Damage spreading and coupling in Markov chains

In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire configuration space. We find dynamical damage-spreading transitions deeply inside the paramagnetic and liquid phases, and show that critical values of the transition temperatures and densities depend on the coupling scheme. We discuss our findings in the light of a classic proof that for arbitrary Monte Carlo algorithms damage spreading can be avoided through non-Markovian coupling schemes.Comment: 6 pages, 8 figure

Transmission of packets on a hierarchical network: Statistics and explosive percolation

We analyze an idealized model for the transmission or flow of particles, or discrete packets of information, in a weight bearing branching hierarchical 2-D networks, and its variants. The capacities add hierarchically down the clusters. Each node can accommodate a limited number of packets, depending on its capacity and the packets hop from node to node, following the links between the nodes. The statistical properties of this system are given by the Maxwell - Boltzmann distribution. We obtain analytical expressions for the mean occupation numbers as functions of capacity, for different network topologies. The analytical results are shown to be in agreement with the numerical simulations. The traffic flow in these models can be represented by the site percolation problem. It is seen that the percolation transitions in the 2-D model and in its variant lattices are continuous transitions, whereas the transition is found to be explosive (discontinuous) for the V- lattice, the critical case of the 2-D lattice. We discuss the implications of our analysis.Comment: 24 pages, 41 figure

Growth and Decay in Life-Like Cellular Automata

We propose a four-way classification of two-dimensional semi-totalistic cellular automata that is different than Wolfram's, based on two questions with yes-or-no answers: do there exist patterns that eventually escape any finite bounding box placed around them? And do there exist patterns that die out completely? If both of these conditions are true, then a cellular automaton rule is likely to support spaceships, small patterns that move and that form the building blocks of many of the more complex patterns that are known for Life. If one or both of these conditions is not true, then there may still be phenomena of interest supported by the given cellular automaton rule, but we will have to look harder for them. Although our classification is very crude, we argue that it is more objective than Wolfram's (due to the greater ease of determining a rigorous answer to these questions), more predictive (as we can classify large groups of rules without observing them individually), and more accurate in focusing attention on rules likely to support patterns with complex behavior. We support these assertions by surveying a number of known cellular automaton rules.Comment: 30 pages, 23 figure

Maximum Likelihood Estimator for Hidden Markov Models in continuous time

The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain.Comment: Warning: due to a flaw in the publishing process, some of the references in the published version of the article are confuse

Growth Kinetics in Systems with Local Symmetry

The phase transition kinetics of Ising gauge models are investigated. Despite the absence of a local order parameter, relevant topological excitations that control the ordering kinetics can be identified. Dynamical scaling holds in the approach to equilibrium, and the growth of typical length scale is characteristic of a new universality class with $L(t)\sim \left(t/\ln t\right)^{1/2}$. We suggest that the asymptotic kinetics of the 2D Ising gauge model is dual to that of the 2D annihilating random walks, a process also known as the diffusion-reaction $A+A\to \hbox{inert}$.Comment: 10 pages in Tex, 2 Postscript figures appended, NSF-ITP-93-4

Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems

Most current methods for identifying coherent structures in spatially-extended systems rely on prior information about the form which those structures take. Here we present two new approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages, and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively), and provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv requirements; write first author for higher-resolution version

Localization dynamics in a binary two-dimensional cellular automaton: the Diffusion Rule

We study a two-dimensional cellular automaton (CA), called Diffusion Rule (DR), which exhibits diffusion-like dynamics of propagating patterns. In computational experiments we discover a wide range of mobile and stationary localizations (gliders, oscillators, glider guns, puffer trains, etc), analyze spatio-temporal dynamics of collisions between localizations, and discuss possible applications in unconventional computing.Comment: Accepted to Journal of Cellular Automat

The time to extinction for an SIS-household-epidemic model

We analyse a stochastic SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission

Randomly Evolving Idiotypic Networks: Structural Properties and Architecture

We consider a minimalistic dynamic model of the idiotypic network of B-lymphocytes. A network node represents a population of B-lymphocytes of the same specificity (idiotype), which is encoded by a bitstring. The links of the network connect nodes with complementary and nearly complementary bitstrings, allowing for a few mismatches. A node is occupied if a lymphocyte clone of the corresponding idiotype exists, otherwise it is empty. There is a continuous influx of new B-lymphocytes of random idiotype from the bone marrow. B-lymphocytes are stimulated by cross-linking their receptors with complementary structures. If there are too many complementary structures, steric hindrance prevents cross-linking. Stimulated cells proliferate and secrete antibodies of the same idiotype as their receptors, unstimulated lymphocytes die. Depending on few parameters, the autonomous system evolves randomly towards patterns of highly organized architecture, where the nodes can be classified into groups according to their statistical properties. We observe and describe analytically the building principles of these patterns, which allow to calculate number and size of the node groups and the number of links between them. The architecture of all patterns observed so far in simulations can be explained this way. A tool for real-time pattern identification is proposed.Comment: 19 pages, 15 figures, 4 table