ESTIMATES FROM EVOLUTIONARY ALGORITHMS THEORY APPLIED TO DIRECTED EVOLUTION

The field of evolutionary computation emerged in the area of computer science due to transfer of ideas from biology and developed independently for several decades, enriched with techniques from probability theory, complexity theory and optimization methods. Our aim is to consider how some recent results form the theory of evolutionary computation may be transferred back into biology. It has been noted that the non-elitist evolutionary algorithms optimizing Royal Road fitness functions may be considered as models of evolutionary search for the synthetic enhancer sequences “from scratch”. This problem asks for a tight cluster of supposedly unknown motifs from the initial random (or partially random) set of DNA sequences using SELEX approaches. We apply the upper bounds on the expected hitting time of a target area of genotypic space in order to upper-bound the expected time to finding a sufficiently fit series of motifs in a SELEX procedure. On the other hand, using the theory of evolutionary computation, we propose an upper bound on the expected proportion of the DNA sequences with sufficiently high fitness at a given round of a SELEX procedure. Both approaches are evaluated in computational experiment, using a Royal Road fitness function as a model of the SELEX procedure for regulatory FIS factor binding site.

A New Framework for Analysis of Coevolutionary Systems—Directed Graph Representation and Random Walks

Studying coevolutionary systems in the context of simplified models (i.e., games with pairwise interactions between coevolving solutions modeled as self plays) remains an open challenge since the rich underlying structures associated with pairwise-comparison-based fitness measures are often not taken fully into account. Although cyclic dynamics have been demonstrated in several contexts (such as intransitivity in coevolutionary problems), there is no complete characterization of cycle structures and their effects on coevolutionary search. We develop a new framework to address this issue. At the core of our approach is the directed graph (digraph) representation of coevolutionary problems that fully captures structures in the relations between candidate solutions. Coevolutionary processes are modeled as a specific type of Markov chains—random walks on digraphs. Using this framework, we show that coevolutionary problems admit a qualitative characterization: a coevolutionary problem is either solvable (there is a subset of solutions that dominates the remaining candidate solutions) or not. This has an implication on coevolutionary search. We further develop our framework that provides the means to construct quantitative tools for analysis of coevolutionary processes and demonstrate their applications through case studies. We show that coevolution of solvable problems corresponds to an absorbing Markov chain for which we can compute the expected hitting time of the absorbing class. Otherwise, coevolution will cycle indefinitely and the quantity of interest will be the limiting invariant distribution of the Markov chain. We also provide an index for characterizing complexity in coevolutionary problems and show how they can be generated in a controlled manner.

The Study on Convergence and Convergence Rate of Genetic Algorithm Based on an Absorbing Markov Chain

The study on convergence of GA is always one of the most important theoretical issues. This paper analyses the sufficient condition which guarantees the convergence of GA. Via analyzing the convergence rate of GA, the average computational complexity can be implied and the optimization efficiency of GA can be judged. This paper proposes the approach to calculating the first expected hitting time and analyzes the bounds of the first hitting time of concrete GA using the proposed approach.

Decentralized search on spheres using small-world Markov chains: expected hitting times and structural properties

We build a family of Markov chains on a sphere using distance-based long-range connection probabilities to model the decentralized message-passing problem that has recently gained significant attention in the small-world literature. Starting at an arbitrary source point on the sphere, the expected message delivery time to an arbitrary target on the sphere is characterized by a particular expected hitting time of our Markov chains. We prove that, within this family, there is a unique efficient Markov chain whose expected hitting time is polylogarithmic in the relative size of the sphere. For all other chains, this expected hitting time is at least polynomial. We conclude by defining two structural properties, called scale invariance and steady improvement, of the probability density function of long-range connections and prove that they are sufficient and necessary for efficient decentralized message delivery.

Decentralized search on spheres using small-world Markov chains: expected hitting times and structural properties

We build a family of Markov chains on a sphere using distance-based long-range connection probabilities to model the decentralized message-passing problem that has recently gained significant attention in the small-world literature. Starting at an arbitrary source point on the sphere, the expected message delivery time to an arbitrary target on the sphere is characterized by a particular expected hitting time of our Markov chains. We prove that, within this family, there is a unique efficient Markov chain whose expected hitting time is polylogarithmic in the relative size of the sphere. For all other chains, this expected hitting time is at least polynomial. We conclude by defining two structural properties, called scale invariance and steady improvement, of the probability density function of long-range connections and prove that they are sufficient and necessary for efficient decentralized message delivery.