Finiteness of the Fixed Point Set for the Simple Genetic Algorithm

The infinite population simple genetic algorithm is a discrete dynamical system model of a genetic algorithm. It is conjectured that trajectories in the model always converge to fixed points. This paper shows that an arbitrarily small perturbation of the fitness will result in a model with a finite number of fixed points. Moreover, every sufficiently small perturbation of fimess preserves the finiteness of the fixed point set. These results allow proofs and constructions that require finiteness of the fixed point set. For example, applying the stable manifold theorem to a fixed point requires the hyperbolicity of the differential of the transition map of the genetic algorithm, which requires (among other things) that the fixed point be isolated

Genetic Algorithms in Time-Dependent Environments

The influence of time-dependent fitnesses on the infinite population dynamics of simple genetic algorithms (without crossover) is analyzed. Based on general arguments, a schematic phase diagram is constructed that allows one to characterize the asymptotic states in dependence on the mutation rate and the time scale of changes. Furthermore, the notion of regular changes is raised for which the population can be shown to converge towards a generalized quasispecies. Based on this, error thresholds and an optimal mutation rate are approximately calculated for a generational genetic algorithm with a moving needle-in-the-haystack landscape. The so found phase diagram is fully consistent with our general considerations.Comment: 24 pages, 14 figures, submitted to the 2nd EvoNet Summerschoo

Subjectivity and complexity of facial attractiveness

The origin and meaning of facial beauty represent a longstanding puzzle. Despite the profuse literature devoted to facial attractiveness, its very nature, its determinants and the nature of inter-person differences remain controversial issues. Here we tackle such questions proposing a novel experimental approach in which human subjects, instead of rating natural faces, are allowed to efficiently explore the face-space and 'sculpt' their favorite variation of a reference facial image. The results reveal that different subjects prefer distinguishable regions of the face-space, highlighting the essential subjectivity of the phenomenon.The different sculpted facial vectors exhibit strong correlations among pairs of facial distances, characterising the underlying universality and complexity of the cognitive processes, and the relative relevance and robustness of the different facial distances.Comment: 15 pages, 5 figures. Supplementary information: 26 pages, 13 figure

Biology & Political Science. Foundational Issues of Political Biology

In their classic formulations, valid to this day, the issue of self-preservation is foundational for both political science and economics. In order to fixate this concept, the Modern theorists relied upon various assumptions about human nature. Due to the advances of biology and evolutionary theory, we are today in the position of explicating these assumptions in the form of stable scientific certainties. A foundational concept in biological theory is that of "fitness". The paper indicates the relationship between the less determined concept of self-preservation and the more rigorous one of fitness. By that, it accomplishes two things: it gives more solidity to the foundation of political theory and political economy, by anchoring them in biology; it opens the path towards a unification between two social sciences and their immediate juxtaposed science, biology. The emphasis of the paper is on political science, aiming to define, on the basis of the above argument, its proper object of study. The notion of fitness extraction is thus defined. A lateral exposition differentiates between political action, thus understood, and economic action, defined more generally as fitness transfer. The distinction is to be eventually furthered in a separate study.Biology; Evolution; Fitness; Foundational Theory; Foundations of Economics; Political Science

Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits

We study the Fokker-Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium. Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of moments of the Fokker-Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.Comment: 28 pages, 4 tables, 5 figure