more equal. The collective dynamics in the signed network are more predictable but generate path dependence. We also find that a balanced structure has no impact on the average market share but is helpful in removing path dependence and promoting system stability.

Non-equilibrium dynamics and phase transitions in Potts model and interacting Ehrenfest urn model

We show that the recently proposed interacting Ehrenfest M-urn model at equilibrium (Cheng et al., 2020) can be exactly mapped to a mean-field M-state Potts model. By exploiting this correspondence, we show that the M-state Potts model with M≥3, with transition rates motivated by the non-equilibrium urn model, can exhibit rich non-equilibrium spin dynamics such as non-equilibrium steady states and non-equilibrium periodic states. Monte Carlo simulations of the 3-state Potts model are performed to demonstrate explicitly the first-order transitions for the equilibrium and non-equilibrium steady states, as well as the far-from-equilibrium periodic states.

Cutoff in the Bernoulli-Laplace urn model with swaps of order n

Cutoff in the Bernoulli-Laplace urn model with swaps of order n

A Continuous-Time Urn Model for a System of Activated Particles

We study a system of M particles with jump dynamics on a network of N sites. The particles can exist in two states, active or inactive. Only the former can jump. The state of each particle depends on its position. A given particle is inactive when it is at a given site, and active when it moves to a change site. Indeed, each sleeping particle activates at a rate λ>0, leaves its initial site, and moves for an exponential random time of parameter μ>0 before uniformly landing at a site and immediately returning to sleep. The behavior of each particle is independent of that of the others. These dynamics conserve the total number of particles; there is no limit on the number of particles at a given site. This system can be represented by a continuous-time Pólya urn with M balls where the colors are the sites, with an additional color to account for particles on the move at a given time t. First, using this Pólya interpretation for fixed M and N, we obtain the average number of particles at each site over time and, therefore, those on the move due to mass conservation. Secondly, we consider a large system in which the number of particles M and the number of sites N grow at the same rate, so that the M/N ratio tends to a scaling constant α>0. Using the moment-generating function technique added to some probabilistic arguments, we obtain the long-term distribution of the number of particles at each site.

Random allocation models in the thermodynamic limit.

We discuss the phase transition and critical exponents in the random allocation model (urn model) for different statistical ensembles. We provide a unified presentation of the statistical properties of the model in the thermodynamic limit, uncover relationships between the thermodynamic potentials, and fill some lacunae in previous results on the singularities of these potentials at the critical point and behavior in the thermodynamic limit. The presentation is intended to be self-contained, so we carefully derive all formulas step by step throughout. Additionally, we comment on a quasiprobabilistic normalization of configuration weights, which was considered in some recent studies.

Azuma-Hoeffding bounds for a class of urn models

We obtain Azuma-Hoeffding bounds, which are exponentially decreasing, for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, strong convergence of proportions is proved as an application of these bounds.

A hypernetwork-based urn model for explaining collective dynamics.

The topological characterization of complex systems has significantly contributed to our understanding of the principles of collective dynamics. However, the representation of general complex networks is not enough for explaining certain problems, such as collective actions. Considering the effectiveness of hypernetworks on modeling real-world complex networks, in this paper, we proposed a hypernetwork-based Pólya urn model that considers the effect of group identity. The mathematical deduction and simulation experiments show that social influence provides a strong imitation environment for individuals, which can prevent the dynamics from being self-correcting. Additionally, the unpredictability of the social system increases with growing social influence, and the effect of group identity can moderate market inequality caused by individual preference and social influence. The present work provides a modeling basis for a better understanding of the logic of collective dynamics.

QBD Processes Associated with Jacobi–Koornwinder Bivariate Polynomials and Urn Models

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.

An improved approach for ecological modeling of social phenomena in Blau space.

Advances in computation have opened new vistas for modeling of sociodemographic niches and related constructs, enabling us to rectify limitations that unavoidably plagued earlier generations of researchers. This is especially true for Blau space, a sociodemographic niche model used to explore competition between social entities over resources, such as memberships. While this approach has been successful in using probabilistic representations of social networks and resource niches, its modeling framework has remained essentially unchanged for over 40 years, and lacks the ability to make predictions about the future states of sociodemographic space. We present a novel Hybrid Blau space (HBS) model, which utilizes a cellular framework and probabilistic urn models to simulate competition over resources while suffering from fewer limitations. We apply this new model to the General Social Survey, running two sets of models from a series of variables (age, education, income, and sex) and utilize an adjustable range of sociodemographic information for local simulation of organizational competition. We also demonstrate the model's predictive ability, as well as introduce new methods of validation and fit.

Large-deviation theory of increasing returns.

An influential theory of increasing returns was proposed by the economist W. B. Arthur in the 1980s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buys the most shared product within this sample. It is known that one of these two companies becomes a monopoly almost surely in the limit of infinite customers. Here we consider a generalization [Dosi, Ermoliev, and Kaniovsky, J. Math. Econom. 23, 1 (1994)10.1016/0304-4068(94)90032-9] in which the new customer follows the indication of the sample with some probability, and buys the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane, and Sudderth, and it includes several models of physical interest as special cases, such as the Elephant Random Walk, Friedman's urn, and other generalized urn models. We provide a large-deviation analysis of this model at the sample-path level, and we provide a formula that allows us to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sublinear. We also find a nonlinear differential equationfor the cumulant-generating function of the market share variable, which can be studied with a suitable perturbation theory.

On the uniform ergodicity rate of a fractional Ehrenfest urn model

On the uniform ergodicity rate of a fractional Ehrenfest urn model

Information cascade final size distributions derived from urn models

Bipolarization is a phenomenon in which either a large or very small information cascade appears randomly when the retweet rate is high. This phenomenon, which has been observed only in simulations, has the potential to significantly advance the prediction of final cascade sizes because forecasters need only focus on the two peaks in the final cascade size distribution rather than considering the effects of various details, such as network structure and user behavioral patterns. The phenomenon also suggests the difficulty of identifying factors that lead to the emergence of large-scale cascades. To verify the existence of bipolarization, this paper theoretically derives mathematical expressions of the cascade final size distribution using urn models, which simplify the diffusion behavior of actual online social networks. Under the assumption of infinite network size, the distribution exhibits power-law behavior, consistent with the results of existing diffusion models and previous Twitter analytical outcomes. Under the assumption of finite network size, bipolarization is observed.

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

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.