Amplifiers for the Moran process

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Reference: Galanis, A, Goebel, A, Goldberg, LA et al., (2016). Amplifiers for the Moran process.Citable link to this page:

Amplifiers for the Moran process

Abstract: 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. Strong amplification is a rather surprising property — it means that in such graphs, the fixation probability of a uniformly-placed initial utant 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 hat are strongly amplifying. The same has been claimed for metafunnels. 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 counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al. A full version [11] containing detailed proofs is available at http://arxiv.org/abs/1512.05632. Theorem-numbering here matches the full version.

Peer Review status:Peer reviewedPublication status:PublishedVersion:Accepted manuscript Funder: Seventh Framework Programme   Conference Details: 43rd International Colloquium on Automata, Languages, and ProgrammingNotes:© Andreas Galanis, Andreas Göbel, Leslie Ann Goldberg, John Lapinskas, David Richerby; licensed under Creative Commons License CC-BY

Bibliographic Details

Publisher: International Colloquium on Automata, Languages, and Programming

Publisher Website: http://www.easyconferences.eu/icalp2016/index.html

Host: 43rd International Colloquium on Automata, Languages, and Programmingsee more from them

Publication Website: http://www.easyconferences.eu/icalp2016/accepted.html

Issue Date: 2016-04Identifiers

Urn: uuid:370cb1f4-d133-4f1b-bc14-66c4af4bc550

Source identifier: 617919

Doi: https://doi.org/10.4230/LIPIcs.ICALP.2016.62 Item Description

Type: Conference;

Version: Accepted manuscriptKeywords: Moran process randomised algorithm on graphs evolutionary dynamics Tiny URL: pubs:617919

Relationships

Author: Galanis, A - institutionUniversity of Oxford Oxford, MPLS, Computer Science - - - Goebel, A - - - Goldberg, LA - institutionUnive

Source: https://ora.ox.ac.uk/objects/uuid:370cb1f4-d133-4f1b-bc14-66c4af4bc550

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