Accessibility Percolation on Cartesian Power Graphs
arXiv:1912.07925 · doi:10.1007/s00285-023-01882-z
Abstract
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 , where is the number of genetic loci and the allele graph 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 displays a transition from 0 to a positive value at a threshold for the fitness difference between the initial and final genotype. We derive a lower bound on for general and show 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.
arXiv comment43 pages, 9 figures, 2 tables
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