Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. Previous works proved that when $G$ is not balanced bipartite, the proportion of balls in the bins converges to a point $w(G)$ almost surely.
We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point $w(G)$ or a closed interval $\mathcal J(G)$.
Sufficient conditions are developed for a class of generalized Polya urn schemes ensuring exchangeability. The extended class includes the Blackwell-MacQueen Polya urn and the urn schemes for the two-parameter Poisson-Dirichlet process and finite dimensional Dirichlet priors among others.
For the plain Polya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls assuming that the initial number of black balls is large. Depending on the initial number of white balls, the limit is either a pure birth process or a diffusion. We also prove analogous results for the q-Polya urn, which is an urn where, when picking a ball, the balls of one color have priority over those of the other.
We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $α>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections.
We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability and derive several exact distributional results for the all-time maximal percentage.
We collect, survey and develop methods of (one-dimensional) stochastic approximation in a framework that seems suitable to handle fairly broad generalizations of Polya urns. To show the applicability of the results we determine the limiting fraction of balls in an urn with balls of two colors. We consider two models generalizing the Polya urn, in the first one ball is drawn and replaced with balls of (possibly) both colors according to which color was drawn. In the second, two balls are drawn simultaneously and replaced along with balls of (possibly) both colors according to what combination o…
We study an urn model introduced in the paper of Chen and Wei, where at each discrete time step $m$ balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m=1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m b…
Consider a generalized time-dependent Pólya urn process defined as follows. Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we distribute $σ_n$ balls randomly to the $d$ urns, proportionally to $f$, where $f$ is a valid reinforcement function. We consider a general class of positive reinforcement functions $\mathcal{R}$ assuming some monotonicity and growth condition. The class $\mathcal{R}$ includes convex functions and the classical case $f(x)=x^α$, $α>1$. The novelty of the paper lies in extending stochastic approximation techniques to the $d$-dimensional case and prov…
For the most general Polya urn schemes, we establish the almost sure convergence of its composition. The only requirement is that there are always enough balls of both colors, so that the extractions can be indefinitely pursued according to the specifications of the model. We also consider the method for determining the probability of fulfilling this requirement, as a function of the initial number of balls of each color.
We consider a special case of the generalized Pólya's urn model introduced by Benaim et al (2013). Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. A question of essential interest for the model is to understand the limiting behavior of the proportion of balls in the bins for different graphs $G$. In this paper, we present two results regarding this question…
Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. Which of the two terms dominates depends on the ratio of the number of time steps …
We consider a version of the classical Pólya urn scheme which incorporates innovations. The space $S$ of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns $C$ balls of the same color and additional balls of different colors given by some finite point process $ξ$ on $S$. When the number of steps goes to infinity, the empirical distribution of the colors in the urn converges to the normalized intensity measure of $ξ$, and we analyze the fluctuations. The ratio $ρ= E(C)/E(R)$ of the average number of copies to the average total number of balls returned p…
It is known that in an irreducible small Pólya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also is balanced, this normal convergence holds with convergence of all moments, thus giving asymptotics of (central) moments.
We consider a Polya urn, started with b black and w white balls, where b>w. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is twice the probability of getting no more than w-1 heads in b+w-1 tosses of a fair coin.
We propose a way of finding a Stein type characterization of a given absolutely continuous distribution $μ$ on $\R$ which is motivated by a regression property satisfied by an exchangeable pair $(W,W')$ where $\calL(W)$ is supposed or known to be close to $μ$. We also develop the exchangeable pairs approach within this setting. This general procedure is then specialized to the class of Beta distributions and as an application, a convergence rate for the relative number of drawn red balls among the first $n$ drawings from a Polya urn is computed.
In this paper, we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the \emph{negatively reinforced} urn scheme. We establish almost sure limit of the random configuration for any \emph{balanced} replacement matrix $R$. In particular, we show that the limiting configuration is uniform on the set of colours, if and only if, $R$ is a \emph{doubly stochastic} matrix. We further establish almost sure limit of the vector of colour counts and prove central…
We consider a time-dependent version of a Pólya urn containing black and white balls. At each time $n$ a ball is drawn from the urn at random and replaced in the urn along with $σ_n$ additional balls of the same colour. The proportion of white balls converges almost surely to a random limit $Θ$, and $\mathcal{D}=\{Θ\in\{0,1\}\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $(σ_n)$, between the regimes ${\mathbb P}(\mathcal{D})=1$ and ${\mathbb P}(\mathcal{D})<1$ was obtained by R. Pemantle in 1990. We describe the phase transition between …
Consider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball immediately is replaced together with a fixed number of balls of the same colour. It is well-known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $Θ(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e., on the i…
This paper introduces and analyzes a particular class of Polya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an "analytic combinatorics" point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006. Through an algebraic generating function to which we apply a multiple coalescing saddle-point method, we are able to give precise asymp…
We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with another ball of the same colour. With probability 1-p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri, and Thörnblad. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there i…
An urn containing specified numbers of balls of distinct ordered colors is considered. A multiple q-Polya urn model is introduced by assuming that the probability of q-drawing a ball of a specific color from the urn varies geometrically, with rate q, both with the number of drawings and the number of balls of the specific color, together with the total number of balls of the preceded colors, drawn in the previous q-drawings. Then, the joint distributions of the numbers of balls of distinct colors drawn (a) in a specific number of q-drawings and (b) until the occurrence of a specific number of …
We consider, as proposed and studied in Hofstad et.\ al.\ \cite{HHKR}, a class of graph-based "interacting urn"-type Polya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency' of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved/reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeatin…
Consider an urn initially containing $b$ black and $w$ white balls. Select a ball at random and observe its color. If it is black, stop. Otherwise, return the white ball together with another white ball to the urn. Continue selecting at random, each time adding a white ball, until a black ball is selected. Let $T_{b,w}$ denote the number of draws until this happens. Surprisingly, the expectation of $T_{b,w}$ is infinite for the "fair" initial scenario $b =w=1$, but finite if $b=2$ and $w=10^9$. In fact, $\mathbb{E}[T_{b,w}]$ is finite if and only if $b\ge 2$, and the variance of $T_{b,w}$ is f…
In [arXiv:1906.10951 (forthcoming on Advances in Applied Probability),arXiv:2011.05933 (published on PLOS ONE)] the authors introduce, study and apply a new variant of the Eggenberger-Polya urn, called the "Rescaled" Polya urn, which, for a suitable choice of the model parameters, 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-…
A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). W…
We study the number of white balls in a classical Pólya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the li…
In the classical Polya urn problem, one begins with $d$ bins, each containing one ball. Additional balls arrive one at a time, and the probability that an arriving ball is placed in a given bin is proportional to $m^γ$, where $m$ is the number of balls in that bin. In this note, we consider the case of $γ= 1$, which corresponds to a process of "proportional preferential attachment" and is a critical point with respect to the limit distribution of the fraction of balls in each bin. It is well known that for $γ< 1$ the fraction of balls in the "heaviest" bin (the bin with the most balls) tends t…
Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power a>0. We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector fie…
We study first passage statistics of the Polya urn model. In this random process, the urn contains two types of balls. In each step, one ball is drawn randomly from the urn, and subsequently placed back into the urn together with an additional ball of the same type. We derive the probability G_n that the two types of balls are equal in number, for the first time, when there is a total of 2n balls. This first passage probability decays algebraically, G_n ~ n^{-2}, when n is large. We also derive the probability that a tie ever happens. This probability is between zero and one, so that a tie may…
Motivated by recent studies of big samples, this work aims at constructing 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 convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result. This triple purpose has been achieved by the introduction of a new variant of the Eggenberger-Polya urn, that we call the "Rescaled" Polya urn. We provide a compl…
A Pólya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots,d\}$ for $d\in \mathbb{N}$. At each time step, a random ball is chosen uniformly in the urn. It is replaced in the urn and, if its colour is $c$, $R_{c,j}$ balls of colour $j$ are also added (for all $1\leq j\leq d$). We introduce a model of measure-valued processes that generalises this construction. This generalisation includes the case when the space of colours is a (possibly infinite) Polish space $\mathcal P$. We see the urn composition at…
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove…
We introduce the Generalized Rescaled Polya (GRP) urn, that provides a generative model for a chi-squared test of goodness of fit for the long-term probabilities of clustered data, with independence between clusters and correlation, due to a reinforcement mechanism, inside each cluster. We apply the proposed test to a data set of Twitter posts about COVID-19 pandemic: in a few words, for a classical chi-squared test the data result strongly significant for the rejection of the null hypothesis (the daily long-run sentiment rate remains constant), but, taking into account the correlation among d…
Knowing how and when trends are formed is a frequently visited research goal. In our work, we focus on the progression of trends through (social) networks. We use a random graph (RG) model to mimic the progression of a trend through the network. The context of the trend is not included in our model. We show that every state of the RG model maps to a state of the Polya process. We find that the limit of the component size distribution of the RG model shows power-law behaviour. These results are also supported by simulations.
We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n is that of X_{n-k_n}, where k_n is sampled independently from a fixed law μon the positive integers. When μhas a roughly power law decay (precisely, when it lies in the domain of attraction of an αstable subordinator, for 0 < α< 1/2) the walk scales to fractional Brownian motion with Hurst parameter α+ 1/2. The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense w…
Let X be a non-deterministic infinite exchangeable sequence with values in {0,1}. We show that X is Hoeffding-decomposable if, and only if, X is either an i.i.d. sequence or a Polya sequence. This completes the results established in Peccati [2004]. The proof uses several combinatorial implications of the correspondence between Hoeffding decomposability and weak independence. Our results must be compared with previous characterizations of i.i.d. and Polya sequences given by Hill et al. [1987] and Diaconis and Yilvisaker [1979].
We 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 if 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 scaling, around the vector of limiting prop…
This paper presents a tighter bound on the degree distribution of arbitrary Pólya urn graph processes, proving that the proportion of vertices with degree $d$ obeys a power-law distribution $P(d) \propto d^{-γ}$ for $d \leq n^{\frac{1}{6}-ε}$ for any $ε> 0$, where $n$ represents the number of vertices in the network. Previous work by Bollobás et al. formalized the well-known preferential attachment model of Barabási and Albert, and showed that the power-law distribution held for $d \leq n^{\frac{1}{15}}$ with $γ= 3$. Our revised bound represents a significant improvement over existing models o…
We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized Pólya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by flipping a coin. We study the asymptotic behaviour of this urn model, and prove a law of large numbers, a central limit theorem and a functional limit theorem for the proportion of balls into the urn. A phase transition is also stated.
The Pólya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the Pólya process. We deal with a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard Pólya process. In one of the scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn," but a broader int…
We introduce and discuss a special type of feedback interacting urn model with deterministic interaction. This is a generalisation of the very well known Eggenberger and Polya (1923) urn model. In our model, balls are added to a particular urn depending on the replacement matrix of that urn and the color of ball chosen from some other urn. This urn model can help in studying how various interacting models might behave in real life in the long run. We have also introduced a special type of interacting urn model with non-deterministic interaction and studied its behaviour. Furthermore, we have p…
A classical Pólya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing Pólya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 …
We consider the generalization of the Pólya urn scheme with possibly infinite many colors as introduced in \cite{Th-Thesis, BaTH2014, BaTh2016, BaTh2017}. For countable many colors, we prove almost sure convergence of the urn configuration under \emph{uniform ergodicity} assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a \emph{branching Markov chain} on a weighted \emph{random recursive tree} as described in \cite{BaTh2017, Sv_2018}. Using this coupling we estimate the covariance between any two selected colors. In particular…
We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.
The generalized Pòlya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the…
We study several exactly solvable Polya-Eggenberger urn models with a \emph{diminishing} character, namely, balls of a specified color, say $x$ are completely drawn after a finite number of draws. The main quantity of interest here is the number of balls left when balls of color $x$ are completely removed. We consider several diminishing urns studied previously in the literature such as the pills problem, the cannibal urns and the OK Corral problem, and derive exact and limiting distributions. Our approach is based on solving recurrences via generating functions and partial differential equati…
This paper extends the link between stochastic approximation (SA) theory and randomized urn models developed in Laruelle, Pag{è}s (2013), and their applications to clinical trials introduced in Bai, HU (1999,2005) and Bai, Hu, Shen (2002). We no longer assume that the drawing rule is uniform among the balls of the urn (which contains d colors), but can be reinforced by a function f. This is a way to model risk aversion. Firstly, by considering that f is concave or convex and by reformulating the dynamics of the urn composition as an SA algorithm with remainder, we derive the a.s. convergence a…
This article describes a purely analytic approach to urn models of the generalized or extended Pólya-Eggenberger type, in the case of two types of balls and constant ``balance,'' that is, constant row sum. The treatment starts from a quasilinear first-order partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences in the case of ``subtractive'' urns are new representations for the probability distribution of the urn's composition a…
