Adaptation Noise And Self Organizing Systems

We analyze a random neighbor version of the OFC stick-slip model. We find that the mean avalanche size is finite as soon as dissipation exists in the bulk but that this size grows exponentially fast when dissipation tends to zero.

An evolution equation for a population of strings evolving under the genetic operators: selection, mutation and crossover is derived. The corresponding equation describing the evolution of schematas is found by performing an exact coarse graining of this equation. In particular exact expressions for schemata reconstruction are derived which allows for a critical appraisal of the ``building-block hypothesis'' of genetic algorithms. A further coarse-graining is made by considering the contribution of all length-l schematas to the evolution of population observables such as fitness growth. As a test function for investigating the emergence of structure in the evolution the increase per generation of the in-schemata fitness averaged over all schematas of length l, Δ l , is introduced. In finding solutions of the evolution equations we concentrate more on the effects of crossover, in particular we consider crossover in the context of Kauffman Nk models with k=0,2. For k=0, with a random initial population, in the first step of evolution the contribution from schemata reconstruction is equal to that of schemata destruction leading to a scale invariant situation where the contribution to fitness of schematas of size l is independent of l. This balance is broken in the next step of evolution leading to a situation where schematas that are either much larger or much smaller than half the string size dominate over those with l≈N/2 . The balance between block destruction and reconstruction is also broken in a k>0 landscape. It is conjectured that the effective degrees of freedom for such landscapes are landscape connective trees that break down into effectively fit smaller blocks, and not the blocks themselves. Numerical simulations confirm this ``connective tree hypothesis'' by showing that correlations drop off with connective distance and not with intrachromosomal distance.

The equivalence of two optimality principles leading to Murray's law has been discussed. The first approach is based on minimization of biological work needed for maintaining the blood flow through the vessels at required level. The second one is the principle of minimal drag and lumen volume. Characteristic features of these principles are considered. An alternative approach leading to Murray's law has been proposed. For that we model the microcirculatory bed in terms of delivering vascular network with symmetrical bifurcation nodes, embedded uniformly into the cellular tissue. It was shown that Murray's law can be regarded as a direct consequence of the organism capacity for controlling the blood flow redistribution over the microcirculatory beds.

Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α , which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes non-trivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α . The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the point-process generation mechanism. In the context of point-process simulation, reduction of this discrepancy requires generating data sets with either a large number of points, or with low jitter in the generation of the points. In the context of fractal data analysis, the results presented here suggest caution when interpreting fractal exponents estimated from experimental data sets.

Several standard processes for searching minima of potential functions, such as thermodynamical strategies (simulated annealing) and biologically motivated selfreproduction strategies, are reduced to Schrödinger problems. The properties of the landscape are encoded in the spectrum of the Hamiltonian. We investigate this relation between landscape and spectrum by means of topological methods which lead to a possible classification of landscapes in the framework of the operator theory. The influence of the dimension d of the search space is discussed. The connection between thermodynamical strategies and biologically motivated selfreproduction strategies is analyzed and interpreted in the above context. Mixing of both strategies is introduced as a new powerful tool of optimization.

The Iterated Prisoner's Dilemma with Choice and Refusal (IPD/CR) is an extension of the Iterated Prisoner's Dilemma with evolution that allows players to choose and to refuse their game partners. From individual behaviors, behavioral population structures emerge. In this report, we examine one particular IPD/CR environment and document the social network methods used to identify population behaviors found within this complex adaptive system. In contrast to the standard homogeneous population of nice cooperators, we have also found metastable populations of mixed strategies within this environment. In particular, the social networks of interesting populations and their evolution are examined.

A variant of Kauffman's model of cellular metabolism is presented. It is a randomly generated network of boolean gates, identical to Kauffman's except for a small bias in favor of boolean gates that depend on at most one input. The bias is asymptotic to 0 as the number of gates increases. Upper bounds on the time until the network reaches a state cycle and the size of the state cycle, as functions of the number of gates n , are derived. If the bias approaches 0 slowly enough, the state cycles will be smaller than n c for some c<1 . This lends support to Kauffman's claim that in his version of random network the average size of the state cycles is approximately n 1/2 .

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point S . If the deterministic dynamics are perturbed by white noise (random perturbations) of strength ϵ , the system state will eventually leave the domain of attraction Ω of S . We analyse the case when, as ϵ→0 , the exit location on the boundary ∂Ω is increasingly concentrated near a saddle point H of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on ∂Ω is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter μ , equal to the ratio | λ s (H)|/ λ u (H) of the stable and unstable eigenvalues of the linearized deterministic flow at H . If μ<1 then the exit location distribution is generically asymptotic as ϵ→0 to a Weibull distribution with shape parameter 2/μ , on the O( ϵ μ/2 ) length scale near H . If μ>1 it is generically asymptotic to a distribution on the O( ϵ 1/2 ) length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of Ω with H as vertex, which is reached from S , in the ϵ→0 limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of H . We deduce from the presence of this forbidden region that the classical Eyring formula for the small- ϵ exponential asymptotics of the mean first exit time is generically inapplicable.

A model of s interacting species is considered with two types of dynamical variables. The fast variables are the populations of the species and slow variables the links of a directed graph that defines the catalytic interactions among them. The graph evolves via mutations of the least fit species. Starting from a sparse random graph, we find that an autocatalytic set (ACS) inevitably appears and triggers a cascade of exponentially increasing connectivity until it spans the whole graph. The connectivity subsequently saturates in a statistical steady state. The time scales for the appearance of an ACS in the graph and its growth have a power law dependence on s and the catalytic probability. At the end of the growth period the network is highly non-random, being localized on an exponentially small region of graph space for large s .

We propose here an autonomous traffic signal control model based on analogy with neural networks. In this model, the length of cycle time period of traffic lights at each signal is autonomously adapted. We find a self-organizing collective behavior of such a model through simulation on a one-dimensional lattice model road: traffic congestion is greatly diffused when traffic signals have such autonomous adaptability with suitably tuned parameters. We also find that effectiveness of the system emerges through interactions between units and shows a threshold transition as a function of proportion of adaptive signals in the model.