Probabilistic Cellular Automata Research Articles

Analysis of quantitative characteristics of microstructures synthesized by method of probabilistic cellular automata

Analysis of quantitative characteristics of microstructures synthesized by method of probabilistic cellular automata

SIRS-модель распространения инфекций с динамическим регулированием численности популяции: Исследование методом вероятностных клеточных автоматов

SIRS-модель распространения инфекций с динамическим регулированием численности популяции: Исследование методом вероятностных клеточных автоматов

Ergodicity of some classes of cellular automata subject to noise

Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise. We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.

AUTOMATON: A Program That Combines a Probabilistic Cellular Automata and a Genetic Algorithm for Global Minimum Search of Clusters and Molecules.

A novel program for the search of global minimum structures of atomic clusters and molecules in the gas phase, AUTOMATON, is introduced in this work. This program involves the following: first, the generation of an initial population, using a simplified probabilistic cellular automaton method, which allows easy control of the adequate distribution of atoms in space; second, the fittest individuals are selected to evolve, through genetic operations (mating and mutations), until the best candidate for a global minimum surfaces. In addition, we propose a simple way to build the descendant structures by establishing a ranking of genes to be inherited. Thus, by means of a chemical formula checker procedure, genes are transferred to the offspring, ensuring that they always have the appropriate type and number of atoms. It is worth noting that a fraction of the fittest group is subject to mutation operations. This program also includes algorithms to identify duplicate structures: one based on geometric similarity and another on the similar distribution of atomic charges. The effectiveness of the program was evaluated in a group of 45 molecules, considering organic and organometallic compounds (benzene, cyclopentadienyl anion, and ferrocene), Zintl ion clusters [Sn9- m- nGe mBi n](4- n)- ( n = 1-4 and m = 0-(9- n)), star-shaped clusters (Li7E5+, E = BH, C, Si, Ge) and a variety of boron-based clusters. The global minimum and the lowest-energy isomers reported in the literature were found for all the cases considered in this article. These results successfully prove AUTOMATON's effectiveness on the identification of energetically preferred structures of a wide variety of chemical species.

Biological control of a predator–prey system through provision of an infected predator

Epidemic transmission has a substantial effect on the dynamics and stability of the predator–prey system, in which the transmission rate plays an important role. The probabilistic cellular automaton (PCA) approach is used to investigate the spatiotemporal dynamics of a predator–prey system with the infected predator. Remarkably, it is impossible to achieve a coexistence state of prey, susceptible predators, and infected predators in a spatial population. This is different from the analysis from a non-spatial population with the mean-field approximation, where Hopf bifurcation arises and the interior equilibrium becomes unstable, and a periodic solution appears with the increasing infection rate. The results show that the introduction of the infected predator with a high transmission rate is beneficial for the persistence of the prey population in space. However, a low transmission rate will promote the coexistence state of the prey and the susceptible predator populations. In summary, it is possible to develop management strategies to manipulate the transmission rate of the infected predator for the benefit of biological control.

Percolation games, probabilistic cellular automata, and the hard-core model

Let each site of the square lattice $$\mathbb {Z}^2$$ be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability $$1-p-q$$ , where $$0<p+q<1$$ . Consider the following game: a token starts at the origin, and two players take turns to move, where a move consists of moving the token from its current site x to either $$x+(0,1)$$ or $$x+(1,0)$$ . A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is drawn with best play—i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on $$\mathbb {Z}^2$$ has no draws. On the other hand, we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of $$\mathbb {Z}^d$$ in all $$d\ge 3$$ . This is proved via a dimension reduction to a hard-core lattice gas in dimension $$d-1$$ . We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice $$\mathbb {Z}^d$$ for $$d\ge 3$$ , but here our method encounters a fundamental obstacle.

A PROBABILISTIC CELLULAR AUTOMATON TO FORECAST PORT CHOICE DECISIONS

The port choice problem consists in predicting the selection of a port, made by an agent who has alternatives to choose from. Most of the literature has tackled this problem assuming a discrete choice model dependent on the ports’ characteristics and agents’ attributes. However, in practice the port choice decision depends also on the choices made by other agents as well as decisions made by these agents in the past. There are only a few examples that incorporate the complexity generated by spatio-temporal interactions between agents. However, those modelling structures are rather cumbersome, precluding their use in practical cases. This article presents a new modelling framework to predict port choice decisions, based on the theory of Cellular Automaton (CA), which is simple in structure and can be quickly calibrated and applied. This framework is a probabilistic CA intended to imitate the decision processes made from multiple shippers that interact with each other. These shippers face similar alternatives of seaports for exporting their products within a certain time span. The port choice here is a dynamic decision that depends on the ports’ characteristics and attributes of each shipper at a given time, as well as the decisions made by their neighbours. The outcome of the interaction is a discrete decision that evolves in time according to the dynamics of the system as a whole. The specified CA was applied to the case of vehicle exports from Brazil and the calibration was performed through a genetic algorithm. The results show that the probabilistic CA is able to replicate the historic behaviour of the port choice decisions in the Brazilian vehicle industry, with a high degree of success. The spatial component of the CA turned out to be of major relevance in the dynamic decision process along with the attributes and geographical location of ports.

Free to move or trapped in your group: Mathematical modeling of information overload and coordination in crowded populations

We present modeling strategies that describe the motion and interaction of groups of pedestrians in obscured spaces. We start off with an approach based on balance equations in terms of measures and then we exploit the descriptive power of a probabilistic cellular automaton model. Based on a variation of the simple symmetric random walk on the square lattice, we test the interplay between population size and an interpersonal attraction parameter for the evacuation of confined and darkened spaces. We argue that information overload and coordination costs associated with information processing in small groups are two key processes that influence the evacuation rate. Our results show that substantial computational resources are necessary to compensate for incomplete information — the more individuals in (information processing) groups the higher the exit rate for low population size. For simple social systems, it is likely that the individual representations are not redundant and large group sizes ensure that this non-redundant information is actually available to a substantial number of individuals. For complex social systems, information redundancy makes information evaluation and transfer inefficient and, as such, group size becomes a drawback rather than a benefit. The effect of group sizes on outgoing fluxes, evacuation times and wall effects is carefully studied with a Monte Carlo framework accounting also for the presence of an internal obstacle.

Clustered Breeding Sites: Shelters for Vector-Borne Diseases.

Here, the propagation of vector-borne diseases is modeled by using a probabilistic cellular automaton. Numerical simulations considering distinct spatial distributions and time variations of the vector abundance are performed, in order to investigate their impacts on the number of infected individuals of the host population. The main conclusion is as follows: in the clustered distributions, the prevalence is lower, but the eradication is more difficult to be achieved, as compared to homogeneous distributions. This result can be relevant in the implementation of preventive surveillance measures.

Gaussian Mean Field Lattice Gas

We study rigorously a lattice gas version of the Sherrington-Kirckpatrick spin glass model. In discrete optimization literature this problem is known as Unconstrained Binary Quadratic Programming (UBQP) and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present an heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.

Dengue fever spreading based on probabilistic cellular automata with two lattices

Modeling and simulation of mosquito-borne diseases have gained attention due to a growing incidence in tropical countries in the past few years. Here, we study the dengue spreading in a population modeled by cellular automata, where there are two lattices to model the human–mosquitointeraction: one lattice for human individuals, and one lattice for mosquitoes in order to enable different dynamics in populations. The disease considered is the dengue fever with one, two or three different serotypes coexisting in population. Although many regions exhibit the incidence of only one serotype, here we set a complete framework to also study the occurrence of two and three serotypes at the same time in a population. Furthermore, the flexibility of the model allows its use to other mosquito-borne diseases, like chikungunya, yellow fever and malaria. An approximation of the cellular automata is proposed in terms of ordinary differential equations; the spreading of mosquitoes is studied and the influence of some model parameters are analyzed with numerical simulations. Finally, a method to combat dengue spreading is simulated based on a reduction of mosquito birth and mosquito bites in population.

Probabilistic Modelling of Microstructural Evolution in Zr Based Bulk Metallic Glass Matrix Composites during Solidification in Additive Manufacturing

Bulk metallic glass and their composites (BMGMCs) are a new class of materials which possess superior mechanical properties as compared to existing conventional materials. Owing to this, they are potential candidates for tomorrow’s structural applications. However, they suffer from poor ductility and little or no toughness which render them brittle and they manifest catastrophic failure under applied force. Their behavior is dubious, unpredictable and requires extensive experimentation to arrive at conclusive results. In present study, an effort has been made to design bulk metallic glass matrix composites by the use of modeling and simulation. A probabilistic cellular automaton (CA) model is developed and described in present study by author which is used in conjunction with earlier developed deterministic model to predict microstructural evolution in Zr based BMGMCs in additive manufacturing liquid melt pool. It is elaborately described with an aim to arrive at quantitative relations which describe process and steps of operations. Results indicate that effect of incorporating all mass transfer and diffusion coefficients under transient conditions and precise determination of probability number play a vital role in refining the model and bringing it closer to a level that it could be compared to actual values. It is shown that proposed tailoring can account for microstructural evolution in metallic glasses.

Modeling Adsorption–Desorption Processes at the Intermolecular Interactions Level

Modeling of the surface adsorption and desorption processes, as well as the diffusion, are of considerable interest for the physical phenomenon under study in ground tests conditions. When imitating physical processes and phenomena, it is important to choose the correct parameters to describe the adsorption of gases and the formation of films on the structural materials surface. In the present research the adsorption-desorption processes on the gas-solid interface are modeled with allowance for diffusion. Approaches are proposed to describe the adsorbate distribution on the solid body surface at the intermolecular interactions level. The potentials of the intermolecular interaction of water–water, water–methane and methane–methane were used to adequately modeling the real physical and chemical processes. The energies calculated by the B3LYP/aug-cc-pVDZ method. Computational algorithms for determining the average molecule area in a dense monolayer, are considered here. Differences in modeling approaches are also given: that of the proposed in this work and the previously approved probabilistic cellular automaton (PCA) method. It has been shown that the main difference is due to certain limitations of the PCA method. The importance of accounting the intermolecular interactions via hydrogen bonding has been indicated. Further development of the adsorption-desorption processes modeling will allow to find the conditions for of surface processes regulation by means of quantity adsorbed molecules control. The proposed approach to representing the molecular system significantly shortens the calculation time in comparison with the use of atom-atom potentials. In the future, this will allow to modeling the multilayer adsorption at a reasonable computational cost.

Modeling and Analysis of Traffic Flow Considering Automatic Gap Control

SUMMARYIn recent years, the number of vehicles with automatic gap control (AGC) (e.g., adaptive cruise control) has increased with the development of vehicle control technology. AGC mainly helps to ease driver fatigue on highways. Vehicles equipped with AGC can maintain an appropriate distance between two vehicles. Therefore, the traffic flow can be improved by AGC‐equipped vehicles. In this research, we develop a traffic flow model that considers AGC‐equipped vehicles, and is based on the Nagel–Schreckenberg model, which is a probabilistic cellular automaton model. The Nagel–Schreckenberg model can reproduce the basic features of traffic flow. On analyzing the mathematical model for the traffic including AGC‐equipped vehicles, we can obtain analytical solutions of the traffic flow and the average velocity.

Investigation of Size and Shape Dependencies of Phase Diagrams of the Ising Nanofilms and Nanotubes on the Honeycomb Lattice Using Cellular Automata Simulation Approach

Size and shape dependencies of phase diagrams of the Ising nanofilms and nanotubes on the honeycomb lattice are investigated by means of probabilistic cellular automata simulation based on Glauber algorithm. The values of reduced critical temperature, K c = k B T c/J (where k B and J are the Boltzmann constant and nearest-neighbor coupling, respectively), for both nanofilms and nanotubes, are obtained at the different sizes of the lattices and the dependency of K c to the number of layers is studied. By increasing the number of layers K c increases but for number of layer more than 8, the critical temperature increases very slowly. We have shown that between two isotropic nanotubes with the same number of spins, the ones with greater diameter (more spins on the edge) have larger critical temperature. For equal size of lattices, the obtained values of K c for nanotube are greater than the nanofilm, but for large sizes, this difference disappears.

Tumor growth modelling by cellular automata

Tumor growth is a complex process that requires mathematical modeling approaches for studying real-life cancer behavior. The use of cellular automata (CA) to represent tumor growth in its avascular stage is explained in this work, and a stochastic CA describing tumor growth is obtained, based on a differential equations system in the range of continuum mechanics. The novelty of this research is the deduction of the neighborhood structure and rules for a probabilistic CA from these differential equations that describe the evolution of the tumor growth. In addition, the influence of the stresses on tumor growth is captured by the CA.

On estimating the impact of information spreading in a consumer market modeled by probabilistic cellular automata and ordinary differential equations

Individuals spread information about their lives, their career, and their recently purchased items. In addition, considering that the customer’s opinions may influence their friendship circle, neighbors, and work colleagues, they constitute a very important factor to consider in future actions of companies. In this paper, we present an approach based on probabilistic cellular automata and ordinary differential equations for modeling the consumer market, inspired by a biological system of an infectious disease-spreading model. However, instead of spreading a disease, consumers spread their opinion to other individuals, thereby having consequences on sales levels. Using marketing, the economic situation, and individual opinion as input variables of the consumer market model, simulation output (generated demand) is then used in the business game, helping the mediator (teacher) to create different scenarios and giving feedback to the student at simulation runtime, even if many players are involved in the game. Analytical and numerical simulations show that the proposed approach allows the teacher to create virtual markets for the players just by setting the parameters that describe a scenario, increasing the dynamism of the game, saving time, and allowing the mediator to focus on other important subjects, such as the teaching concepts of business strategies and decision support. Moreover, simulations are conducted to demonstrate the applicability of the proposed model in real life problems, such as the electro-domestic market.

Convergence Time and Phase Transition in a Non-monotonic Family of Probabilistic Cellular Automata

In dynamical systems, some of the most important questions are related to phase transitions and convergence time. We consider a one-dimensional probabilistic cellular automaton where their components assume two possible states, zero and one, and interact with their two nearest neighbors at each time step. Under the local interaction, if the component is in the same state as its two neighbors, it does not change its state. In the other cases, a component in state zero turns into a one with probability $$\alpha ,$$ and a component in state one turns into a zero with probability $$1-\beta $$ . For certain values of $$\alpha $$ and $$\beta $$ , we show that the process will always converge weakly to $$\delta _{0},$$ the measure concentrated on the configuration where all the components are zeros. Moreover, the mean time of this convergence is finite, and we describe an upper bound in this case, which is a linear function of the initial distribution. We also demonstrate an application of our results to the percolation PCA. Finally, we use mean-field approximation and Monte Carlo simulations to show coexistence of three distinct behaviours for some values of parameters $$\alpha $$ and $$\beta $$ .

Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects.

We investigate one-dimensional elementary probabilistic cellular automata (PCA) whose dynamics in first-order mean-field approximation yields discrete logisticlike growth models for a single-species unstructured population with nonoverlapping generations. Beginning with a general six-parameter model, we find constraints on the transition probabilities of the PCA that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof. Several possible models display a negative cubic term that can be interpreted as a weak Allee factor. We also investigate the conditions under which a one-parameter PCA derived from the more general six-parameter model can generate valid population growth dynamics. Numerical simulations illustrate the behavior of some of the PCA found.

Construction of phase diagrams for nanoscaled Ising thin films on the honeycomb lattice using cellular automata simulation approach

ABSTRACTThe phase diagrams of the three-layer Ising model on the honeycomb lattice with a diluted surface have been constructed using the probabilistic cellular automata based on Glauber algorithm. The effects of the exchange interactions on the phase diagrams have been investigated. A general mathematical expression for the critical temperature is obtained in terms of relative coupling r = J1/J and Δs = (Js/J) − 1, where J and Js represent the nearest neighbor coupling within inner- and surface-layers, respectively, and each magnetic site in the surface-layer is coupled with the nearest neighbor site in the inner-layer via the exchange coupling J1. In the case of antiferromagnetic coupling between surface-layer and inner-layer, system reveals many interesting phenomena, such as the possibility of existence of compensation line before the critical temperature.