Phase-type distributions in stochastic automata networks

Stochastic automata networks (Sans) are high-level formalisms for modeling very large and complex Markov chains in a compact and structured manner. To date, the exponential distribution has been the only distribution used to model the passage of time in the evolution of the different San components. In this paper we show how phase-type distributions may be incorporated into Sans thereby providing the wherewithal by which arbitrary distributions can be used which in turn leads to an improved ability for more accurately modeling numerous real phenomena.     

Comparative analysis of asynchronous cellular automata in stochastic pharmaceutical modelling

In pharmaceutical modelling, cellular automata have been used as an established tool to represent molecular changes through discrete structural interactions. The data quality provided by such modelling is found suitable for the early drug design phase where flexibility is paramount. While both synchronous (CA) and asynchronous (ACA) types of automata have been used, analysis of their nature and comparative influence on model outputs is lacking. In this paper, we outline a representative probabilistic CA for modelling complex controlled drug formulations and investigate its transition from synchronous to asynchronous update algorithms. The key investigation points include quantification of model dynamics through three distinct scenarios, parallelisation performance and the ability to describe different release phenomena, namely erosion, diffusion and swelling. The choice of the appropriate update mechanism impacts the perceived realism of the simulation as well as the applicability of large-scale simulations.     

Vulnerability for systems described by cellular automata: Application to flood phenomena

In this paper, we introduce the vulnerability and vulnerability index for systems described by cellular automata. A zone will be vulnerable to the space-time expansion of a given property during a time horizon if the property trajectory reaches the zone at a certain time. The space-time expansion of the property known as spreadability depends closely on the property definition, the cells properties and the transition function which governs the cellular automata evolution. These constraints exhibit two types of spreadability (inclusion sense and area sense) which affect vulnerable zones. When multiple zones are vulnerable during a given time horizon, we need criteria which describe whose are the most vulnerable. Then we consider iterative, average and global vulnerability index whose use depends on the considered phenomena. As an application we consider the flood phenomena. For this we use the two scale cellular automaton for flow dynamics modeling (2CAFDYM) over a given terrain. This application requires an adaptation of the general vulnerability index in order to take into account the both horizontal and vertical distributions of surface water. As practice application, we consider simulation for a basin in northern Morocco using a simulation software we have designed in Java Object Oriented Programming. Digital terrain model, geological maps and satellite image are used for input data.     

Exponential stability analysis of stochastic delayed cellular neural network

In this paper, the stability of stochastic delayed cellular neural networks are studied. Via the Lyapunov function method and some analysis techniques, we obtain some new criteria of exponential 1-stability and mean square exponential stability.     

Prevalence of odometers in cellular automata

We consider left permutive cellular automata  with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points  such that is topologically conjugate to an odometer, the ``' map on the countable product of finite cyclic groups. This set is a dense subset of an appropriate subspace. We identify the odometer in several cases.

     

Computation of shortest path in cellular automata

In this paper, we show how to find the shortest path between given nodes of a mesh with weighted edges. We use a cellular automaton which exhibits autowave patterns, where a wave of auto-excitation originates in source node, spreads around the mesh, and modifies states of the cells to make a stationary pattern isomorphed to the shortest path from source node to destination one. For different formulations of the shortest path problem, and various cellular automata (which solve it), we proved the bounds of complexity. For the sake of clarity, we present several examples of cellular automata computation of the shortest path. They are illustrated in detail.     

Critical Fields in Ferromagnetic Thin Films: Identification of Four Regimes

We are interested in critical fields for ferromagnetic elements: At which strength of the external field does a branch of stationary magnetizations become unstable, and what is the unstable mode? We consider samples which are infinite in the direction of the external field and have a rectangular cross-section, of much smaller thickness than width, as an idealization of a thin film element. For this geometry Aharoni [Phys. Stat. Sol. 16, 3-42 (1966)] claims that there are only three different regimes: The unstable mode is either of coherent rotation type, of buckling type, or of curling type. We discover a large fourth parameter regime with an unstable mode displaying an oscillation in the infinite direction. We prove the existence of exactly four regimes by rigorously analyzing the scaling of the Rayleigh quotient of the Hessian of the energy functional. The parameters are the film width, the film thickness, and the exchange length.     

Stability of vertices in random boolean cellular automata

Based on computer simulations, Kauffman (Physica D, 10, 145-156, 1984) made several generalizations about a random Boolean cellular automaton which he invented as a model of cellular metabolism. Here we give the first rigorous proofs of two of Kauffman's generalizations: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automaton's behavior is small compared to that of a random mapping with the same number of states; and reversal of the states of a large fraction of the vertices does not affect the cycle to which the automaton moves.     

Mean square exponential stability and periodic solutions of stochastic delay cellular neural networks

This paper mainely concerns the exponential stability analysis and the existence of periodic solution problems for a class of stochastic cellular neural networks with discrete delays (SDCNNs). Above all, Poincare contraction theory is utilized to derive the conditions guaranteeing the existence of periodic solutions of SDCNNs. Next, Lyapunov function, stochastic analysis theory and Young inequality approach is developed to derive some theorems which gives several sufficient conditions such that periodic solutions of SDCNNs are mean square exponential stable. These sufficient conditions only including those governing parameters of SDCNNs can be easily checked by simple algebraic methods. Finally, two examples are given to demonstrate that the proposed criteria are useful and effective.     

Exhaustive study of the noise-induced phase transition in a stochastic model of self-catalyzed reactions

We completely investigate the stationary distribution density in the space of relative concentrations for the three-parameter stochastic Horsthemke–Lefever model of a binary self-catalyzed cyclic chemical reaction with perturbations produced by thermal fluctuations of reagents taken into account. This model is a stationary diffusion random process generated by a stochastic equation with the Stratonovich differential, whose marginal distribution density admits a bifurcation restructuring from the unimodal to the bimodal phase with increasing noise intensity, which is interpreted physically as a dynamical phase transition induced by fluctuations in the system.     

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

The use of “control parameters” as applied to describe the dynamics of complex mathematical systems within models of real social systems is discussed. Whereas single control parameters cannot sufficiently characterize the dynamics of such systems it is suggested that domains of values of certain sets of parameters are appropriately denoting necessary conditions for highly disordered dynamics of social systems. Various of those control parameters permit a straightforward interpretation in terms of properties of social rules and structures. © 1999 John Wiley & Sons, Inc.     

Hierarchical models of interacting diffusions: Multiple time scale phenomena,phase transition and pattern of cluster-formation

Summary The purpose of this paper is to explore the connection between multiple space-time scale behaviour for block averages and phase transitions, respectively formation of clusters, in infinite systems with locally interacting components. The essential object is the associated Markov chain which describes the joint distribution of the block averages at different time scales. A fixed-point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions.     

Predicting crashes in a model of evolving networks

We consider an evolving network of interacting species which exhibits self‐organization. The system is characterized by repeated crashes in which a large number of species are extinct and subsequent recoveries. We investigate the macroscopic properties of this system before such crashes, concentrating on the variance of the relative population sizes of species and its evolution over time. A simple score function is constructed to determine the probability of a crash within a certain time interval to be used as a predictor for crashes. © 2004 Wiley Periodicals, Inc. Complexity 9: 24–30, 2004     

The social impact in a high-risk community: A cellular automata model

This research examines the spread of criminal behavior and hard drug consumption using a mathematical approach called cellular automata (CA). This CA model is based on two behavioral concepts. Firstly, peer association impacts criminal involvement. Secondly, addiction can heighten criminal activity. The model incorporates four types of actors who interact in a high-risk social community and one intervention method. The actors exert a social influence on each other by encouraging or discouraging drug use and criminal behavior. The intervention method called Incapacitation has a probabilistic impact on the individuals in the model. The results identify the threshold where positive influences on a population reduce the number of high-rate offenders in the community. These results are discussed to further the knowledge about the social influences in a high-risk community and how these influences can effect decisions on offender management.     

Modelling nonlinear sequence generators in terms of linear cellular automata

In this work, a wide family of LFSR-based sequence generators, the so-called clock-controlled shrinking generators (CCSGs), has been analyzed and identified with a subset of linear cellular automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.     

Exponential synchronization of stochastic fuzzy cellular neural networks with time delay in the leakage term and reaction-diffusion

Separate studies have been published on the stability of fuzzy cellular neural networks with time delay in the leakage term and synchronization issue of coupled chaotic neural networks with stochastic perturbation and reaction-diffusion effects. However, there have not been studies that integrate the two fields. Motivated by the achievements from both fields, this paper considers the exponential synchronization problem of coupled chaotic fuzzy cellular neural networks with stochastic noise perturbation, time delay in the leakage term and reaction-diffusion effects using linear feedback control. Lyapunov stability theory combining with stochastic analysis approaches are employed to derive sufficient criteria ensuring the coupled chaotic fuzzy neural networks to be exponentially synchronized. This paper also presents an illustrative example and uses simulated results of this example to show the feasibility and effectiveness of the proposed scheme.     

The emergence of dynamical complexity: An exploration using elementary cellular automata

This work concerns the interaction between two classical problems: the forecasting of the dynamical behaviors of elementary cellular automata (ECA) from its intrinsic mathematical laws and the conditions that determine the emergence of complex dynamics. To approach these problems, and inspired by the theory of reversible logical gates, we decompose the ECA laws in a “spectrum” of dyadic Boolean gates. Emergent properties due to interactions are captured generating another spectrum of logical gates. The combined analysis of both spectra shows the existence of characteristic bias in the distribution of Boolean gates for ECA belonging to different dynamical classes. These results suggest the existence of signatures capable to indicate the propensity to develop complex dynamics. Logical gates “exclusive‐or” and “equivalence” are among these signatures of complexity. An important conclusion is that within ECA space, interactions are not capable to generate signatures of complexity in the case these signatures are absent in the intrinsic law of the automaton. © 2004 Wiley Periodicals, Inc. Complexity 9: 33–42, 2004