Narrow Results

The short-time and long-time dynamics of the Bak–Sneppen model of biological evolution are investigated using the damage spreading technique. By defining a proper Hamming distance measure, we are able to make it exhibit an initial power-law growth which, for finite size systems, is followed by a decay towards equilibrium. In this sense, the dynamics of self-organized critical states is shown to be similar to the one observed at the usual critical point of continuous phase transitions and at the onset of chaos of nonlinear low-dimensional dynamical maps. The transient, pre-asymptotic and asymptotic exponential relaxation of the Hamming distance between two initially uncorrelated equilibrium configurations is also shown to be fit within a single mathematical framework.

Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.

The method of Damage Spreading was used to simulate the influence that a single persons' change of opionion has on the consensus opinion built up in a population if one assumes opinions to form according to the Sznajd Model. The results confirm the intuitive assumption that there is hardly any chance for one person to change the consensus, that the effect of this change dies out after a certain time and its range decreases with time. The consensus times were compared and it turned out that the consensus can be delayed or accelerated by this slight modification and that the amount of the difference in the consensus times obeys a certain power law as well as the lifetime of the effect. Moreover, two scaling laws concerning temporal and spatial aspects could be observed up to a certain size of population.

Monte Carlo simulations have been used to carry out a comparative study of the block distribution function and damage spreading characteristics of a half-filled lattice gas model (conserved order parameter) and a simple Ising model (non-conserved order parameter). Spin-exchange heat-bath dynamics were used to check the validity of equilibrium predictions for the lattice gas model using a histogram ratio analysis. Histogram reweighting was used to extract the interface energy for the lattice gas model, and the critical exponent β =1/8 was successfully estimated by using sub-block scaling and sub-block damage scaling.

We describe systems using Kauffman and similar networks. They are directed functioning networks consisting of finite number of nodes with finite number of discrete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of "structural tendencies". Along the way we expand Kauffman networks, which were a synonym of Boolean networks, to more than two signal variants and we find a phenomenon during network growth which we interpret as "complexity threshold". For simulation we define a simplified algorithm which allows us to omit the problem of periodic attractors. We estimate that living and human designed systems are chaotic (in Kauffman sense) which can be named — complex. Such systems grow in adaptive evolution. These two simple assumptions lead to certain statistical effects, i.e., structural tendencies observed in classic biology but still not explained and not investigated on theoretical way. For example, terminal modifications or terminal predominance of additions where terminal means: near system outputs. We introduce more than two equally probable variants of signal, therefore our networks generally are not Boolean networks. They grow randomly by additions and removals of nodes imposed on Darwinian elimination. Fitness is defined on external outputs of system. During growth of the system we observe a phase transition to chaos (threshold of complexity) in damage spreading. Above this threshold we identify mechanisms of structural tendencies which we investigate in simulation for a few different networks types, including scale-free BA networks.

We study the asymmetric influential effects in the damage spreading processes, by investigating the opinion dynamics of the Krause–Hegselmann consensus model, wherein the agents sit on the nodes of small-world networks. One agent is randomly selected as the influential agent, which can influence the opinions of other agents with probability p but not vice versa. The damage consists in a sharp change of the opinion of one agent (a randomly selected agent or the influential one) in the initial random opinion configuration. We find that for small values of the influential probability p, there is a damage spreading transition at low values of the confidence bound parameter ε. Interestingly, this transition vanishes for large values of p when the influential agent is damaged in the initial state. We find as well that, there is a critical value of the confidence bound parameter above which the initial perturbation manages to propagate to the whole system. The relationship between the amount of the damage and the influential activity of the perturbed agent is also discussed.

Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature T_c of the undiluted Ising model, but above the critical temperature of the diluted system. Using damage spreading (or rather damage "healing"), we are able to measure down to much lower final magnetizations (10^–9) and to much larger times than previous authors. Nevertheless, we do not yet find the predicted asymptotic behavior in the Griffiths phase T < T_c. But we can at least exclude a stretched exponential decay as found in previous papers, for T < T_c. Finally, we discuss the case T = T_c where a stretched exponential decay can be proven to hold, at least for p < p_c. We indeed do see a stretched exponential for p = p_c (and T = T_c), but we show that it cannot describe the asymptotic behavior either.

By studying correlation functions of two replicas with the damage spreading method, the spin glass transition temperature of the 3D ±J EA model is evaluated by using Campbell's criterion D (T_g, t → ∞) = 1/2 or , where D and are the Hamming distance and the overlap of the two replicas, respectively. The two replicas evolve with the same series of the random numbers and with initial spin-configurations related to each other by (for all i). From our simulational data for system sizes up to L = 64, the spin glass transition temperature is estimated as T_g = 1.18 ± 0.06. The replica simulational method and the damage spreading method are compared with each other. The relaxation exponents of the magnetization and the correlation functions are also estimated in the vicinity of T_g on a large lattice and the dynamical critical exponent z is estimated as z ≃ 6.0.

We study the effects of random faults on the behavior of one-dimensional, non-uniform cellular automata (CA), where the local update rule need not be identical for all grid sites. The CA systems examined were obtained via an approach known as cellular programming, which involves the evolution of non-uniform CAs to perform non-trivial computational tasks. Using the "system replicas" methodology, involving a comparison between a perfect, non-perturbed version of the CA and a faulty one, we find that our evolved systems exhibit graceful degradation in performance, able to tolerate a certain level of faults. We then "zoom" into the fault-tolerant zone, where "good" computational behavior is exhibited, introducing measures to fine-tune our understanding of the faulty CAs' operation. We study the error level as a function of time and space, as well as the recuperation time needed to recover from faults. Our investigation reveals an intricate interplay between temporal and spatial factors, with the presence of different rules in the grid giving rise to complex dynamics. Studies along this line may have applications to future computing systems that will contain thousands or even millions of computing elements, rendering crucial the issue of resilience.

After improving the Monte Carlo statistics, Derrida's exponent θ for the never damaged sites in Ising models at finite temperatures is shown to be compatible with θ(T=0) =0.22 in two dimensions. In three and more dimensions the number of these sites decays differently from zero temperature.

We study numerically the nature of the retrieval attractors in an asymmetrically diluted Hopfield neural network through the damage propagation technique. We consider the damage evolution of two replicas, initially very close to each other and having both finite projection with one memorized pattern. By analyzing the asymptotic behavior of the damage, we characterize the dynamical nature of the retrieval attractors. We found, in the recognition phase, a dynamical phase transition separating chaotic and fixed point retrieval trajectories. We also present a conjugate field h associated with the damage, which destroys the dynamical transition and whose corresponding susceptibility presents a sharp peak at the critical parameter.

The temperature dependence of the dynamics of the fraction F(t) of persistent spins in the triangular Q-state Potts model is investigated by large scale Monte Carlo simulations. After extending Derrida's approach of measuring the fraction of spins that remain in one phase to allQ low-temperature phases, it is shown that the exponent θ of algebraic decay of F(t) is independent of temperature.

We investigate the asymptotic density of living cells as dependent on an initial density, the results of scattering of gliders on small stable configurations and the damage spreading in an open system, subjected to a stream of gliders with the rules of the cellular automaton "Game of Life." We find that a damage increases approximately with the second power of time through about the first ten steps; later its time dependence becomes almost linear.

The damage spreading of the Ising model on the homo- and hetero-cell lattices (here the topological hexagonal lattice and the 4–8 lattice) with competing Glauber (G-) and Kawasaki (K-) dynamics is studied and the results are compared. For the homo-cell lattice, we pay attention to the pure K-dynamics or the cases in which the K-dynamics is dominant. We get four main conclusions related to the K-dynamics through our calculations: (1) the damage always spreads as long as Kawasaki dynamics is dominant during the competition of two dynamics; (2) the relation for the saturation damage, 〈s〉_∞ = 0.5, holds for K-dynamics whatever the updating rules are; (3) 〈s〉_∞ = 0.5 is independent of the structures of the system; (4) the DS process under pure K-dynamics converges very slowly, especially for T = 0 K-dynamics. For the hetero-cell lattice, we are interested in the long-range interaction between spins and the different interaction strength for the spins in the hetero-cells. To include the long-range spin interaction, we consider the spin interactions up to next-nearest neighbors. It is shown that the inclusion of the next-nearest-neighbor interaction enhances the transition temperature greatly. The explanation and discussion for the results are presented.

The damage spreading of the Ising model on a set of trivalent structures is studied with competing Glauber and Kawasaki dynamics. We find that the damage transition temperature T_d increases as the number of rectangles increases at the pure Glauber dynamics case and T_d decreases sharply as the probability of occurrence of Kawasaki dynamics increases. A heuristic argument is provided to explain the numerical results.