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78
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive const ..."
Abstract

Cited by 89 (15 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Fitness landscapes and evolvability
 Evolutionary Computation
, 2002
"... In this paper, we develop techniques based on evolvability statistics of the fitness landscape surrounding sampled solutions. Averaging the measures over a sample of equal fitness solutions allows us to build up fitness evolvability portraits of the fitness landscape, which we show can be used to co ..."
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Cited by 37 (2 self)
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In this paper, we develop techniques based on evolvability statistics of the fitness landscape surrounding sampled solutions. Averaging the measures over a sample of equal fitness solutions allows us to build up fitness evolvability portraits of the fitness landscape, which we show can be used to compare both the ruggedness and neutrality in a set of tunably rugged and tunably neutral landscapes. We further show that the techniques can be used with solution samples collected through both random sampling of the landscapes and online sampling during optimization. Finally, we apply the techniques to two real evolutionary electronics search spaces and highlight differences between the two search spaces, comparing with the time taken to find good solutions through search.
Combinatorial Landscapes
 SIAM REVIEW
, 2002
"... Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, ne ..."
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Cited by 33 (2 self)
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Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.
Landscapes  Complex Optimization Problems and Biopolymer Structures
 Computers Chem
, 1993
"... The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by statistical ..."
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Cited by 31 (16 self)
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The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by statistical measures like number of optima, lengths of walks, and correlation functions. The evolution of a quasispecies on such landscapes exhibits three dynamical regimes depending on the replication fidelity: Above the "localization threshold" the population is centered around a (local) optimum. Between localization and "dispersion threshold" the population is still centered around a consensus sequence, which, however, changes in time. For very large mutation rates the population spreads in sequence space like a gas. The critical mutation rates separating the three domains depend strongly on characteristics properties of the fitness landscapes. Statistical characteristics of RNA landscapes are acces...
The algebraic theory of recombination spaces
, 2000
"... A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the s ..."
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Cited by 29 (15 self)
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A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourierdecomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for Pstructures. For binary string recombination the elementary landscapes are exactly the pspin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...
Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm
, 1995
"... In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 ..."
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Cited by 24 (3 self)
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In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 x 100 can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than 20 000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero field is 1.317.
The power of quantum systems on a line
 Proc. 48th IEEE Symposium on the Foundations of Computer Science (FOCS
, 2007
"... We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implicati ..."
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Cited by 21 (5 self)
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We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMAcomplete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one dimensional MAX2SAT with nearest neighbor constraints, is in P. The proof of the QMAcompleteness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Since it is unlikely that quantum computers can efficiently solve QMA problems, our construction gives a onedimensional system which, at low temperatures, takes an exponential time to relax to its thermal equilibrium state. This makes it a candidate for a onedimensional spin glass. 1
Disordered Ising systems and random cluster representations
 In Probability and Phase Transition
, 1994
"... Abstract. We discuss the Fortuin–Kasteleyn (FK) random cluster representation for Ising models with no external field and with pair interactions which need not be ferromagnetic. In the ferromagnetic case, the close connections between FK percolation and Ising spontaneous magnetization and the availa ..."
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Cited by 18 (2 self)
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Abstract. We discuss the Fortuin–Kasteleyn (FK) random cluster representation for Ising models with no external field and with pair interactions which need not be ferromagnetic. In the ferromagnetic case, the close connections between FK percolation and Ising spontaneous magnetization and the availability of comparison inequalities to independent percolation have been applied to certain disordered systems, such as dilute Ising ferromagnets and quantum Ising models in random environments; we review some of these applications. For nonferromagnetic disordered systems, such as spin glasses, the state of the art is much more primitive. We discuss some of the many open problems for spin glasses and show how the FK representation leads to one small result, that there is uniqueness of the spin glass Gibbs distribution above the critical temperature of the associated ferromagnet. Key words: FK representations, spin glasses, disordered Ising models, percolation. 1. The FK Random Cluster Representation
Amplitude spectra of fitness landscapes
 J. COMPLEX SYSTEMS
, 1998
"... Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used f ..."
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Cited by 18 (9 self)
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Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used for classi cation and comparison purposes. We consider here three very di erent types of landscapes using both mutation and recombination to de ne the topological structure of the con guration spaces. A reliable procedure for estimating the amplitude spectra is presented. The method is based on certain correlation functions that are easily obtained from empirical studies of the landscapes.
Analyzing probabilistic models in hierarchical boa on traps and spin glasses
 Genetic and Evolutionary Computation Conference (GECCO2007), I
, 2007
"... The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common t ..."
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Cited by 17 (15 self)
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The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common test problems: concatenated traps and 2D Ising spin glasses with periodic boundary conditions. We argue that although Bayesian networks with local structures can encode complex probability distributions, analyzing these models in hBOA is relatively straightforward and the results of such analyses may provide practitioners with useful information about their problems. The results show that the probabilistic models in hBOA closely correspond to the structure of the underlying problem, the models do not change significantly in subsequent iterations of BOA, and creating adequate probabilistic models by hand is not straightforward even with complete knowledge of the optimization problem. Categories and Subject Descriptors