Christopher R. Stephens

Schemata evolution and building blocks

In the light of a recently derived evolution equation for genetic algorithms we consider the schema theorem and the building block hypothesis. We derive a schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The equation makes manifest the content of the building block hypothesis showing how fit schemata are constructed from fit sub-schemata. However, we show that, generically, there is no preference for short, low-order schemata. In the case where schema reconstruction is favored over schema destruction, large schemata tend to be favored. As a corollary of the evolution equation we prove Geiringers theorem.

Approximating Mexican highways with slime mould

Plasmodium of Physarum polycephalum is a single cell visible by unaided eye. During its foraging behavior the cell spans spatially distributed sources of nutrients with a protoplasmic network. The geometrical structure of the protoplasmic networks allows the plasmodium to optimize transport of nutrients between remote parts of its body. Assuming major Mexican cities are sources of nutrients that need to be distributed across Mexico, how much does the structure of the Physarum protoplasmic network correspond to the structure of Mexican Federal highway network? To address the issue we undertook a series of laboratory experiments with living P. polycephalum. We represent geographical locations of major cities (19 locations) by oat flakes, place a piece of plasmodium in the area corresponding to Mexico city, record the plasmodium’s foraging behavior and extract topology of the resulting nutrient transport networks. Results of our experiments show that the protoplasmic network formed by Physarum is isomorphic, subject to limitations imposed, to a network of principal highways. Ideas and results in the paper may contribute towards future developments in bio-inspired road planning.

Self-adaptation in evolving systems

A theoretical and experimental analysis is made of the effects of self-adaptation in a simple evolving system. Specifically, we consider the effects of coding the mutation and crossover probabilities of a genetic algorithm evolving in certain model fitness landscapes. The resultant genotypephenotype mapping is degenerate in fitness space, there being no direct selective advantage for one probability versus another. Thus there is a symmetry between various genotypes that all correspond to the same phenotype. We show that the action of mutation and crossover lifts this degeneracy, that is, the genetic operators induce a breaking of the genotype-phenotype symmetry, thus leading to a preference for those genotypes that propagate most successfully into future generations. We demonstrate that this induced symmetry breaking allows the system to self-adapt in a time-dependent environment.

Using biotic interaction networks for prediction in biodiversity and emerging diseases.

Networks offer a powerful tool for understanding and visualizing inter-species ecological and evolutionary interactions. Previously considered examples, such as trophic networks, are just representations of experimentally observed direct interactions. However, species interactions are so rich and complex it is not feasible to directly observe more than a small fraction. In this paper, using data mining techniques, we show how potential interactions can be inferred from geographic data, rather than by direct observation. An important application area for this methodology is that of emerging diseases, where, often, little is known about inter-species interactions, such as between vectors and reservoirs. Here, we show how using geographic data, biotic interaction networks that model statistical dependencies between species distributions can be used to infer and understand inter-species interactions. Furthermore, we show how such networks can be used to build prediction models. For example, for predicting the most important reservoirs of a disease, or the degree of disease risk associated with a geographical area. We illustrate the general methodology by considering an important emerging disease - Leishmaniasis. This data mining methodology allows for the use of geographic data to construct inferential biotic interaction networks which can then be used to build prediction models with a wide range of applications in ecology, biodiversity and emerging diseases.

Limitations of Existing Mutation Rate Heuristics and How a Rank GA Overcomes Them

Using a set of different search metrics and a set of model landscapes we theoretically and empirically study how ldquooptimalrdquo mutation rates for the simple genetic algorithm (SGA) depend not only on the fitness landscape, but also on population size and population state. We discuss the limitations of current mutation rate heuristics, showing that any fixed mutation rate can be expected to be suboptimal in terms of balancing exploration and exploitation. We then develop a mutation rate heuristic that offers a better balance by assigning different mutation rates to different subpopulations. When the mutation rate is assigned through a ranking of the population, according to fitness for example, we call the resulting algorithm a Rank GA. We show how this Rank GA overcomes the limitations of other heuristics on a set of model problems showing under what circumstances it might be expected to outperform a SGA with any choice of mutation rate.

Effective Fitness as an Alternative Paradigm for Evolutionary Computation I: General Formalism

In evolutionary computation the concept of a fitness landscape has played an important role, evolution itself being portrayed as a hill-climbing process on a rugged landscape. In this article we review the recent development of an alternative paradigm for evolution on a fitness landscape—effective fitness. It is shown that in general, in the presence of other genetic operators such as mutation and recombination, hill-climbing is the exception rather than the rule; a discrepancy that has its origin in the different ways in which the concept of fitness appears—as a measure of the number of fit offspring, or as a measure of the probability to reach reproductive age. Effective fitness models the former not the latter and gives an intuitive way to understand population dynamics as flows on an effective fitness landscape when genetic operators other than reproductive selection play an important role. Additionally, we will show that when the genotype-phenotype map is degenerate, i.e. there exists a synonym symmetry, it can be used to quantify the degree of symmetry breaking of the map, thus allowing for a quantitative explanation of phenomena such as self-adaptation, bloat and evolutionary robustness.

Landscapes and Effective Fitness

The concept of a fitness landscape arose in theoretical biology, while that of effective fitness has its origin in evolutionary computation. Both have emerged as useful conceptual tools with which to understand the dynamics of evolutionary processes, especially in the presence of complex genotype-phenotype relations. In this contribution we attempt to provide a unified discussion of these two approaches, discussing both their advantages and disadvantages in the context of some simple models. We also discuss how fitness and effective fitness change under various transformations of the configuration space of the underlying genetic model, concentrating on coarse graining transformations and on a particular coordinate transformation that provides an appropriate basis for illuminating the structure and consequences of recombination.

Effective Fitness as an Alternative Paradigm for Evolutionary Computation II: Examples and Applications

In paper I of this series we reviewed the recent development of an alternative paradigm for evolution on a fitness landscape–effective fitness–which offers an intuitive way to understand population dynamics as flows on an effective fitness landscape when genetic operators other than reproductive selection play an important role. In this article we demonstrate the utility of the concept using several simple analytical models and some more complex models that we simulate numerically. In particular, we show that effective fitness offers a qualitative and quantitative framework within which the phenomenon of induced symmetry breaking of the genotype-phenotype map may be understood. As explicit examples we consider: the violation of the building block hypothesis in non-epistatic landscapes; self-adaptation of genetic algorithms in time-dependent fitness landscapes and the appearance of evolutionary robustness as an emergent property in the evolution of language. In all cases we demonstrate that effective fitness offers a framework within which these diverse phenomena can be understood and in principle quantitatively studied.

Allele Diffusion in Linear Genetic Programming and Variable-Length Genetic Algorithms with Subtree Crossover

In this paper we study, theoretically, the search biases produced by GP subtree crossover when applied to linear representations, such as those used in linear GP or in variable length GAs. The study naturally leads to generalisations of Geiringers theorem and of the notion of linkage equilibrium, which, until now, were applicable only to fixed-length representations. This indicates the presence of a diffusion process by which, even in the absence of selective pressure and mutation, the alleles in a particular individual tend not just to be swapped with those of other individuals in the population, but also to diffuse within the representation of each individual. More precisely, crossover attempts to push the population towards distributions of primitives where each primitive is equally likely to be found in any position in any individual.

An Estimation of Distribution Algorithm Based on Maximum Entropy

Estimation of distribution algorithms (EDA) are similar to genetic algorithms except that they replace crossover and mutation with sampling from an estimated probability distribution. We develop a framework for estimation of distribution algorithms based on the principle of maximum entropy and the conservation of schema frequencies. An algorithm of this type gives better performance than a standard genetic algorithm (GA) on a number of standard test problems involving deception and epistasis (i.e. Trap and NK).