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Absorption time of the Moran process

Díaz, Josep and Goldberg, Leslie Ann and Richerby, David and Serna, Maria (2016) 'Absorption time of the Moran process.' Random Structures and Algorithms, 49 (1). 137 - 159. ISSN 1042-9832

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Abstract

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n⁴) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is Ω(n log n) and O(n²). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.

Item Type: Article
Divisions: Faculty of Science and Health > Computer Science and Electronic Engineering, School of
Depositing User: Elements
Date Deposited: 06 Oct 2020 13:56
Last Modified: 06 Oct 2020 14:15
URI: http://repository.essex.ac.uk/id/eprint/25615

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