Adaptation Noise And Self Organizing Systems

In this review article we explore several recent advances in the quantitative modeling of financial markets. We begin with the Efficient Markets Hypothesis and describe how this controversial idea has stimulated a number of new directions of research, some focusing on more elaborate mathematical models that are captable of rationalizing the empirical facrts, others taking a completely different different tack in rejecting rationality altogether. One of the most promising directions is to view financial markets from a biological perspective and, specifically, with an evolutionary framework in which markets, instruments, institutions, and investors interact and evolve dynamically according to the "law" of economic selection. Under this view, financial agents compete and adapt, but they do not necessarily do so in an optimal fashion. Evolutionary and ecological models of financial markets is truly a new frontier whose exploration has just begun.

The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function f , not of variables, having a self-reference term f∘f , introduced by recalling that operation in a biological system is often applied to itself, as is typically seen in rules in the natural language or genes. Starting from an inarticulate network, two types of fixed points are formed as an invariant structure with iterations. The function is folded with time, until it has finite or infinite piecewise-flat segments of fixed points, regarded as articulation. For an initial logistic map, attracted functions are classified into step, folded step, fractal, and random phases, according to the degree of folding. Oscillatory dynamics are also found, where function values are mapped to several fixed points periodically. The significance of our results to prototype categorization in language is discussed.

Functional dynamics, introduced in a previous paper, is analyzed, focusing on the formation of a hierarchical rule to determine the dynamics of the functional value. To study the periodic (or non-fixed) solution, the functional dynamics is separated into fixed and non-fixed parts. It is shown that the fixed parts generate a 1-dimensional map by which the dynamics of the functional values of some other parts are determined. Piecewise-linear maps with multiple branches are generally created, while an arbitrary one-dimensional map can be embedded into this functional dynamics if the initial function coincides with the identity function over a finite interval. Next, the dynamics determined by the one-dimensional map can again generate a `meta-map', which determines the dynamics of the generated map. This hierarchy of meta-rules can continue recursively. It is also shown that the dynamics can produce `meta-chaos' with an orbital instability that is stronger than exponential. The relevance of the generated hierarchy to biological and language systems is discussed, in relation with the formation of syntax of a network.

We study a family of deterministic models for highway traffic flow which generalize cellular automaton rule 184. This family is parametrized by the speed limit m and another parameter k that represents a ``degree of aggressiveness'' in driving, strictly related to the distance between two consecutive cars. We compare two driving strategies with identical maximum throughput: ``conservative'' driving with high speed limit and ``aggressive'' driving with low speed limit. Those two strategies are evaluated in terms of accident probability. We also discuss fundamental diagrams of generalized traffic rules and examine limitations of maximum achievable throughput. Possible modifications of the model are considered.

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form \cite{Solomon96a} w i (t+1)=λ(t) w i (t)+a w ¯ (t)−b w i (t) w ¯ (t) is studied by computer simulations. The variables w i , i=1,...N , are the individual system components and w ¯ (t)= 1 N ∑ i w i (t) is their average. The parameters a and b are constants, while λ(t) is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution P(w,t) of the system components w i , turns out to fulfill a (truncated) Pareto power-law P(w,t)∼ w −1−α . The time evolution of w ¯ (t) presents intermittent fluctuations parametrized by a truncated Lévy distribution of index α , showing a connection between the distribution of the w i 's at a given time and the temporal fluctuations of their average.

Finite sample size corrections to the reparametrization-invariant solution of the inverse problem of probability are computed, and shown to converge uniformly to the correct distribution.

A simple mathematical model is proposed to study the effect of the average trend of a population on the opinion of each individual, when a group decision has to be made by voting. It is shown that if such effect is strong enough a transition to coherent behavior occurs, in which the whole population converges to a single opinion. This transition has the character of a first-order critical phenomenon.

We propose a new Ising spin glass model on Z d of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2 N , where N=N(d) is the number of distinct global components in the ``invasion forest''. We prove that N(d)=∞ if the invasion connectivity function is square summable. We argue that the critical dimension separating N=1 and N=∞ is d c =8 . When N(d)=∞ , we consider free or periodic boundary conditions on cubes of side length L and show that frustration leads to chaotic L dependence with {\it all} pairs of ground states occuring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.

The time evolution of a simple model for crossover is discussed. A variant of this model with an improved exploration behavior in phase space is derived as a subset of standard one- and multi-point crossover operations. This model is solved analytically in the flat fitness case. Numerical simulations compare the way of phase space exploration of different genetic operators. In the case of a non-flat fitness landscape, numerical solutions of the evolution equations point out ways to estimate premature convergence.

The main steps of automatic control methodology include the hierarchical representation of management system and the formal definitions of input variables, object and goal of control of each management level. A Petri net model of individual traffic source is presented. It is noted that the current set of traffic parameters recommended by ATM-forum is not enough to synthesize optimal traffic control system. The feature of traffic self-similarity can be used to effectively solve optimal control task. An example of an optimal control scheme for cell discarding algorithm is presented.