A fitness landscape is a mapping from the space of genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, which implies that the fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph A ^L, where L is the number of genetic loci and the allele graph A encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in L displays a sharp transition from 0 to 1 at a threshold value \beta^\ast of the fitness difference between the initial and final genotype. We derive a lower bound on \beta ^\ast for general A and conjecture that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs . A key tool of our analysis is the concept of quasi-accessibility, which eliminates the need to account for the self-avoidance of accessible paths .
A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph \mathcal{A ^L, where L is the number of genetic loci and the allele graph \mathcal{A encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in L displays a transition from 0 to a positive value at a threshold \beta_c for the fitness difference between the initial and final genotype. We derive a lower bound on \beta _c for general \mathcal{A that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs .
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