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arXiv1998

Collective Behaviour and Diversity in Economic Communities: Some Insights from an Evolutionary Game

Many complex adaptive systems contain a large diversity of specialized components. The specialization at the level of the microscopic degrees of freedom, and diversity at the level of the system as a whole are phenomena that appear during the course of evolution of the system. We present a mathematical model to describe these evolutionary phenomena in economic communities. The model is a generalization of the replicator equation. The economic motivation for the model and its relationship with some other game theoretic models applied to ecology and sociobiology is discussed. Some results about the attractors of this dynamical system are described. We argue that while the microscopic variables -- the agents comprising the community -- act locally and independently, time evolution produces a collective behaviour in the system characterized by individual specialization of the agents as well as global diversity in the community. This occurs for generic values of the parameters and initial conditions provided the community is sufficiently large, and can be viewed as a kind of self-organization in the system. The context dependence of acceptable innovations in the community appears naturally in this framework.

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arXiv1998

Dynamics of Individual Specialization and Global Diversification in Communities

We discuss a model of an economic community consisting of $N$ interacting agents. The state of each agent at any time is characterized, in general, by a mixed strategy profile drawn from a space of $s$ pure strategies. The community evolves as agents update their strategy profiles in response to payoffs received from other agents. The evolution equation is a generalization of the replicator equation. We argue that when $N$ is sufficiently large and the payoff matrix elements satisfy suitable inequalities, the community evolves to retain the full diversity of available strategies even as individual agents specialize to pure strategies.

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arXiv1998

Neural network design for J function approximation in dynamic programming

This paper shows that a new type of artificial neural network (ANN) -- the Simultaneous Recurrent Network (SRN) -- can, if properly trained, solve a difficult function approximation problem which conventional ANNs -- either feedforward or Hebbian -- cannot. This problem, the problem of generalized maze navigation, is typical of problems which arise in building true intelligent control systems using neural networks. (Such systems are discussed in the chapter by Werbos in K.Pribram, Brain and Values, Erlbaum 1998.) The paper provides a general review of other types of recurrent networks and alternative training techniques, including a flowchart of the Error Critic training design, arguable the only plausible approach to explain how the brain adapts time-lagged recurrent systems in real-time. The C code of the test is appended. As in the first tests of backprop, the training here was slow, but there are ways to do better after more experience using this type of network.

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arXiv1998

On the 1/f fluctuations in the nonlinear systems affected by noise

We investigate a problem of the necessary and sufficient conditions for appearance of the 1/f fluctuations in the simple systems affected by the external random perturbations, i.e. the power spectral density of the flux of particles moving in some contours and perturbed by the external forces. In some cases we observe the 1/f behavior but only in some range of frequencies and parameters of the systems.

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arXiv1998

How fast do structures emerge in hypercycle-systems?

A general framework for the simulation of reaction-diffusion systems with probabilistic cellular automata is presented. The basic reaction probabilities of the chemical model translate directly into the transition rules of the automaton, thus allowing a clear comparison between simulation results and analytic calculations. This framework is then applied to simulations of hypercycle-systems in up to three dimensions. Furthermore, a new measurement quantity is introduced and applied to the hypercycle-systems in two and three dimensions. It can be shown that this quantity can be interpreted as a measure for the macroscopic order of the hypercycle systems.

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arXiv1998

Simple model of 1/f noise

Simple analytically solvable model of 1/f noise is proposed. The model consists of one or few particles moving in the closed contour. The drift period of the particle round the contour fluctuates about some average value, e.g. due to the external random perturbations of the system's parameters. The model contains only one relaxation rate, however, the power spectral density of the current of particles reveals an exact 1/f spectrum in any desirable wide range of frequency and can be expressed by the Hooge formula. It is likely that the analysis and the generalizations of the model can strongly influence on the understanding of the origin of 1/f noise.

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arXiv1999

Nondecay probability of a metastable state: almost exact analytical description in wide range of noise intensity

This paper presents a complete description of noise-induced decay of a metastable state in a wide range of noise intensity. Recurrent formulas of exact moments of decay time valid for arbitrary noise intensity have been obtained. The nondecay probability of a metastable state was found to be really close to the exponent even for the case when the potential barrier height is comparable or smaller than the noise intensity.

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arXiv1998

Development and Evolution of Neural Networks in an Artificial Chemistry

We present a model of decentralized growth for Artificial Neural Networks (ANNs) inspired by the development and the physiology of real nervous systems. In this model, each individual artificial neuron is an autonomous unit whose behavior is determined only by the genetic information it harbors and local concentrations of substrates modeled by a simple artificial chemistry. Gene expression is manifested as axon and dendrite growth, cell division and differentiation, substrate production and cell stimulation. We demonstrate the model's power with a hand-written genome that leads to the growth of a simple network which performs classical conditioning. To evolve more complex structures, we implemented a platform-independent, asynchronous, distributed Genetic Algorithm (GA) that allows users to participate in evolutionary experiments via the World Wide Web.

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arXiv1998

A dynamical theory of speciation on holey adaptive landscapes

The metaphor of holey adaptive landscapes provides a pictorial representation of the process of speciation as a consequence of genetic divergence. In this metaphor, biological populations diverge along connected clusters of well-fit genotypes in a multidimensional adaptive landscape and become reproductively isolated species when they come to be on opposite sides of a ``hole'' in the adaptive landscape. No crossing of any adaptive valleys is required. I formulate and study a series of simple models describing the dynamics of speciation on holey adaptive landscapes driven by mutation and random genetic drift. Unlike most previous models that concentrate only on some stages of speciation, the models studied here describe the complete process of speciation from initiation until completion. The evolutionary factors included are selection (reproductive isolation), random genetic drift, mutation, recombination, and migration. In these models, pre- and post-mating reproductive isolation is a consequence of cumulative genetic change. I study possibilities for speciation according to allopatric, parapatric, peripatric and vicariance scenarios. The analytic theory satisfactorily matches results of individual-based simulations reported by Gavrilets et al. (1998). It is demonstrated that rapid speciation including simultaneous emergence of several new species is a plausible outcome of the evolutionary dynamics of subdivided populations. I consider effects of population size, population subdivision, and local adaptation on the dynamics of speciation. I briefly discuss some implications of the dynamics on holey adaptive landscapes for molecular evolution.

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arXiv1998

Rapid parapatric speciation on holey adaptive landscapes

A classical view of speciation is that reproductive isolation arises as a by-product of genetic divergence. Here, individual-based simulations are used to evaluate whether the mechanisms implied by this view may result in rapid speciation if the only source of genetic divergence are mutation and random genetic drift. Distinctive features of the simulations are the consideration of the complete process of speciation (from initiation until completion), and of a large number of loci, which was only one order of magnitude smaller than that of bacteria. It is demonstrated that rapid speciation on the time scale of hundreds of generations is plausible without the need for extreme founder events, complete geographic isolation, the existence of distinct adaptive peaks or selection for local adaptation. The plausibility of speciation is enhanced by population subdivision. Simultaneous emergence of more than two new species from a subdivided population is highly probable. Numerical examples relevant to the theory of centrifugal speciation and to the conjectures about the fate of ``ring species'' and ``sexual continuums'' are presented.

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arXiv1998

Life as Complex Systems --- Viewpoint from Intra-Inter Dynamics

Basic problems in complex systems are surveyed in connection with Life. As a key issue for complex systems, complementarity between syntax/rule/parts and semantics/behavior/whole is stressed. To address the issue, a constructive approach for a biological system is proposed. As a construction in a computer, intra-inter dynamics is presented for cell biology, where the following five general features are drawn from our model experiments; intrinsic diversification, recursive type formation, rule generation, formation of internal representation, and macroscopic robustness. Significance of the constructed logic to the biology of existing organisms is also discussed.

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arXiv1998

Cooperative mechanism of self-regulation in hierarchical living systems

We study the problem of how a ``living'' system complex in structure can respond perfectly to local changes in the environment. Such a system is assumed to consist of a distributed ``living'' medium and a hierarchical ``supplying'' network that provides this medium with ``nutritious'' products. Because of the hierarchical organization each element of the supplying network has to behave in a self-consistent way for the system can adapt to changes in the environment. We propose a cooperative mechanism of self-regulation by which the system as a whole can react perfectly. This mechanism is based on an individual response of each element to the corresponding small piece of the information on the state of the ``living'' medium. The conservation of flux through the supplying network gives rise to a certain processing of information and the self-consistent behavior of the elements, leading to the perfect self-regulation. The corresponding equations governing the ``living'' medium state are obtained.

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arXiv1998

The Evolutionary Design of Collective Computation in Cellular Automata

We investigate the ability of a genetic algorithm to design cellular automata that perform computations. The computational strategies of the resulting cellular automata can be understood using a framework in which ``particles'' embedded in space-time configurations carry information and interactions between particles effect information processing. This structural analysis can also be used to explain the evolutionary process by which the strategies were designed by the genetic algorithm. More generally, our goals are to understand how machine-learning processes can design complex decentralized systems with sophisticated collective computational abilities and to develop rigorous frameworks for understanding how the resulting dynamical systems perform computation.

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arXiv1998

Dynamics of clade diversification on the morphological hypercube

Understanding the relationship between taxonomic and morphological changes is important in identifying the reasons for accelerated morphological diversification early in the history of animal phyla. Here, a simple general model describing the joint dynamics of taxonomic diversity and morphological disparity is presented and applied to the data on the diversification of blastozoans. I show that the observed patterns of deceleration in clade diversification can be explicable in terms of the geometric structure of the morphospace and the effects of extinction and speciation on morphological disparity without invoking major declines in the size of morphological transitions or taxonomic turnover rates. The model allows testing of hypotheses about patterns of diversification and estimation of rates of morphological evolution. In the case of blastozoans, I find no evidence that major changes in evolutionary rates and mechanisms are responsible for the deceleration of morphological diversification seen during the period of this clade's expansion. At the same time, there is evidence for a moderate decline in overall rates of morphological diversification concordant with a major change (from positive to negative values) in the clade's growth rate.

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arXiv1998

Decline in extinction rates and scale invariance in the fossil record

We show that the decline in the extinction rate during the Phanerozoic can be accurately parameterized by a logarithmic fit to the cumulative total extinction. This implies that extinction intensity is falling off approximately as the reciprocal of time. We demonstrate that this observation alone is sufficient to explain the existence of the proposed power-law forms in the distribution of the sizes of extinction events and in the power spectrum of Phanerozoic extinction, results which previously have been explained by appealing to self-organized critical theories of evolutionary dynamics.

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arXiv1998

Nonlinear theory of autooscillations of quasiplanar interface during directional solidification

Within the framework of the frozen temperature approximation we develop a strongly-nonlinear theory of one-dimensional pattern formation during directional solidification of binary mixture under nonequilibrium segregation. In the case of small partition coefficient the full problem is reduced to the system of two ordinary differential equations describing the interface motion in terms of its velocity and position coordinate. The type of the oscillatory instability bifurcation is studied in detail in different limits. For the subcrytical bifurcaton relaxation interface oscillations are analyzed analytically and numerically. We show that these oscillation exibit a number of anomalous properies. In particular, such oscillations can be weakly- or strongly-dissipative depending on the physical parameters and the amplitude of the strongly-dissipative oscillations is determined not only by the form of the corresponding nullcline but also by the behavior of the system for small values of the interface velocity. Characteristic parameters of the superlattice occuring in the growing crystal are estimated.

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arXiv1998

Optimizing Epochal Evolutionary Search: Population-Size Independent Theory

Epochal dynamics, in which long periods of stasis in population fitness are punctuated by sudden innovations, is a common behavior in both natural and artificial evolutionary processes. We use a recent quantitative mathematical analysis of epochal evolution to estimate, as a function of population size and mutation rate, the average number of fitness function evaluations to reach the global optimum. This is then used to derive estimates of and bounds on evolutionary parameters that minimize search effort.

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arXiv1998

Optimizing Epochal Evolutionary Search: Population-Size Dependent Theory

Epochal dynamics, in which long periods of stasis in an evolving population are punctuated by a sudden burst of change, is a common behavior in both natural and artificial evolutionary processes. We analyze the population dynamics for a class of fitness functions that exhibit epochal behavior using a mathematical framework developed recently. In the latter the approximations employed led to a population-size independent theory that allowed us to determine optimal mutation rates. Here we extend this approach to include the destabilization of epochs due to finite-population fluctuations and show that this dynamical behavior often occurs around the optimal parameter settings for efficient search. The resulting, more accurate theory predicts the total number of fitness function evaluations to reach the global optimum as a function of mutation rate, population size, and the parameters specifying the fitness function. We further identify a generalized error threshold, smoothly bounding the two-dimensional regime of mutation rates and population sizes for which evolutionary search operates efficiently.

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