Amplifiers for the Moran process
The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutatio...
| Main Authors: | , , , , |
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| Format: | Journal article |
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Association for Computing Machinery
2017
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| _version_ | 1826301014080225280 |
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| author | Galanis, A Goebel, A Goldberg, L Lapinskas, J Richerby, D |
| author_facet | Galanis, A Goebel, A Goldberg, L Lapinskas, J Richerby, D |
| author_sort | Galanis, A |
| collection | OXFORD |
| description | The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the paper. Strong amplification is a rather surprising property — it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified”. Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families — superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this paper, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars”. When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n−1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight — there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counter-example which clarifies the literature concerning the isothermal theorem of Lieberman et al. |
| first_indexed | 2024-03-07T05:25:56Z |
| format | Journal article |
| id | oxford-uuid:e08c741f-8f47-4e51-96ef-9c4709c3ea03 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:25:56Z |
| publishDate | 2017 |
| publisher | Association for Computing Machinery |
| record_format | dspace |
| spelling | oxford-uuid:e08c741f-8f47-4e51-96ef-9c4709c3ea032022-03-27T09:48:04ZAmplifiers for the Moran processJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e08c741f-8f47-4e51-96ef-9c4709c3ea03Symplectic Elements at OxfordAssociation for Computing Machinery2017Galanis, AGoebel, AGoldberg, LLapinskas, JRicherby, DThe Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the paper. Strong amplification is a rather surprising property — it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified”. Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families — superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this paper, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars”. When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n−1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight — there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counter-example which clarifies the literature concerning the isothermal theorem of Lieberman et al. |
| spellingShingle | Galanis, A Goebel, A Goldberg, L Lapinskas, J Richerby, D Amplifiers for the Moran process |
| title | Amplifiers for the Moran process |
| title_full | Amplifiers for the Moran process |
| title_fullStr | Amplifiers for the Moran process |
| title_full_unstemmed | Amplifiers for the Moran process |
| title_short | Amplifiers for the Moran process |
| title_sort | amplifiers for the moran process |
| work_keys_str_mv | AT galanisa amplifiersforthemoranprocess AT goebela amplifiersforthemoranprocess AT goldbergl amplifiersforthemoranprocess AT lapinskasj amplifiersforthemoranprocess AT richerbyd amplifiersforthemoranprocess |