Absorption time of the Moran process1. Issue 1 (12th January 2016)

• Record Type: Journal Article
• Authors: Díaz, Josep; Goldberg, Leslie Ann; Richerby, David; Serna, Maria
• Is Part Of: Random structures & algorithms. Volume 49:Issue 1(2016)
• Page Start: 137
• Page End: 159
• Publication Date: 2016-01-12
• Subjects: evolutionary dynamics -- Moran process -- Markov chains -- digraphs -- absorption time; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20617
• Languages: English
• ISSNs: 1042-9832

Abstract

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 ( n 4 ) 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 ( n 2 ) . 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 137–159, 2016