Natural selection drives evolving populations up the fitness landscape, the projection from nucleotide sequence space to organismal reproductive success. While it has long been appreciated that topographic complexities on fitness landscapes can arise only as a consequence of epistatic interactions between mutations, evolutionary genetics has mainly focused on epistasis between pairs of mutations. Here we propose a generalization to the classical population genetic treatment of pairwise epistasis that yields expressions for epistasis among arbitrary subsets of mutations of all orders (pairwise, three-way, etc.). Our approach reveals substantial higher-order epistasis in almost every published fitness landscape. Furthermore we demonstrate that higher-order epistasis is critically important in two systems we know best. We conclude that higher-order epistasis deserves empirical and theoretical attention from evolutionary geneticists.

The effect of a mutation on the organism often depends on what other mutations are already present in its genome. Geneticists refer to such mutational interactions as epistasis. Pairwise epistatic effects have been recognized for over a century, and their evolutionary implications have received theoretical attention for nearly as long. However, pairwise epistatic interactions themselves can vary with genomic background. This is called higher-order epistasis, and its consequences for evolution are much less well understood. Here, we assess the influence that higher-order epistasis has on the topography of 16 published, biological fitness landscapes. We find that on average, their effects on fitness landscape declines with order, and suggest that notable exceptions to this trend may deserve experimental scrutiny. We conclude by highlighting opportunities for further theoretical and experimental work dissecting the influence that epistasis of all orders has on fitness landscape topography and on the efficiency of evolution by natural selection.Electronic supplementary materialThe online version of this article (10.1007/s10955-018-1975-3) contains supplementary material, which is available to authorized users.

This paper addresses the problem of discovering the structure of a fitness function from binary strings to the reals under the assumption of bounded epistasis. Two loci (string positions) are epistatically linked if the effect of changing the allele (value) at one locus depends on the allele at the other locus. Similarly, a group of loci are epistatically linked if the effect of changing the allele at one locus depends on the alleles at all other loci of the group. Under the assumption that the size of such groups of loci are bounded, and assuming that the function is given only as a “black box function”, this paper presents and analyzes a randomized algorithm that finds the complete epistatic structure of the function in the form of the Walsh coefficients of the function.

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Classically, epistasis is either computed exactly by Walsh coefficients or estimated by sampling. Exact computation is usually of theoretical interest since the computation typically grows exponentially with the number of bits in the domain. Given an evaluation function, epistasis also can be estimated by sampling. However this approach gives us little insight into the origin of the epistasis and is prone to sampling error. This paper presents theorems establishing the bounds of epistasis for problems that can be stated as mathematical expressions. This leads to substantial computational savings for bounding the difficulty of a problem. Furthermore, working with these theorems in a mathematical context, one can gain insight into the mathematical origins of epistasis and how a problem's epistasis might be reduced. We present several new measures for epistasis and give empirical evidence and examples to demonstrate the application of the theorems. In particular, we show that some functions display “parity” such that by picking a well-defined representation, all Walsh coefficients of either odd or even index become zero, thereby reducing the nonlinearity of the function.

In this paper we introduce embedded landscapes as an extension of NK landscapes and MAXSAT problems. This extension is valid for problems where the representation can be expressed as a simple sum of subfunctions over subsets of the representation domain. This encompasses many additive constraint problems and problems expressed as the interaction of subcomponents, where the critical features of the subcomponents are represented by subsets of bits in the domain. We show that embedded landscapes of fixed maximum epistasis K are exponentially sparse in epistatic space with respect to all possible functions. We show we can compute many important statistical features of these functions in polynomial time including all the epistatic interactions and the statistical moments of hyperplanes about the function mean and hyperplane mean. We also show that embedded landscapes of even small fixed K can be NP-complete. We can conclude that knowing the epistasis and many of the hyperplane statistics is not enough to solve the exponentially difficult part of these general problems and that the difficulty of the problem lies not in the epistasis itself but in the interaction of the epistatic parts.

Abstract-Replay attacks on security protocols have been discussed for quite some time in the literature. However, the efforts to address these attacks have been largely incomplete, lacking generality and many times in fact, proven unsuccessful. In this paper we address these issues and prove the efficacy of a simple and general scheme in defending a protocol against these attacks. We believe that our work will be particularly useful in security critical applications and to protocol analyzers that are unable to detect some or all of the attacks in this class.Index Terms-security protocols, replay attacks, adapted strand spaces, run identifiers, component numbers.

Abstract. This paper addresses the problem of determining the epistatic linkage of a function from binary strings to the reals. There is a close relationship between the Walsh coefficients of the function and "probes" (or perturbations) of the function. This relationship leads to two linkage detection algorithms that generalize earlier algorithms of the same type. A rigorous complexity analysis is given of the first algorithm. The second algorithm not only detects the epistatic linkage, but also computes all of the Walsh coefficients. This algorithm is much more efficient than previous algorithms for the same purpose.