The Application of Chaos Theory to Psychology

Chaos theory has successfully explained various phenomena in the natural sciences and has subsequently been heralded by some as the new paradigm for science. Chaos and its concepts are beginning to be applied to psychology by researchers from cognitive, developmental and clinical psychology. This paper seeks to provide an overview of this work and evaluate the application of chaos to psychology. Chaos is briefly explained before existing applications of chaos in psychology and possible implications are examined. Finally, problems of applying chaos are evaluated and conclusions drawn regarding the usefulness of chaos in psychology.

Extreme fluctuations in stochastic network coordination with time delays

We study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks. We obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network. We then obtain the scaling behavior of the extreme fluctuations with system size, as well as the distribution of the extremes on complex networks, and compare them to those on regular one-dimensional lattices. For large complex networks, when the delay is not too close to the critical one, fluctuations at the nodes effectively decouple, and the limit distributions converge to the Fisher-Tippett-Gumbel density. In contrast, fluctuations in low-dimensional spatial graphs are strongly correlated, and the limit distribution of the extremes is the Airy density. Finally, we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes.

Network coordination and synchronization in a noisy environment with time delays

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in complex networks. We consider two types of time delays: transmission delays between interacting nodes and local delays at each node (due to processing, cognitive, or execution delays). By investigating the underlying fluctuations for several delay schemes, we obtain the synchronizability threshold (phase boundary) and the scaling behavior of the width of the synchronization landscape, in some cases for arbitrary networks and in others for specific weighted networks. Numerical computations allow the behavior of these networks to be explored when direct analytical results are not available. We comment on the implications of these findings for simple locally or globally weighted network couplings and possible trade-offs present in such systems.

Network synchronization in a noisy environment with time delays: fundamental limits and trade-offs

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. Using the known exact threshold value from the theory of differential equations with delays, we provide the synchronizability threshold for an arbitrary network. Further, by constructing the scaling theory of the underlying fluctuations, we establish the absolute limit of synchronization efficiency in a noisy environment with uniform time delays, i.e., the minimum attainable value of the width of the synchronization landscape. Our results also have strong implications for optimization and trade-offs in network synchronization with delays.

Learning in multilevel games with incomplete information. I

A model is presented of learning automata playing stochastic games at two levels. The high level represents the choice of the game environment and corresponds to a group decision. The low level represents the choice of action within the selected game environment. Both of these decision processes are affected by delays in the information state due to inherent latencies or to the delayed broadcast of state changes. Analysis of the intrinsic properties of this Markov process is presented along with simulated iterative behavior and expected iterative behavior. The results show that simulation agrees with expected behavior for small step lengths in the iterative map. A Feigenbaum diagram and numerical computation of the Lyapunov exponents show that, for very small penalty parameters, the system exhibits chaotic behavior.

An application of a computational ecology model to a routing method in computer networks

The paper proposes a network routing method based on a computational ecology model. The computational ecology model is a mathematical model proposed by B.A. Huberman and T. Hogg (1988), which represents a macro action of multi-agent systems. We formulate routing on a computer network as a resource allocation problem, where packets and links are regarded as agents and resources, respectively. Then, we apply an extended computational ecology model for this problem. Agents conflict so as to get more payoffs from links. As a result, they get the same payoffs, and a good resource allocation is achieved. In each node, each packet selects a link according to the selection rate decided through conflicts, and routing is accomplished autonomously with adaptability on the computer network. Moreover, we improve fault-tolerance of the system by local information exchanges. Finally, we examine the efficiency of the proposed method by computer simulation.

Price and niche wars in a free-market economy of software agents

One scenario of the future of computation populates the Internet with vast numbers of software agents providing, trading, and using a rich variety of information goods and services in an open, free-market economy. An essential task in such an economy is the retailing or brokering of information: gathering it from the right producers and distributing it to the right consumers. This article investigates one crucial aspect of brokers' dynamical behavior, their price-setting mechanisms, in the context of a simple information-filtering economy. We consider only the simplest cases in which a broker sets its price and product parameters based solely on the system's current state, without explicit prediction of the future. Analytical and numerical results show that the system's dynamical behavior in such "myopic" cases is generally an unending cycle of disastrous competitive "wars" in price/product space. These in turn are directly attributable to the existence of multiple peaks in the brokers' profitability landscapes, a feature whose generality is likely to extend far beyond our model.

Criticality and parallelism in combinatorial optimization

Local search methods constitute one of the most successful approaches to solving large-scale combinatorial optimization problems. As these methods are increasingly parallelized, optimization performance initially improves but then abruptly degrades to no better than that of random search beyond a certain point. The existence of this transition is demonstrated for a family of generalized spin-glass models and the traveling salesman problem. Finite-size scaling is used to characterize size-dependent effects near the transition, and analytical insight is obtained through a mean-field approximation.

Adaptation in a stochastic Prisoner's Dilemma with delayed information

Players in a Prisoner's Dilemma are modeled as learning automata that receive feedback from the environment and coadaptively adjust their strategies. Theory and simulations show the coevolutionary dynamics of the reward-inaction and reward-penalty schemes. The players are assumed to be physically distributed or, at least, in an environment where the effects of decisions are lagged. These systems include biological and social systems with constraints on instantaneous information or where environmental responses do not necessarily reflect the true state of the system. Linear stability analysis determines the conditions for persistent oscillations in the players' mixed strategies. Using a parameterized stochastic version of the dilemma, the results indicate that if the environment modifies the payoffs, and thus 'releases' the prisoners from their dilemma, the prisoners become prone to instabilities in their strategies given sufficient delays. Again, the prisoners fail to coordinate their actions.