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15 pages, 679 KiB

Open AccessArticle

Phase Transition in Ant Colony Optimization

by Shintaro Mori, Shogo Nakamura, Kazuaki Nakayama and Masato Hisakado

Physics 2024, 6(1), 123-137; https://doi.org/10.3390/physics6010009 - 18 Jan 2024

Cited by 3 | Viewed by 1586

Ant colony optimization (ACO) is a stochastic optimization algorithm inspired by the foraging behavior of ants. We investigate a simplified computational model of ACO, wherein ants sequentially engage in binary decision-making tasks, leaving pheromone trails contingent upon their choices. The quantity of pheromone [...] Read more.

Ant colony optimization (ACO) is a stochastic optimization algorithm inspired by the foraging behavior of ants. We investigate a simplified computational model of ACO, wherein ants sequentially engage in binary decision-making tasks, leaving pheromone trails contingent upon their choices. The quantity of pheromone left is the number of correct answers. We scrutinize the impact of a salient parameter in the ACO algorithm, specifically, the exponent

α

, which governs the pheromone levels in the stochastic choice function. In the absence of pheromone evaporation, the system is accurately modeled as a multivariate nonlinear Pólya urn, undergoing phase transition as

α

varies. The probability of selecting the correct answer for each question asymptotically approaches the stable fixed point of the nonlinear Pólya urn. The system exhibits dual stable fixed points for

α \geq α_{c}

and a singular stable fixed point for

α < α_{c}

where

α_{c}

is the critical value. When pheromone evaporates over a time scale

τ

, the phase transition does not occur and leads to a bimodal stationary distribution of probabilities for

α \geq α_{c}

and a monomodal distribution for

α < α_{c}

. Full article

(This article belongs to the Special Issue In Honor of Professor Serge Galam for His 70th Birthday and Forty Years of Sociophysics)

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13 pages, 302 KiB

Open AccessArticle

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

by Rafik Aguech and Hanene Mohamed

Mathematics 2023, 11(24), 4967; https://doi.org/10.3390/math11244967 - 15 Dec 2023

Viewed by 1242

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

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11 pages, 292 KiB

Open AccessArticle

Bayesian Bootstrap in Multiple Frames

by Daniela Cocchi, Lorenzo Marchi and Riccardo Ievoli

Stats 2022, 5(2), 561-571; https://doi.org/10.3390/stats5020034 - 15 Jun 2022

Cited by 3 | Viewed by 2247

Abstract

Multiple frames are becoming increasingly relevant due to the spread of surveys conducted via registers. In this regard, estimators of population quantities have been proposed, including the multiplicity estimator. In all cases, variance estimation still remains a matter of debate. This paper explores [...] Read more.

Multiple frames are becoming increasingly relevant due to the spread of surveys conducted via registers. In this regard, estimators of population quantities have been proposed, including the multiplicity estimator. In all cases, variance estimation still remains a matter of debate. This paper explores the potential of Bayesian bootstrap techniques for computing such estimators. The suitability of the method, which is compared to the existing frequentist bootstrap, is shown by conducting a small-scale simulation study and a case study. Full article

(This article belongs to the Special Issue Re-sampling Methods for Statistical Inference of the 2020s)

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10 pages, 307 KiB

Open AccessArticle

Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes

by Lancelot F. James

Mathematics 2022, 10(4), 561; https://doi.org/10.3390/math10040561 - 11 Feb 2022

Cited by 2 | Viewed by 1706

Abstract

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for

0 < α < 1,

and

θ > - α

, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters [...] Read more.

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for

0 < α < 1,

and

θ > - α

, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters

(α, 1 - α)

to a mass partition having a Poisson–Dirichlet distribution with parameters

(α, θ)

leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters

(α, θ + r)

for

r = 0, 1, 2, \dots .

Furthermore, these generate a Markovian sequence of

α

-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer

n,

which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where

n = 0, 1 .

Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

11 pages, 692 KiB

Open AccessArticle

The Rescaled Pólya Urn and the Wright—Fisher Process with Mutation

by Giacomo Aletti and Irene Crimaldi

Mathematics 2021, 9(22), 2909; https://doi.org/10.3390/math9222909 - 15 Nov 2021

Cited by 2 | Viewed by 2021

Abstract

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation [...] Read more.

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright—Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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19 pages, 385 KiB

Open AccessArticle

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

by Sandra Fortini, Sonia Petrone and Hristo Sariev

Mathematics 2021, 9(22), 2845; https://doi.org/10.3390/math9222845 - 10 Nov 2021

Cited by 5 | Viewed by 2611

Abstract

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP [...] Read more.

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP

{(μ_{n})}_{n \geq 0}

on a Polish space

X

, the normalized sequence

{(μ_{n} / μ_{n} (X))}_{n \geq 0}

agrees with the marginal predictive distributions of some random process

{(X_{n})}_{n \geq 1}

. Moreover,

μ_{n} = μ_{n - 1} + R_{X_{n}}

,

n \geq 1

, where

x \mapsto R_{x}

is a random transition kernel on

X

; thus, if

μ_{n - 1}

represents the contents of an urn, then

X_{n}

denotes the color of the ball drawn with distribution

μ_{n - 1} / μ_{n - 1} (X)

and

R_{X_{n}}

—the subsequent reinforcement. In the case

R_{X_{n}} = W_{n} δ_{X_{n}}

, for some non-negative random weights

W_{1}, W_{2}, \dots

, the process

{(X_{n})}_{n \geq 1}

is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of

{(X_{n})}_{n \geq 1}

under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

16 pages, 2188 KiB

Open AccessFeature PaperArticle

Taylor’s Law in Innovation Processes

by Francesca Tria, Irene Crimaldi, Giacomo Aletti and Vito D. P. Servedio

Entropy 2020, 22(5), 573; https://doi.org/10.3390/e22050573 - 19 May 2020

Cited by 10 | Viewed by 4746

Abstract

Taylor’s law quantifies the scaling properties of the fluctuations of the number of innovations occurring in open systems. Urn-based modeling schemes have already proven to be effective in modeling this complex behaviour. Here, we present analytical estimations of Taylor’s law exponents in such [...] Read more.

Taylor’s law quantifies the scaling properties of the fluctuations of the number of innovations occurring in open systems. Urn-based modeling schemes have already proven to be effective in modeling this complex behaviour. Here, we present analytical estimations of Taylor’s law exponents in such models, by leveraging on their representation in terms of triangular urn models. We also highlight the correspondence of these models with Poisson–Dirichlet processes and demonstrate how a non-trivial Taylor’s law exponent is a kind of universal feature in systems related to human activities. We base this result on the analysis of four collections of data generated by human activity: (i) written language (from a Gutenberg corpus); (ii) an online music website (Last.fm); (iii) Twitter hashtags; (iv) an online collaborative tagging system (Del.icio.us). While Taylor’s law observed in the last two datasets agrees with the plain model predictions, we need to introduce a generalization to fully characterize the behaviour of the first two datasets, where temporal correlations are possibly more relevant. We suggest that Taylor’s law is a fundamental complement to Zipf’s and Heaps’ laws in unveiling the complex dynamical processes underlying the evolution of systems featuring innovation. Full article

(This article belongs to the Special Issue Statistical Mechanics of Complex Systems)

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27 pages, 493 KiB

Open AccessArticle

Saddlepoint Approximation for Data in Simplices: A Review with New Applications

by Riccardo Gatto

Stats 2019, 2(1), 121-147; https://doi.org/10.3390/stats2010010 - 18 Feb 2019

Cited by 1 | Viewed by 2679

Abstract

This article provides a review of the saddlepoint approximation for a M-statistic of a sample of nonnegative random variables with fixed sum. The sample vector follows the multinomial, the multivariate hypergeometric, the multivariate Polya or the Dirichlet distributions. The main objective is to [...] Read more.

This article provides a review of the saddlepoint approximation for a M-statistic of a sample of nonnegative random variables with fixed sum. The sample vector follows the multinomial, the multivariate hypergeometric, the multivariate Polya or the Dirichlet distributions. The main objective is to provide a complete presentation in terms of a single and unambiguous notation of the common mathematical framework of these four situations: the simplex sample space and the underlying general urn model. Some important applications are reviewed and special attention is given to recent applications to models of circular data. Some novel applications are developed and studied numerically. Full article

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19 pages, 1280 KiB

Open AccessArticle

Zipf’s, Heaps’ and Taylor’s Laws are Determined by the Expansion into the Adjacent Possible

by Francesca Tria, Vittorio Loreto and Vito D. P. Servedio

Entropy 2018, 20(10), 752; https://doi.org/10.3390/e20100752 - 30 Sep 2018

Cited by 27 | Viewed by 7146

Abstract

Zipf’s, Heaps’ and Taylor’s laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the scaling of fluctuations for systems as diverse as [...] Read more.

Zipf’s, Heaps’ and Taylor’s laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the scaling of fluctuations for systems as diverse as written texts and cities, ecological systems and stock markets. Many modeling schemes have been proposed in literature to explain those laws, but only recently a modeling framework has been introduced that accounts for the emergence of those laws without deducing the emergence of one of the laws from the others or without ad hoc assumptions. This modeling framework is based on the concept of adjacent possible space and its key feature of being dynamically restructured while its boundaries get explored, i.e., conditional to the occurrence of novel events. Here, we illustrate this approach and show how this simple modeling framework, instantiated through a modified Pólya’s urn model, is able to reproduce Zipf’s, Heaps’ and Taylor’s laws within a unique self-consistent scheme. In addition, the same modeling scheme embraces other less common evolutionary laws (Hoppe’s model and Dirichlet processes) as particular cases. Full article

(This article belongs to the Special Issue Economic Fitness and Complexity)

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