Adaptation Noise And Self Organizing Systems

Multiplicative random processes in (not necessaryly equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the normalized elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuations. These predictions are validated by Monte Carlo simulations and experimental data. We speculate that stochastic multiplicative dynamics might be the natural origin for the emergence of criticality and scale hierarchies without fine-tuning.

Recent Fourier analyses of fossil extinction data have indicated that the power spectrum of extinction during the Phanerozoic may take the form of 1/f noise, a result which, it has been suggested, could be indicative of the presence of "critical dynamics" in the processes giving rise to extinction. In this paper we examine extinction power spectra in some detail, using family-level data from a variety of different sources. We find that although the average form of the power spectrum roughly obeys the 1/f law, the spectrum can be represented more accurately by dividing it into two regimes: a low-frequency one which is well fit by an exponential, and a high-frequency one in which it follows a power law with a 1/f^2 form. We give explanations for the occurrence of each of these behaviours and for the position of the cross-over between them.

We describe the results of analytic calculations and computer simulations of adaptive predictors (predictive agents) responding to an evolving chaotic environment and to one another. Our simulations are designed to quantify adaptation and to explore co-adaptation for a simple calculable model of a complex adaptive system. We first consider the ability of a single agent, exposed to a chaotic environment, to model, control, and predict the future states of that environment. We then introduce a second agent which, in attempting to model and control both the chaotic environment and the first agent, modifies the extent to which that agent can identify patterns and exercise control. We find that (i) optimal adaptive predictors have an optimal memory and an optimal complexity, which are small for a rapidly changing map dynamics and (ii) that the predictive power can be increased by imposing chaos or random noise onto the map dynamics. The competition between the two predictive agents can lead either to chaos, or to metastable emergent behavior, best described as a leader-follower relationship. Our results suggest a correlation between optimal adaptation, optimal complexity, and emergent behavior, and provide preliminary support for the concept of optimal co-adaptation near the edge of chaos.

Partner selection is an important process in many social interactions, permitting individuals to decrease the risks associated with cooperation. In large populations, defectors may escape punishment by roving from partner to partner, but defectors in smaller populations risk social isolation. We investigate these possibilities for an evolutionary prisoner's dilemma in which agents use expected payoffs to choose and refuse partners. In comparison to random or round-robin partner matching, we find that the average payoffs attained with preferential partner selection tend to be more narrowly confined to a few isolated payoff regions. Most ecologies evolve to essentially full cooperative behavior, but when agents are intolerant of defections, or when the costs of refusal and social isolation are small, we also see the emergence of wallflower ecologies in which all agents are socially isolated. In between these two extremes, we see the emergence of ecologies whose agents tend to engage in a small number of defections followed by cooperation thereafter. The latter ecologies exhibit a plethora of interesting social interaction patterns. Keywords: Evolutionary Game; Iterated Prisoner's Dilemma; Partner Choice and Refusal; Artificial Life; Genetic Algorithm; Finite Automata.

Given a sequence of N positive real numbers { a 1 , a 2 ,..., a N } , the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of a j over the two sets is minimized. In the case that the a j 's are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random anti-ferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large N limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like N −3/2 .

We observe the propagation of information in a system of self-replicating strings of code (``Artificial Life'') as a function of fitness and mutation rate. Comparison with theoretical predictions based on the reaction-diffusion equation shows that the response of the artificial system to fluctuations (\eg velocity of the information wave as a function of relative fitness) closely follows that of natural systems. We find that the relaxation time of the system depends on the speed of propagation of information and the size of the system. This analysis offers the possibility of determining the minimal system size for observation of non-equilibrium effects at fixed mutation rate.

As a model of proportion regulation in differentiation process of biological system, globally coupled activator-inhibitor systems are studied. Formation and destabilization of one and two cluster state are predicted analytically. Numerical simulations show that the proportion of units of clusters is chosen within a finite range and it is selected depend on the initial condition.

Simulating a cellular automaton (CA) for t time-steps into the future requires t^2 serial computation steps or t parallel ones. However, certain CAs based on an Abelian group, such as addition mod 2, are termed ``linear'' because they obey a principle of superposition. This allows them to be predicted efficiently, in serial time O(t) or O(log t) in parallel. In this paper, we generalize this by looking at CAs with a variety of algebraic structures, including quasigroups, non-Abelian groups, Steiner systems, and others. We show that in many cases, an efficient algorithm exists even though these CAs are not linear in the previous sense; we term them ``quasilinear.'' We find examples which can be predicted in serial time proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log t, log t log log t and log^2 t. We also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition, and exhibit several novel ``vector-valued'' CAs.

We derive the exact equations of motion for the random neighbor version of the Olami-Feder-Christensen earthquake model in the infinite-size limit. We solve them numerically, and compare with simulations of the model for large numbers of sites. We find perfect agreement. But we do not find any scaling or phase transitions, except in the conservative limit. This is in contradiction to claims by Lise & Jensen (Phys. Rev. Lett. 76, 2326 (1996)) based on approximate solutions of the same model. It indicates again that scaling in the Olami-Feder-Christensen model is only due to partial synchronization driven by spatial inhomogeneities. Finally, we point out that our method can be used also for other SOC models, and treat in detail the random neighbor version of the Feder-Feder model.

We study the problem of a random walk on a lattice in which bonds connecting nearest neighbor sites open and close randomly in time, a situation often encountered in fluctuating media. We present a simple renormalization group technique to solve for the effective diffusive behavior at long times. For one-dimensional lattices we obtain better quantitative agreement with simulation data than earlier effective medium results. Our technique works in principle in any dimension, although the amount of computation required rises with dimensionality of the lattice.