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A key quantity is the fixation time of a single mutant with relative reproduction rate\n                    <jats:italic>r<\/jats:italic>\n                    which invades a population of residents. We say that the fixation time is \u201cfast\u201d if it is at most a polynomial function in terms of the population size\n                    <jats:italic>N<\/jats:italic>\n                    . Here we study fixation times of advantageous mutants (\n                    <jats:italic>r<\/jats:italic>\n                    &gt; 1) and neutral mutants (\n                    <jats:italic>r<\/jats:italic>\n                    = 1) on\n                    <jats:italic>directed<\/jats:italic>\n                    graphs, which are those graphs that have at least some one-way connections. We obtain three main results. First, we prove that for any directed graph the fixation time is fast, provided that\n                    <jats:italic>r<\/jats:italic>\n                    is sufficiently large. Second, we construct an efficient algorithm that gives an upper bound for the fixation time for any graph and any\n                    <jats:italic>r<\/jats:italic>\n                    \u2265 1. Third, we identify a broad class of directed graphs with fast fixation times for any\n                    <jats:italic>r<\/jats:italic>\n                    \u2265 1. This class includes previously studied amplifiers of selection, such as Superstars and Metafunnels. We also show that on some graphs the fixation time is not a monotonically declining function of\n                    <jats:italic>r<\/jats:italic>\n                    ; in particular, neutral fixation can occur faster than fixation for small selective advantages.\n                  <\/jats:p>","DOI":"10.1371\/journal.pcbi.1012299","type":"journal-article","created":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T13:59:25Z","timestamp":1721311165000},"page":"e1012299","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":3,"title":["Fixation times on directed graphs"],"prefix":"10.1371","volume":"20","author":[{"given":"David A.","family":"Brewster","sequence":"first","affiliation":[]},{"given":"Martin A.","family":"Nowak","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1097-9684","authenticated-orcid":true,"given":"Josef","family":"Tkadlec","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2024,7,18]]},"reference":[{"key":"pcbi.1012299.ref001","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1038\/217624a0","article-title":"Evolutionary rate at the molecular level","volume":"217","author":"M Kimura","year":"1968","journal-title":"Nature"},{"key":"pcbi.1012299.ref002","first-page":"60","volume-title":"Mathematical proceedings of the cambridge philosophical society","author":"PAP Moran","year":"1958"},{"key":"pcbi.1012299.ref003","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21822-9","volume-title":"Mathematical population genetics: theoretical introduction","author":"WJ Ewens","year":"2004"},{"issue":"3","key":"pcbi.1012299.ref004","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1006\/tpbi.1994.1032","article-title":"The importance of being discrete (and spatial)","volume":"46","author":"R Durrett","year":"1994","journal-title":"Theoretical population biology"},{"issue":"2","key":"pcbi.1012299.ref005","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1093\/genetics\/164.2.767","article-title":"Fixation probability and time in subdivided populations","volume":"164","author":"MC Whitlock","year":"2003","journal-title":"Genetics"},{"key":"pcbi.1012299.ref006","doi-asserted-by":"crossref","first-page":"477","DOI":"10.2307\/2408196","article-title":"Fixation probabilities and fixation times in a subdivided population","author":"M Slatkin","year":"1981","journal-title":"Evolution"},{"issue":"6222","key":"pcbi.1012299.ref007","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1126\/science.1261487","article-title":"Termite mounds can increase the robustness of dryland ecosystems to climatic change","volume":"347","author":"JA Bonachela","year":"2015","journal-title":"Science"},{"issue":"64926492","key":"pcbi.1012299.ref008","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1038\/371065a0","article-title":"Habitat destruction and the extinction debt","volume":"371","author":"D Tilman","year":"1994","journal-title":"Nature"},{"issue":"3","key":"pcbi.1012299.ref009","doi-asserted-by":"crossref","first-page":"735","DOI":"10.2307\/5627","article-title":"The Spatial Dynamics of Host\u2013Parasitoid Systems","volume":"61","author":"HN Comins","year":"1992","journal-title":"Journal of Animal Ecology"},{"issue":"63416341","key":"pcbi.1012299.ref010","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1038\/353255a0","article-title":"Spatial structure and chaos in insect population dynamics","volume":"353","author":"MP Hassell","year":"1991","journal-title":"Nature"},{"issue":"6","key":"pcbi.1012299.ref011","doi-asserted-by":"crossref","first-page":"1943","DOI":"10.2307\/1941447","article-title":"The Problem of Pattern and Scale in Ecology: The Robert H. 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