Phase transition in random adaptive walks on correlated fitness landscapes

Park, Su‐Chan; Szendro, Ivan G.; Neidhart, Johannes; Krug, Joachim

doi:10.1103/physreve.91.042707

Cited by 27 publications

(78 citation statements)

References 42 publications

(109 reference statements)

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“…A b (c → ∞) = 0. Thus, in contrast to the case b = 1, there is no phase transition as a function of c in the sense of [14]. For b = 1 the solution of (16) is A 1 (c) = 1 − c/a which reflects the phase transition at c = a and is confirmed by the exact solution presented in [14].…”

Section: Gumbel Classmentioning

confidence: 53%

“…The special role of the exponential distribution reflects the wellknown fact that record values from exponentially-tailed distributions are asymptotically equally spaced [2]. Correspondingly, phase-transition like phenomena can be expected for other tail shapes if the constant δ < 0 is replaced by an offset that varies with the number of records, and some preliminary results along these lines were reported in [14]. The purpose of this paper is to provide a comprehensive analysis of the generalized δ-exceedance record problem with a monotonically varying, negative offset.…”

Section: Goal and Outline Of The Papermentioning

confidence: 91%

“…In [14] we considered RAW's in a setting where i.i.d. random fitness values are added to a deterministic fitness profile.…”

Section: Adaptive Walksmentioning

confidence: 99%

“…[14], the mean value z l of the l'th record will turn out to play an important role in understanding the record process. To derive a recursion relation for z l we consider the probability density of Y l , which is denoted by Q l .…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…To derive a recursion relation for z l we consider the probability density of Y l , which is denoted by Q l . It is a straightforward generalization of the case with b = 1 [14] to obtain the recursion relation…”

Section: Statement Of the Problemmentioning

confidence: 99%

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δ-exceedance records and random adaptive walks

Park¹,

Krug²

2016

J. Phys. A: Math. Theor.

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Abstract. We study a modified record process where the k'th record in a series of independent and identically distributed random variables is defined recursively through the condition Y k > Y k−1 − δ k−1 with a deterministic sequence δ k > 0 called the handicap. For constant δ k ≡ δ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences δ k , focusing in particular on the case when δ k matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when δ k is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where δ k corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.

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Section: Gumbel Classmentioning

confidence: 53%

Section: Goal and Outline Of The Papermentioning

confidence: 91%

“…In [14] we considered RAW's in a setting where i.i.d. random fitness values are added to a deterministic fitness profile.…”

Section: Adaptive Walksmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

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δ-exceedance records and random adaptive walks

Park¹,

Krug²

2016

J. Phys. A: Math. Theor.

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Universality Classes of Interaction Structures for NK Fitness Landscapes

Hwang¹,

Schmiegelt²,

Ferretti³

et al. 2018

J Stat Phys

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Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of L binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of k ≤ L loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of L, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large k. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in L for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the

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Greedy adaptive walks on a correlated fitness landscape

Park

Neidhart

Krug

2016

Journal of Theoretical Biology

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We study adaptation of a haploid asexual population on a fitness landscape defined over binary genotype sequences of length L. We consider greedy adaptive walks in which the population moves to the fittest among all single mutant neighbors of the current genotype until a local fitness maximum is reached. The landscape is of the rough mount Fuji type, which means that the fitness value assigned to a sequence is the sum of a random and a deterministic component. The random components are independent and identically distributed random variables, and the deterministic component varies linearly with the distance to a reference sequence. The deterministic fitness gradient c is a parameter that interpolates between the limits of an uncorrelated random landscape (c = 0) and an effectively additive landscape (c → ∞). When the random fitness component is chosen from the Gumbel distribution, explicit expressions for the distribution of the number of steps taken by the greedy walk are obtained, and it is shown that the walk length varies non-monotonically with the strength of the fitness gradient when the starting point is sufficiently close to the reference sequence. Asymptotic results for general distributions of the random fitness component are obtained using extreme value theory, and it is found that the walk length attains a non-trivial limit for L → ∞, different from its values for c = 0 and c = ∞, if c is scaled with L in an appropriate combination.

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Phase transition in random adaptive walks on correlated fitness landscapes

Cited by 27 publications

References 42 publications

δ-exceedance records and random adaptive walks

δ-exceedance records and random adaptive walks

Universality Classes of Interaction Structures for NK Fitness Landscapes

Greedy adaptive walks on a correlated fitness landscape

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